FlyLee/deepscaler-randomsampled-10k · Datasets at AI快站

problem stringlengths 11 4.03k	answer stringlengths 0 157	solution stringlengths 0 11.1k
Find $PB$ given that $AP$ is a tangent to $\Omega$, $\angle PAB=\angle PCA$, and $\frac{PB}{PA}=\frac{4}{7}=\frac{PA}{PB+6}$.	\frac{32}{11}	Since $AP$ is a tangent to $\Omega$, we know that $\angle PAB=\angle PCA$, so $\triangle PAB \sim \triangle PCA$, so we get that $$\frac{PB}{PA}=\frac{4}{7}=\frac{PA}{PB+6}$$ Solving, we get that $7PB=4PA$, so $$4(PB+6)=7PA=\frac{49}{4}PB \Rightarrow \frac{33}{4}PB=24 \Rightarrow PB=\frac{32}{11}$$
The minimum value of the function \( y = \sin^4{x} + \cos^4{x} + \sec^4{x} + \csc^4{x} \).	\frac{17}{2}
Given in $\triangle ABC$, $\tan A$ and $\tan B$ are the two real roots of the equation $x^2 + ax + 4 = 0$: (1) If $a = -8$, find the value of $\tan C$; (2) Find the minimum value of $\tan C$, and specify the corresponding values of $\tan A$ and $\tan B$.	\frac{4}{3}
The real numbers $a$, $b$, and $c$ satisfy the equation $({a}^{2}+\frac{{b}^{2}}{4}+\frac{{c}^{2}}{9}=1)$. Find the maximum value of $a+b+c$.	\sqrt{14}
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$ holds for all positive integers $x, y$.	f(x) = x	Let's analyze the problem by working with the given inequality: \[ f(x) + y f(f(x)) \le x(1 + f(y)) \] for all positive integers \(x, y\). To find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) satisfying this inequality, we will first test some small values and then generalize our findings. Step 1: Consider specific cases. 1. For \(y = 1\): \[ f(x) + f(f(x)) \le x(1 + f(1)) \] Rearranging gives: \[ f(x) + f(f(x)) \le x + x f(1) \] 2. For \(x = 1\): \[ f(1) + y f(f(1)) \le 1(1 + f(y)) = 1 + f(y) \] Which simplifies to: \[ y f(f(1)) \le 1 + f(y) - f(1) \] Since \(y\) is arbitrary, this implies \(f(f(1)) = 0\) unless \(f(x) = x\). Step 2: Consider \(f(x) = x\). Assume \(f(x) = x\) for all \(x \in \mathbb{N}\). Substitute into the original inequality: \[ x + y f(x) \le x(1 + f(y)) \] which becomes: \[ x + yx \le x + xy \] This simplifies to: \[ x + yx \le x + xy \] The inequality holds, which suggests that \(f(x) = x\) is indeed a solution. Step 3: Consider the nature of the function. Assuming any function \(f(x)\) other than the identity leads to contradictions in maintaining the inequality universally. For instance, choosing other forms may not satisfy for larger values of \(x, y\), due to the restrictive nature of the inequality, especially where powers of \(f\) terms (like \(f(f(x))\)) would arise. Therefore, after analyzing potential options, we determine that the function: \[ f(x) = x \] is the only solution that satisfies the given inequality for all \(x, y \in \mathbb{N}\). Thus, the solution to the problem is: \[ \boxed{f(x) = x} \]
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 18$ and $CP = 6.$ If $\tan \angle APD = 2,$ find $AB.$	18
The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence?	1000
Given two fixed points $A(-1,0)$ and $B(1,0)$, and a moving point $P(x,y)$ on the line $l$: $y=x+3$, an ellipse $C$ has foci at $A$ and $B$ and passes through point $P$. Find the maximum value of the eccentricity of ellipse $C$.	\dfrac{\sqrt{5}}{5}
Given real numbers \\(x\\) and \\(y\\) satisfy the equation \\((x-3)^{2}+y^{2}=9\\), find the minimum value of \\(-2y-3x\\) \_\_\_\_\_\_.	-3\sqrt{13}-9
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$ .	3012
What is the area, in square units, of a triangle whose vertices are at $(4, -1)$, $(10, 3)$ and $(4, 5)$?	18
The sum of the first n terms of the sequence $\{a_n\}$ is $S_n$. If the terms of the sequence $\{a_n\}$ are arranged according to the following rule: $$\frac {1}{2}, \frac {1}{3}, \frac {2}{3}, \frac {1}{4}, \frac {2}{4}, \frac {3}{4}, \frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \ldots, \frac {1}{n}, \frac {2}{n}, \ldots, \frac {n-1}{n}, \ldots$$ If there exists a positive integer k such that $S_{k-1} < 10$ and $S_k > 10$, then $a_k = \_\_\_\_\_\_$.	\frac{6}{7}
Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is:	$x >\frac{10}{3}$	To find the set of all values of $x$ that satisfy the inequality $2x - 3 > 7 - x$, we start by simplifying the inequality: 1. Combine like terms: \[ 2x - 3 > 7 - x \] Add $x$ to both sides to get all $x$ terms on one side: \[ 2x + x > 7 + 3 \] Simplify: \[ 3x > 10 \] 2. Solve for $x$: Divide both sides by 3 to isolate $x$: \[ x > \frac{10}{3} \] This tells us that $x$ must be greater than $\frac{10}{3}$. Therefore, the correct answer is: \[ \boxed{\textbf{(D)}\ x > \frac{10}{3}} \]
If $a, b$, and $c$ are random real numbers from 0 to 1, independently and uniformly chosen, what is the average (expected) value of the smallest of $a, b$, and $c$?	1/4	Let $d$ be a fourth random variable, also chosen uniformly from $[0,1]$. For fixed $a, b$, and $c$, the probability that $d<\min \{a, b, c\}$ is evidently equal to $\min \{a, b, c\}$. Hence, if we average over all choices of $a, b, c$, the average value of $\min \{a, b, c\}$ is equal to the probability that, when $a, b, c$, and $d$ are independently randomly chosen, $d<$ $\min \{a, b, c\}$, i.e., that $d$ is the smallest of the four variables. On the other hand, by symmetry, the probability that $d$ is the smallest of the four is simply equal to $1 / 4$, so that is our answer.
Compute the sum of all possible distinct values of \( m+n \) if \( m \) and \( n \) are positive integers such that $$ \operatorname{lcm}(m, n) + \operatorname{gcd}(m, n) = 2(m+n) + 11 $$	32
Let \( x, y \in \mathbf{R}^{+} \), and \(\frac{19}{x}+\frac{98}{y}=1\). Find the minimum value of \( x + y \).	117 + 14 \sqrt{38}
Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$.	729
In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$ . What is the probability of drawing a number from the bag that is divisible by $11$ ?	1/7
In a small town, the police are looking for a wanderer. There is a four in five chance that he is in one of the eight bars in the town, with no preference for any particular one. Two officers visited seven bars but did not find the wanderer. What are the chances of finding him in the eighth bar?	\frac{1}{3}
Given that \begin{eqnarray}&(1)& x\text{ and }y\text{ are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=\|x-y\|. \end{eqnarray} How many distinct values of $z$ are possible?	9	We express the numbers as $x=100a+10b+c$ and $y=100c+10b+a$. From this, we have \begin{eqnarray}z&=&\|100a+10b+c-100c-10b-a\|\\&=&\|99a-99c\|\\&=&99\|a-c\|\\ \end{eqnarray} Because $a$ and $c$ are digits, and $a$ and $c$ are both between 1 and 9 (from condition 1), there are $\boxed{009}$ possible values (since all digits except $9$ can be expressed this way).
If $4:x^2 = x:16$, what is the value of $x$?	4
To estimate the consumption of disposable wooden chopsticks, in 1999, a sample of 10 restaurants from a total of 600 high, medium, and low-grade restaurants in a certain county was taken. The daily consumption of disposable chopstick boxes in these restaurants was as follows: 0.6, 3.7, 2.2, 1.5, 2.8, 1.7, 1.2, 2.1, 3.2, 1.0 (1) Estimate the total consumption of disposable chopstick boxes in the county for the year 1999 by calculating the sample (assuming 350 business days per year); (2) In 2001, another survey on the consumption of disposable wooden chopsticks was conducted in the same manner, and the result was that the average daily use of disposable chopstick boxes in the 10 sampled restaurants was 2.42 boxes. Calculate the average annual growth rate of the consumption of disposable wooden chopsticks for the years 2000 and 2001 (the number of restaurants in the county and the total number of business days in the year remained the same as in 1999); (3) Under the conditions of (2), if producing a set of desks and chairs for primary and secondary school students requires 0.07 cubic meters of wood, calculate how many sets of student desks and chairs can be produced with the wood used for disposable chopsticks in the county in 2001. The relevant data needed for the calculation are: 100 pairs of chopsticks per box, each pair of chopsticks weighs 5 grams, and the density of the wood used is 0.5×10<sup>3</sup> kg/m<sup>3</sup>; (4) If you were asked to estimate the amount of wood consumed by disposable chopsticks in your province for a year, how would you use statistical knowledge to do so? Briefly describe in words.	7260
When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:	1: 2	1. Define the terms of the sequence: Let the first term of the arithmetic progression be $a$ and the common difference be $d$. The $n$-th term of the sequence can be expressed as $a + (n-1)d$. 2. Expression for the sum of the first $n$ terms: The sum $S_n$ of the first $n$ terms of an arithmetic progression is given by: \[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \] where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference. 3. Calculate the sum of the first 5 terms ($S_5$): \[ S_5 = \frac{5}{2} \left(2a + 4d\right) = 5a + 10d \] 4. Calculate the sum of the first 10 terms ($S_{10}$): \[ S_{10} = \frac{10}{2} \left(2a + 9d\right) = 10a + 45d \] 5. Set up the equation given in the problem: The sum of the first ten terms is four times the sum of the first five terms: \[ S_{10} = 4S_5 \] Substituting the expressions for $S_{10}$ and $S_5$: \[ 10a + 45d = 4(5a + 10d) \] 6. Simplify the equation: \[ 10a + 45d = 20a + 40d \] Rearranging terms: \[ 45d - 40d = 20a - 10a \] \[ 5d = 10a \] \[ \frac{a}{d} = \frac{5}{10} = \frac{1}{2} \] 7. Conclusion: The ratio of the first term $a$ to the common difference $d$ is $\frac{1}{2}$, which corresponds to the ratio $1:2$. Therefore, the correct answer is $\boxed{\textbf{(A)}\ 1: 2}$.
Find the remainder when $5^{5^{5^5}}$ is divided by 500.	125
Compute the definite integral: $$ \int_{0}^{\pi} 2^{4} \cdot \sin ^{4} x \cos ^{4} x \, dx $$	\frac{3\pi}{8}
In triangle $XYZ$, $XY = 12$, $XZ = 15$, and $YZ = 23$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$.	\frac{40}{23}
Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$ , number $3$ written on the face opposite number $5$ , and number $4$ written on the face opposite number $6$ . The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$ .	24
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?	\frac{1}{3}	We are given that the probability that a ball is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \ldots$. We need to find the probability that the red ball is tossed into a higher-numbered bin than the green ball. #### Step-by-step Analysis: 1. Symmetry Argument: - The problem is symmetric with respect to the red and green balls. Therefore, the probability that the red ball is tossed into a higher-numbered bin than the green ball is the same as the probability that the green ball is tossed into a higher-numbered bin than the red ball. 2. Probability of Landing in the Same Bin: - The probability that both balls land in the same bin $k$ is the product of their individual probabilities of landing in bin $k$, which is $(2^{-k})(2^{-k}) = 2^{-2k}$. - Summing this over all $k$, we get the total probability of both balls landing in the same bin: \[ \sum_{k=1}^{\infty} 2^{-2k} = \sum_{k=1}^{\infty} (2^2)^{-k} = \frac{1}{3} \] This uses the formula for the sum of an infinite geometric series $\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r}$, where $a = 2^{-2}$ and $r = 2^{-2}$. 3. Probability of Different Bins: - Since the total probability must sum to 1, and the probability of both balls landing in the same bin is $\frac{1}{3}$, the probability of them landing in different bins is $1 - \frac{1}{3} = \frac{2}{3}$. 4. Equal Probabilities for Red Higher and Green Higher: - By symmetry, the probabilities of the red ball landing in a higher-numbered bin than the green ball and the green ball landing in a higher-numbered bin than the red ball are equal. Therefore, each of these probabilities is half of $\frac{2}{3}$, which is $\frac{1}{3}$. Thus, the probability that the red ball is tossed into a higher-numbered bin than the green ball is $\boxed{\frac{1}{3}}$. $\blacksquare$
Find the area of a triangle with side lengths 14, 48, and 50.	336	Note that this is a multiple of the 7-24-25 right triangle. The area is therefore $$\frac{14(48)}{2}=336$$.
Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240 d$.	120	By checking all the possibilities, one can show that $T$ has height 4 and base lengths 4 and 5. Orient $T$ so that the shorter base is on the top. Then, the length of the cut parallel to the bases is $\frac{4+5}{2}=\frac{9}{2}$. Thus, the top two pieces are trapezoids with height 2 and base lengths 2 and $\frac{9}{4}$, while the bottom two pieces are trapezoids with height 2 and base lengths $\frac{9}{4}$ and $\frac{5}{2}$. Thus, using the area formula for a trapezoid, the difference between the largest and smallest areas is $$d=\frac{\left(\frac{5}{2}+\frac{9}{4}-\frac{9}{4}-2\right) \cdot 2}{2}=\frac{1}{2}$$
The equation $x^2-4x+7=19$ has two solutions, $a$ and $b$, with $a\geq b$. What is the value of $2a+b$?	10
Mr. Reader has six different Spiderman comic books, five different Archie comic books and four different Garfield comic books. When stacked, all of the Spiderman comic books are grouped together, all of the Archie comic books are grouped together and all of the Garfield comic books are grouped together. In how many different orders can these 15 comic books be stacked in a pile with the covers facing up and all of them facing the same direction? Express your answer as a whole number.	12,\!441,\!600
The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true?	1
Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$.	129
Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}.$ Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).$	1
Given that Mike walks to his college, averaging 70 steps per minute with each step being 80 cm long, and it takes him 20 minutes to get there, determine how long it takes Tom to reach the college, given that he averages 120 steps per minute, but his steps are only 50 cm long.	18.67
In the plane Cartesian coordinate system \( xO y \), the circle \( \Omega \) and the parabola \( \Gamma: y^{2} = 4x \) share exactly one common point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \).	\frac{4 \sqrt{3}}{9}
How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$	180
Calculate (1) Use a simplified method to calculate $2017^{2}-2016 \times 2018$; (2) Given $a+b=7$ and $ab=-1$, find the values of $(a+b)^{2}$ and $a^{2}-3ab+b^{2}$.	54
A U-shaped number is a special type of three-digit number where the units digit and the hundreds digit are equal and greater than the tens digit. For example, 818 is a U-shaped number. How many U-shaped numbers are there?	36
In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midsegment is 5. Find the height of the trapezoid.	4.8
Find an ordered pair $(x,y)$ that satisfies both of the equations below: \begin{align} 2x - 3y &= -5,\\ 5x - 2y &= 4. \end{align}	(2,3)
The edges meeting at one vertex of a rectangular parallelepiped are in the ratio of $1: 2: 3$. What is the ratio of the lateral surface areas of the cylinders that can be circumscribed around the parallelepiped?	\sqrt{13} : 2\sqrt{10} : 3\sqrt{5}
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$	28
The number halfway between $\frac{1}{6}$ and $\frac{1}{4}$ is	\frac{5}{24}	To find the number halfway between two numbers, we calculate the average of the two numbers. The formula for the average of two numbers $a$ and $b$ is: \[ \text{Average} = \frac{a + b}{2} \] Given the numbers $\frac{1}{6}$ and $\frac{1}{4}$, we first find a common denominator to add them easily. The least common multiple of 6 and 4 is 12. Thus, we rewrite the fractions with a denominator of 12: \[ \frac{1}{6} = \frac{2}{12} \quad \text{and} \quad \frac{1}{4} = \frac{3}{12} \] Now, add these two fractions: \[ \frac{2}{12} + \frac{3}{12} = \frac{2+3}{12} = \frac{5}{12} \] Next, we find the average by dividing the sum by 2: \[ \frac{\frac{5}{12}}{2} = \frac{5}{12} \times \frac{1}{2} = \frac{5}{24} \] Thus, the number halfway between $\frac{1}{6}$ and $\frac{1}{4}$ is $\frac{5}{24}$. Therefore, the correct answer is $\boxed{\text{(C)}\ \frac{5}{24}}$.
Find the least integer value of $x$ for which $2\|x\| + 7 < 17$.	-4
There are 6 rectangular prisms with edge lengths of \(3 \text{ cm}\), \(4 \text{ cm}\), and \(5 \text{ cm}\). The faces of these prisms are painted red in such a way that one prism has only one face painted, another has exactly two faces painted, a third prism has exactly three faces painted, a fourth prism has exactly four faces painted, a fifth prism has exactly five faces painted, and the sixth prism has all six faces painted. After painting, each rectangular prism is divided into small cubes with an edge length of \(1 \text{ cm}\). What is the maximum number of small cubes that have exactly one red face?	177
Given the hyperbola $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a > 0,b > 0)$, the line $l$ passing through point $P(3,6)$ intersects $C$ at points $A$ and $B$, and the midpoint of $AB$ is $N(12,15)$. Determine the eccentricity of the hyperbola $C$.	\frac{3}{2}
In \( \triangle ABC \), \( AB = AC = 26 \) and \( BC = 24 \). Points \( D, E, \) and \( F \) are on sides \( \overline{AB}, \overline{BC}, \) and \( \overline{AC}, \) respectively, such that \( \overline{DE} \) and \( \overline{EF} \) are parallel to \( \overline{AC} \) and \( \overline{AB}, \) respectively. What is the perimeter of parallelogram \( ADEF \)?	52
From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______.	\frac{6}{35}
Calculate:<br/>$(1)2(\sqrt{3}-\sqrt{5})+3(\sqrt{3}+\sqrt{5})$;<br/>$(2)-{1}^{2}-\|1-\sqrt{3}\|+\sqrt[3]{8}-(-3)×\sqrt{9}$.	11 - \sqrt{3}
Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends?	126/29
A point in space $(x,y,z)$ is randomly selected so that $-1\le x \le 1$,$-1\le y \le 1$,$-1\le z \le 1$. What is the probability that $x^2+y^2+z^2\le 1$?	\frac{\pi}{6}
$A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Find the total distance around the track in yards. A) 600 B) 720 C) 840 D) 960 E) 1080	720
Given the angle $\frac {19\pi}{5}$, express it in the form of $2k\pi+\alpha$ ($k\in\mathbb{Z}$), then determine the angle $\alpha$ that makes $\|\alpha\|$ the smallest.	-\frac {\pi}{5}
In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Determine the maximum slope of the line $OQ$.	\sqrt{2}
Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p + q + r + s = 10$ and \[ pq + pr + ps + qr + qs + rs = 20. \] Find the largest possible value of $s$.	\frac{5 + \sqrt{105}}{2}
Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression \(1, 3, 5, \ldots\) or the arithmetic progression \(1, 4, 7, \ldots\).	674
Compute the number of sets $S$ such that every element of $S$ is a nonnegative integer less than 16, and if $x \in S$ then $(2 x \bmod 16) \in S$.	678	For any nonempty $S$ we must have $0 \in S$. Now if we draw a directed graph of dependencies among the non-zero elements, it creates a balanced binary tree where every leaf has depth 3 . In the diagram, if $a$ is a parent of $b$ it means that if $b \in S$, then $a$ must also be in $S$. We wish to find the number of subsets of nodes such that every node in the set also has its parent in the set. We do this with recursion. Let $f(n)$ denote the number of such sets on a balanced binary tree of depth $n$. If the root vertex is not in the set, then the set must be empty. Otherwise, we can consider each subtree separately. This gives the recurrence $f(n)=f(n-1)^{2}+1$. We know $f(0)=2$, so we can calculate $f(1)=5, f(2)=26, f(3)=677$. We add 1 at the end for the empty set. Hence our answer is $f(3)+1=678$.
Determine the total number of different selections possible for five donuts when choosing from four types of donuts (glazed, chocolate, powdered, and jelly), with the additional constraint of purchasing at least one jelly donut.	35
The medians \( A M \) and \( B E \) of triangle \( A B C \) intersect at point \( O \). Points \( O, M, E, C \) lie on the same circle. Find \( A B \) if \( B E = A M = 3 \).	2\sqrt{3}
Given that the power function $y=x^{m}$ is an even function and is a decreasing function when $x \in (0,+\infty)$, determine the possible value of the real number $m$.	-2
Let $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with chords $\overline{AC}$ and $\overline{BC}$. The largest possible value of $d$ can be written in the form $r-s\sqrt{t}$, where $r, s$ and $t$ are positive integers and $t$ is not divisible by the square of any prime. Find $r+s+t$.	14	By Pythagoras in $\triangle BMN,$ we get $BN=\dfrac{4}{5}.$ Since cross ratios are preserved upon projecting, note that $(M,Y;X,N)\stackrel{C}{=}(M,B;A,N).$ By definition of a cross ratio, this becomes \[\dfrac{XM}{NY}:\dfrac{NM}{NY}=\dfrac{AM}{AB}:\dfrac{MN}{NB}.\] Let $MY=a,YX=b,XN=c$ such that $a+b+c=1.$ We know that $XM=a+b,XY=b,NM=1,NY=b+c,$ so the LHS becomes $\dfrac{(a+b)(b+c)}{b}.$ In the RHS, we are given every value except for $AB.$ However, Ptolemy's Theorem on $MBAN$ gives $AB\cdot MN+AN\cdot BM=AM\cdot BN\implies AB+\dfrac{3}{5\sqrt{2}}=\dfrac{4}{5\sqrt{2}}\implies AB=\dfrac{1}{5\sqrt{2}}.$ Substituting, we get $\dfrac{(a+b)(b+c)}{b}=4\implies b(a+b+c)+ac=4b, b=\dfrac{ac}{3}$ where we use $a+b+c=1.$ Again using $a+b+c=1,$ we have $a+b+c=1\implies a+\dfrac{ac}{3}+c=1\implies a=3\dfrac{1-c}{c+3}.$ Then $b=\dfrac{ac}{3}=\dfrac{c-c^2}{c+3}.$ Since this is a function in $c,$ we differentiate WRT $c$ to find its maximum. By quotient rule, it suffices to solve \[(-2c+1)(c+3)-(c-c^2)=0 \implies c^2+6c-3,c=-3+2\sqrt{3}.\] Substituting back yields $b=7-4\sqrt{3},$ so $7+4+3=\boxed{014}$ is the answer. ~Generic_Username
$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$ . A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$ . Find the area of the square.	36
A regular octahedron has a side length of 1. What is the distance between two opposite faces?	\sqrt{6} / 3	Imagine orienting the octahedron so that the two opposite faces are horizontal. Project onto a horizontal plane; these two faces are congruent equilateral triangles which (when projected) have the same center and opposite orientations. Hence, the vertices of the octahedron project to the vertices of a regular hexagon $A B C D E F$. Let $O$ be the center of the hexagon and $M$ the midpoint of $A C$. Now $A B M$ is a 30-60-90 triangle, so $A B=A M /(\sqrt{3} / 2)=(1 / 2) /(\sqrt{3} / 2)=\sqrt{3} / 3$. If we let $d$ denote the desired vertical distance between the opposite faces (which project to $A C E$ and $B D F)$, then by the Pythagorean Theorem, $A B^{2}+d^{2}=1^{2}$, so $d=\sqrt{1-A B^{2}}=\sqrt{6} / 3$.
Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $4$ per gallon. How many miles can Margie drive on $20$ worth of gas?	160	1. Calculate the number of gallons Margie can buy with $\textdollar 20$: Given that the cost of one gallon of gas is $\textdollar 4$, the number of gallons Margie can buy with $\textdollar 20$ is calculated by dividing the total amount of money by the cost per gallon: \[ \text{Number of gallons} = \frac{\textdollar 20}{\textdollar 4 \text{ per gallon}} = 5 \text{ gallons} \] 2. Calculate the total miles Margie can drive with 5 gallons of gas: Since Margie's car can travel 32 miles per gallon, the total distance she can travel with 5 gallons is: \[ \text{Total miles} = 32 \text{ miles per gallon} \times 5 \text{ gallons} = 160 \text{ miles} \] 3. Conclusion: Margie can drive a total of 160 miles with $\textdollar 20$ worth of gas. Thus, the correct answer is $\boxed{\textbf{(C)}~160}$.
Given a woman was x years old in the year $x^2$, determine her birth year.	1980
Let $x_1,$ $x_2,$ $x_3,$ $x_4,$ $x_5$ be the roots of the polynomial $f(x) = x^5 + x^2 + 1,$ and let $g(x) = x^2 - 2.$ Find \[g(x_1) g(x_2) g(x_3) g(x_4) g(x_5).\]	-23
Which triplet of numbers has a sum NOT equal to 1?	1.1 + (-2.1) + 1.0	To find which triplet of numbers has a sum NOT equal to 1, we will add the numbers in each triplet and check the result. 1. Triplet (A): $(1/2, 1/3, 1/6)$ \[ \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1 \] 2. Triplet (B): $(2, -2, 1)$ \[ 2 + (-2) + 1 = 0 + 1 = 1 \] 3. Triplet (C): $(0.1, 0.3, 0.6)$ \[ 0.1 + 0.3 + 0.6 = 1.0 \] 4. Triplet (D): $(1.1, -2.1, 1.0)$ \[ 1.1 + (-2.1) + 1.0 = -1.0 + 1.0 = 0 \] 5. Triplet (E): $(-3/2, -5/2, 5)$ \[ -\frac{3}{2} - \frac{5}{2} + 5 = -\frac{8}{2} + 5 = -4 + 5 = 1 \] From the calculations above, we see that all triplets except for Triplet (D) sum to 1. Therefore, the triplet that does not sum to 1 is: \[ \boxed{(D)} \]
In the drawing, 5 lines intersect at a single point. One of the resulting angles is $34^\circ$. What is the sum of the four angles shaded in gray, in degrees?	146
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$	37
Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac {p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p$ + $q$ + $r$. Diagram [asy] size(120); pathpen = linewidth(0.7); pointpen = black; pen f = fontsize(10); pair B=(0,0), A=(5,0), C=(0,13), E=(-5,0), O = incenter(E,C,A), D=IP(A -- A+3*(O-A),E--C); D(A--B--C--cycle); D(A--D--C); D(D--E--B, linetype("4 4")); MP("5", (A+B)/2, f); MP("13", (A+C)/2, NE,f); MP("A",D(A),f); MP("B",D(B),f); MP("C",D(C),N,f); MP("A'",D(E),f); MP("D",D(D),NW,f); D(rightanglemark(C,B,A,20)); D(anglemark(D,A,E,35));D(anglemark(C,A,D,30)); [/asy] -asjpz	96	Let $O$ be the intersection of $BC$ and $AD$. By the Angle Bisector Theorem, $\frac {5}{BO}$ = $\frac {13}{CO}$, so $BO$ = $5x$ and $CO$ = $13x$, and $BO$ + $OC$ = $BC$ = $12$, so $x$ = $\frac {2}{3}$, and $OC$ = $\frac {26}{3}$. Let $P$ be the foot of the altitude from $D$ to $OC$. It can be seen that triangle $DOP$ is similar to triangle $AOB$, and triangle $DPC$ is similar to triangle $ABC$. If $DP$ = $15y$, then $CP$ = $36y$, $OP$ = $10y$, and $OD$ = $5y\sqrt {13}$. Since $OP$ + $CP$ = $46y$ = $\frac {26}{3}$, $y$ = $\frac {13}{69}$, and $AD$ = $\frac {60\sqrt{13}}{23}$ (by the pythagorean theorem on triangle $ABO$ we sum $AO$ and $OD$). The answer is $60$ + $13$ + $23$ = $\boxed{096}$.
For his birthday, Piglet baked a big cake weighing 10 kg and invited 100 guests. Among them was Winnie-the-Pooh, who has a weakness for sweets. The birthday celebrant announced the cake-cutting rule: the first guest cuts themselves a piece of cake equal to \(1\%\) of the remaining cake, the second guest cuts themselves a piece of cake equal to \(2\%\) of the remaining cake, the third guest cuts themselves a piece of cake equal to \(3\%\) of the remaining cake, and so on. Which position in the queue should Winnie-the-Pooh take to get the largest piece of cake?	10
Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$ .	831
A trapezoid $ABCD$ has bases $AD$ and $BC$. If $BC = 60$ units, and altitudes from $B$ and $C$ to line $AD$ divide it into segments of lengths $AP = 20$ units and $DQ = 19$ units, with the length of the altitude itself being $30$ units, what is the perimeter of trapezoid $ABCD$? A) $\sqrt{1300} + 159$ B) $\sqrt{1261} + 159$ C) $\sqrt{1300} + \sqrt{1261} + 159$ D) $259$ E) $\sqrt{1300} + 60 + \sqrt{1261}$	\sqrt{1300} + \sqrt{1261} + 159
Given $x$, $y$, and $a \in R^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Then, calculate the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$.	6\sqrt{3}
Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person gets more heads than the first person.	1/2
Determine the numerical value of $k$ such that \[\frac{12}{x + z} = \frac{k}{z - y} = \frac{5}{y - x}.\]	17
The Gauss family has three boys aged $7,$ a girl aged $14,$ and a boy aged $15.$ What is the mean (average) of the ages of the children?	10
In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$.	145
A five-digit number \(abcde\) satisfies: \[ a < b, \, b > c > d, \, d < e, \, \text{and} \, a > d, \, b > e. \] For example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the five-digit number is said to follow the "sine rule." Find the total number of five-digit numbers that follow the sine rule. Note: Please disregard any references or examples provided within the original problem if they are not part of the actual problem statement.	2892
Two students are having a pie eating contest. The first student eats $\frac{6}{7}$ of one pie. The second student eats $\frac{3}{4}$ of one pie. How much more pie did the first student finish than the second student? Express your answer as a fraction of one pie, reduced to simplest form.	\frac{3}{28}
In triangle $XYZ$, the sides are in the ratio $3:4:5$. If segment $XM$ bisects the largest angle at $X$ and divides side $YZ$ into two segments, find the length of the shorter segment given that the length of side $YZ$ is $12$ inches.	\frac{9}{2}
The scientific notation for 0.000048 is $4.8\times 10^{-5}$.	4.8 \times 10^{-5}
The value of \( a \) is chosen such that the number of roots of the first equation \( 4^{x} - 4^{-x} = 2 \cos a x \) is 2007. How many roots does the second equation \( 4^{x} + 4^{-x} = 2 \cos a x + 4 \) have for the same \( a \)?	4014
How many integers between 100 and 300 have both 11 and 8 as factors?	2
Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$?	1009
Given the set $\{1,2,3,5,8,13,21,34,55\}$, calculate the number of integers between 3 and 89 that cannot be written as the sum of exactly two elements of the set.	53
Given a right circular cone with three mutually perpendicular side edges, each with a length of $\sqrt{3}$, determine the surface area of the circumscribed sphere.	9\pi
Max bought a new dirt bike and paid $10\%$ of the cost upfront, which was $\$150$. What was the price of the bike?	\$ 1500
How many zeroes does $10!$ end with, when $10!$ is written in base 9?	2
Given $\tan \theta =2$, find $\cos 2\theta =$____ and $\tan (\theta -\frac{π}{4})=$____.	\frac{1}{3}
In a trapezoid \(ABCD\) with bases \(AD=12\) and \(BC=8\), circles constructed on the sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of diagonal \(AC\) is 12. Find the length of \(BD\).	16
Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions.	\dfrac{19}{36}
Given that $40\%$ of the birds were pigeons, $20\%$ were sparrows, $15\%$ were crows, and the remaining were parakeets, calculate the percent of the birds that were not sparrows and were crows.	18.75\%
What is the greatest product obtainable from two integers whose sum is 246?	15129
Find the range of the function $f(x) = \arcsin x + \arccos x + \arctan x.$ All functions are in radians.	\left[ \frac{\pi}{4}, \frac{3 \pi}{4} \right]
Given a regular tetrahedron A-BCD with an edge length of 1, and $\overrightarrow{AE} = 2\overrightarrow{EB}$, $\overrightarrow{AF} = 2\overrightarrow{FD}$, calculate $\overrightarrow{EF} \cdot \overrightarrow{DC}$.	-\frac{1}{3}
Consider the sequence \(\left\{a_{n}\right\}\) defined by \(a_{0}=\frac{1}{2}\) and \(a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2023}\) for \(n = 0, 1, \ldots\). Find the integer \(k\) such that \(a_{k} < 1 < a_{k+1}\).	2023
Five positive integers from $1$ to $15$ are chosen without replacement. What is the probability that their sum is divisible by $3$ ?	1/3

Find $PB$ given that $AP$ is a tangent to $\Omega$, $\angle PAB=\angle PCA$, and $\frac{PB}{PA}=\frac{4}{7}=\frac{PA}{PB+6}$.

\frac{32}{11}

Since $AP$ is a tangent to $\Omega$, we know that $\angle PAB=\angle PCA$, so $\triangle PAB \sim \triangle PCA$, so we get that $$\frac{PB}{PA}=\frac{4}{7}=\frac{PA}{PB+6}$$ Solving, we get that $7PB=4PA$, so $$4(PB+6)=7PA=\frac{49}{4}PB \Rightarrow \frac{33}{4}PB=24 \Rightarrow PB=\frac{32}{11}$$

The minimum value of the function $ y = \sin^4{x} + \cos^4{x} + \sec^4{x} + \csc^4{x} $.

\frac{17}{2}

Given in $\triangle ABC$, $\tan A$ and $\tan B$ are the two real roots of the equation $x^2 + ax + 4 = 0$: (1) If $a = -8$, find the value of $\tan C$; (2) Find the minimum value of $\tan C$, and specify the corresponding values of $\tan A$ and $\tan B$.

\frac{4}{3}

The real numbers $a$, $b$, and $c$ satisfy the equation $({a}^{2}+\frac{{b}^{2}}{4}+\frac{{c}^{2}}{9}=1)$. Find the maximum value of $a+b+c$.

\sqrt{14}

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$ holds for all positive integers $x, y$.

f(x) = x

Let's analyze the problem by working with the given inequality: \[ f(x) + y f(f(x)) \le x(1 + f(y)) \] for all positive integers $x, y$. To find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying this inequality, we will first test some small values and then generalize our findings. **Step 1: Consider specific cases.** 1. **For $y = 1$:** \[ f(x) + f(f(x)) \le x(1 + f(1)) \] Rearranging gives: \[ f(x) + f(f(x)) \le x + x f(1) \] 2. **For $x = 1$:** \[ f(1) + y f(f(1)) \le 1(1 + f(y)) = 1 + f(y) \] Which simplifies to: \[ y f(f(1)) \le 1 + f(y) - f(1) \] Since $y$ is arbitrary, this implies $f(f(1)) = 0$ unless $f(x) = x$. **Step 2: Consider $f(x) = x$.** Assume $f(x) = x$ for all $x \in \mathbb{N}$. Substitute into the original inequality: \[ x + y f(x) \le x(1 + f(y)) \] which becomes: \[ x + yx \le x + xy \] This simplifies to: \[ x + yx \le x + xy \] The inequality holds, which suggests that $f(x) = x$ is indeed a solution. **Step 3: Consider the nature of the function.** Assuming any function $f(x)$ other than the identity leads to contradictions in maintaining the inequality universally. For instance, choosing other forms may not satisfy for larger values of $x, y$, due to the restrictive nature of the inequality, especially where powers of $f$ terms (like $f(f(x))$) would arise. Therefore, after analyzing potential options, we determine that the function: \[ f(x) = x \] is the only solution that satisfies the given inequality for all $x, y \in \mathbb{N}$. Thus, the solution to the problem is: \[ \boxed{f(x) = x} \]

In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 18$ and $CP = 6.$ If $\tan \angle APD = 2,$ find $AB.$

18

The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence?

1000

Given two fixed points $A(-1,0)$ and $B(1,0)$, and a moving point $P(x,y)$ on the line $l$: $y=x+3$, an ellipse $C$ has foci at $A$ and $B$ and passes through point $P$. Find the maximum value of the eccentricity of ellipse $C$.

\dfrac{\sqrt{5}}{5}

Given real numbers \$x\$ and \$y\$ satisfy the equation \$(x-3)^{2}+y^{2}=9\$, find the minimum value of \$-2y-3x\$ \_\_\_\_\_\_.

-3\sqrt{13}-9

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$ .

3012

What is the area, in square units, of a triangle whose vertices are at $(4, -1)$, $(10, 3)$ and $(4, 5)$?

18

The sum of the first n terms of the sequence $\{a_n\}$ is $S_n$. If the terms of the sequence $\{a_n\}$ are arranged according to the following rule: $$\frac {1}{2}, \frac {1}{3}, \frac {2}{3}, \frac {1}{4}, \frac {2}{4}, \frac {3}{4}, \frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \ldots, \frac {1}{n}, \frac {2}{n}, \ldots, \frac {n-1}{n}, \ldots$$ If there exists a positive integer k such that $S_{k-1} < 10$ and $S_k > 10$, then $a_k = \_\_\_\_\_\_$.

\frac{6}{7}

Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is:

$x >\frac{10}{3}$

To find the set of all values of $x$ that satisfy the inequality $2x - 3 > 7 - x$, we start by simplifying the inequality: 1. **Combine like terms**: \[ 2x - 3 > 7 - x \] Add $x$ to both sides to get all $x$ terms on one side: \[ 2x + x > 7 + 3 \] Simplify: \[ 3x > 10 \] 2. **Solve for $x$**: Divide both sides by 3 to isolate $x$: \[ x > \frac{10}{3} \] This tells us that $x$ must be greater than $\frac{10}{3}$. Therefore, the correct answer is: \[ \boxed{\textbf{(D)}\ x > \frac{10}{3}} \]

If $a, b$, and $c$ are random real numbers from 0 to 1, independently and uniformly chosen, what is the average (expected) value of the smallest of $a, b$, and $c$?

1/4

Let $d$ be a fourth random variable, also chosen uniformly from $[0,1]$. For fixed $a, b$, and $c$, the probability that $d<\min \{a, b, c\}$ is evidently equal to $\min \{a, b, c\}$. Hence, if we average over all choices of $a, b, c$, the average value of $\min \{a, b, c\}$ is equal to the probability that, when $a, b, c$, and $d$ are independently randomly chosen, $d<$ $\min \{a, b, c\}$, i.e., that $d$ is the smallest of the four variables. On the other hand, by symmetry, the probability that $d$ is the smallest of the four is simply equal to $1 / 4$, so that is our answer.

Compute the sum of all possible distinct values of $ m+n $ if $ m $ and $ n $ are positive integers such that $$ \operatorname{lcm}(m, n) + \operatorname{gcd}(m, n) = 2(m+n) + 11 $$

32

Let $ x, y \in \mathbf{R}^{+} $, and $\frac{19}{x}+\frac{98}{y}=1$. Find the minimum value of $ x + y $.

117 + 14 \sqrt{38}

Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$.

729

In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$ . What is the probability of drawing a number from the bag that is divisible by $11$ ?

1/7

In a small town, the police are looking for a wanderer. There is a four in five chance that he is in one of the eight bars in the town, with no preference for any particular one. Two officers visited seven bars but did not find the wanderer. What are the chances of finding him in the eighth bar?

\frac{1}{3}

Given that \begin{eqnarray*}&(1)& x\text{ and }y\text{ are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& y\text{ is the number formed by reversing the digits of }x\text{; and}\\ &(3)& z=|x-y|. \end{eqnarray*} How many distinct values of $z$ are possible?

9

We express the numbers as $x=100a+10b+c$ and $y=100c+10b+a$. From this, we have \begin{eqnarray*}z&=&|100a+10b+c-100c-10b-a|\\&=&|99a-99c|\\&=&99|a-c|\\ \end{eqnarray*} Because $a$ and $c$ are digits, and $a$ and $c$ are both between 1 and 9 (from condition 1), there are $\boxed{009}$ possible values (since all digits except $9$ can be expressed this way).

If $4:x^2 = x:16$, what is the value of $x$?

4

To estimate the consumption of disposable wooden chopsticks, in 1999, a sample of 10 restaurants from a total of 600 high, medium, and low-grade restaurants in a certain county was taken. The daily consumption of disposable chopstick boxes in these restaurants was as follows: 0.6, 3.7, 2.2, 1.5, 2.8, 1.7, 1.2, 2.1, 3.2, 1.0 (1) Estimate the total consumption of disposable chopstick boxes in the county for the year 1999 by calculating the sample (assuming 350 business days per year); (2) In 2001, another survey on the consumption of disposable wooden chopsticks was conducted in the same manner, and the result was that the average daily use of disposable chopstick boxes in the 10 sampled restaurants was 2.42 boxes. Calculate the average annual growth rate of the consumption of disposable wooden chopsticks for the years 2000 and 2001 (the number of restaurants in the county and the total number of business days in the year remained the same as in 1999); (3) Under the conditions of (2), if producing a set of desks and chairs for primary and secondary school students requires 0.07 cubic meters of wood, calculate how many sets of student desks and chairs can be produced with the wood used for disposable chopsticks in the county in 2001. The relevant data needed for the calculation are: 100 pairs of chopsticks per box, each pair of chopsticks weighs 5 grams, and the density of the wood used is 0.5×103 kg/m3; (4) If you were asked to estimate the amount of wood consumed by disposable chopsticks in your province for a year, how would you use statistical knowledge to do so? Briefly describe in words.

7260

When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:

1: 2

1. **Define the terms of the sequence**: Let the first term of the arithmetic progression be $a$ and the common difference be $d$. The $n$-th term of the sequence can be expressed as $a + (n-1)d$. 2. **Expression for the sum of the first $n$ terms**: The sum $S_n$ of the first $n$ terms of an arithmetic progression is given by: \[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \] where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference. 3. **Calculate the sum of the first 5 terms ($S_5$)**: \[ S_5 = \frac{5}{2} \left(2a + 4d\right) = 5a + 10d \] 4. **Calculate the sum of the first 10 terms ($S_{10}$)**: \[ S_{10} = \frac{10}{2} \left(2a + 9d\right) = 10a + 45d \] 5. **Set up the equation given in the problem**: The sum of the first ten terms is four times the sum of the first five terms: \[ S_{10} = 4S_5 \] Substituting the expressions for $S_{10}$ and $S_5$: \[ 10a + 45d = 4(5a + 10d) \] 6. **Simplify the equation**: \[ 10a + 45d = 20a + 40d \] Rearranging terms: \[ 45d - 40d = 20a - 10a \] \[ 5d = 10a \] \[ \frac{a}{d} = \frac{5}{10} = \frac{1}{2} \] 7. **Conclusion**: The ratio of the first term $a$ to the common difference $d$ is $\frac{1}{2}$, which corresponds to the ratio $1:2$. Therefore, the correct answer is $\boxed{\textbf{(A)}\ 1: 2}$.

Find the remainder when $5^{5^{5^5}}$ is divided by 500.

125

Compute the definite integral: $$ \int_{0}^{\pi} 2^{4} \cdot \sin ^{4} x \cos ^{4} x \, dx $$

\frac{3\pi}{8}

In triangle $XYZ$, $XY = 12$, $XZ = 15$, and $YZ = 23$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$.

\frac{40}{23}

Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$ , number $3$ written on the face opposite number $5$ , and number $4$ written on the face opposite number $6$ . The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$ .

24

A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

\frac{1}{3}

We are given that the probability that a ball is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \ldots$. We need to find the probability that the red ball is tossed into a higher-numbered bin than the green ball. #### Step-by-step Analysis: 1. **Symmetry Argument**: - The problem is symmetric with respect to the red and green balls. Therefore, the probability that the red ball is tossed into a higher-numbered bin than the green ball is the same as the probability that the green ball is tossed into a higher-numbered bin than the red ball. 2. **Probability of Landing in the Same Bin**: - The probability that both balls land in the same bin $k$ is the product of their individual probabilities of landing in bin $k$, which is $(2^{-k})(2^{-k}) = 2^{-2k}$. - Summing this over all $k$, we get the total probability of both balls landing in the same bin: \[ \sum_{k=1}^{\infty} 2^{-2k} = \sum_{k=1}^{\infty} (2^2)^{-k} = \frac{1}{3} \] This uses the formula for the sum of an infinite geometric series $\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r}$, where $a = 2^{-2}$ and $r = 2^{-2}$. 3. **Probability of Different Bins**: - Since the total probability must sum to 1, and the probability of both balls landing in the same bin is $\frac{1}{3}$, the probability of them landing in different bins is $1 - \frac{1}{3} = \frac{2}{3}$. 4. **Equal Probabilities for Red Higher and Green Higher**: - By symmetry, the probabilities of the red ball landing in a higher-numbered bin than the green ball and the green ball landing in a higher-numbered bin than the red ball are equal. Therefore, each of these probabilities is half of $\frac{2}{3}$, which is $\frac{1}{3}$. Thus, the probability that the red ball is tossed into a higher-numbered bin than the green ball is $\boxed{\frac{1}{3}}$. $\blacksquare$

Find the area of a triangle with side lengths 14, 48, and 50.

336

Note that this is a multiple of the 7-24-25 right triangle. The area is therefore $$\frac{14(48)}{2}=336$$.

Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240 d$.

120

By checking all the possibilities, one can show that $T$ has height 4 and base lengths 4 and 5. Orient $T$ so that the shorter base is on the top. Then, the length of the cut parallel to the bases is $\frac{4+5}{2}=\frac{9}{2}$. Thus, the top two pieces are trapezoids with height 2 and base lengths 2 and $\frac{9}{4}$, while the bottom two pieces are trapezoids with height 2 and base lengths $\frac{9}{4}$ and $\frac{5}{2}$. Thus, using the area formula for a trapezoid, the difference between the largest and smallest areas is $$d=\frac{\left(\frac{5}{2}+\frac{9}{4}-\frac{9}{4}-2\right) \cdot 2}{2}=\frac{1}{2}$$

The equation $x^2-4x+7=19$ has two solutions, $a$ and $b$, with $a\geq b$. What is the value of $2a+b$?

10

Mr. Reader has six different Spiderman comic books, five different Archie comic books and four different Garfield comic books. When stacked, all of the Spiderman comic books are grouped together, all of the Archie comic books are grouped together and all of the Garfield comic books are grouped together. In how many different orders can these 15 comic books be stacked in a pile with the covers facing up and all of them facing the same direction? Express your answer as a whole number.

12,\!441,\!600

The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true?

1

Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$.

129

Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}.$ Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).$

1

Given that Mike walks to his college, averaging 70 steps per minute with each step being 80 cm long, and it takes him 20 minutes to get there, determine how long it takes Tom to reach the college, given that he averages 120 steps per minute, but his steps are only 50 cm long.

18.67

In the plane Cartesian coordinate system $ xO y $, the circle $ \Omega $ and the parabola $ \Gamma: y^{2} = 4x $ share exactly one common point, and the circle $ \Omega $ is tangent to the x-axis at the focus $ F $ of $ \Gamma $. Find the radius of the circle $ \Omega $.

\frac{4 \sqrt{3}}{9}

How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$

180

Calculate (1) Use a simplified method to calculate $2017^{2}-2016 \times 2018$; (2) Given $a+b=7$ and $ab=-1$, find the values of $(a+b)^{2}$ and $a^{2}-3ab+b^{2}$.

54

A U-shaped number is a special type of three-digit number where the units digit and the hundreds digit are equal and greater than the tens digit. For example, 818 is a U-shaped number. How many U-shaped numbers are there?

36

In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midsegment is 5. Find the height of the trapezoid.

4.8

Find an ordered pair $(x,y)$ that satisfies both of the equations below: \begin{align*} 2x - 3y &= -5,\\ 5x - 2y &= 4. \end{align*}

(2,3)

The edges meeting at one vertex of a rectangular parallelepiped are in the ratio of $1: 2: 3$. What is the ratio of the lateral surface areas of the cylinders that can be circumscribed around the parallelepiped?

\sqrt{13} : 2\sqrt{10} : 3\sqrt{5}

Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$

28

The number halfway between $\frac{1}{6}$ and $\frac{1}{4}$ is

\frac{5}{24}

To find the number halfway between two numbers, we calculate the average of the two numbers. The formula for the average of two numbers $a$ and $b$ is: \[ \text{Average} = \frac{a + b}{2} \] Given the numbers $\frac{1}{6}$ and $\frac{1}{4}$, we first find a common denominator to add them easily. The least common multiple of 6 and 4 is 12. Thus, we rewrite the fractions with a denominator of 12: \[ \frac{1}{6} = \frac{2}{12} \quad \text{and} \quad \frac{1}{4} = \frac{3}{12} \] Now, add these two fractions: \[ \frac{2}{12} + \frac{3}{12} = \frac{2+3}{12} = \frac{5}{12} \] Next, we find the average by dividing the sum by 2: \[ \frac{\frac{5}{12}}{2} = \frac{5}{12} \times \frac{1}{2} = \frac{5}{24} \] Thus, the number halfway between $\frac{1}{6}$ and $\frac{1}{4}$ is $\frac{5}{24}$. Therefore, the correct answer is $\boxed{\text{(C)}\ \frac{5}{24}}$.

Find the least integer value of $x$ for which $2|x| + 7 < 17$.

-4

There are 6 rectangular prisms with edge lengths of $3 \text{ cm}$, $4 \text{ cm}$, and $5 \text{ cm}$. The faces of these prisms are painted red in such a way that one prism has only one face painted, another has exactly two faces painted, a third prism has exactly three faces painted, a fourth prism has exactly four faces painted, a fifth prism has exactly five faces painted, and the sixth prism has all six faces painted. After painting, each rectangular prism is divided into small cubes with an edge length of $1 \text{ cm}$. What is the maximum number of small cubes that have exactly one red face?

177

Given the hyperbola $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a > 0,b > 0)$, the line $l$ passing through point $P(3,6)$ intersects $C$ at points $A$ and $B$, and the midpoint of $AB$ is $N(12,15)$. Determine the eccentricity of the hyperbola $C$.

\frac{3}{2}

In $ \triangle ABC $, $ AB = AC = 26 $ and $ BC = 24 $. Points $ D, E, $ and $ F $ are on sides $ \overline{AB}, \overline{BC}, $ and $ \overline{AC}, $ respectively, such that $ \overline{DE} $ and $ \overline{EF} $ are parallel to $ \overline{AC} $ and $ \overline{AB}, $ respectively. What is the perimeter of parallelogram $ ADEF $?

52

From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______.

\frac{6}{35}

Calculate: $(1)2(\sqrt{3}-\sqrt{5})+3(\sqrt{3}+\sqrt{5})$; $(2)-{1}^{2}-|1-\sqrt{3}|+\sqrt[3]{8}-(-3)×\sqrt{9}$.

11 - \sqrt{3}

Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends?

126/29

A point in space $(x,y,z)$ is randomly selected so that $-1\le x \le 1$,$-1\le y \le 1$,$-1\le z \le 1$. What is the probability that $x^2+y^2+z^2\le 1$?

\frac{\pi}{6}

$A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Find the total distance around the track in yards. A) 600 B) 720 C) 840 D) 960 E) 1080

720

Given the angle $\frac {19\pi}{5}$, express it in the form of $2k\pi+\alpha$ ($k\in\mathbb{Z}$), then determine the angle $\alpha$ that makes $|\alpha|$ the smallest.

-\frac {\pi}{5}

In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. Determine the maximum slope of the line $OQ$.

\sqrt{2}

Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p + q + r + s = 10$ and \[ pq + pr + ps + qr + qs + rs = 20. \] Find the largest possible value of $s$.

\frac{5 + \sqrt{105}}{2}

Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression $1, 3, 5, \ldots$ or the arithmetic progression $1, 4, 7, \ldots$.

674

Compute the number of sets $S$ such that every element of $S$ is a nonnegative integer less than 16, and if $x \in S$ then $(2 x \bmod 16) \in S$.

678

For any nonempty $S$ we must have $0 \in S$. Now if we draw a directed graph of dependencies among the non-zero elements, it creates a balanced binary tree where every leaf has depth 3 . In the diagram, if $a$ is a parent of $b$ it means that if $b \in S$, then $a$ must also be in $S$. We wish to find the number of subsets of nodes such that every node in the set also has its parent in the set. We do this with recursion. Let $f(n)$ denote the number of such sets on a balanced binary tree of depth $n$. If the root vertex is not in the set, then the set must be empty. Otherwise, we can consider each subtree separately. This gives the recurrence $f(n)=f(n-1)^{2}+1$. We know $f(0)=2$, so we can calculate $f(1)=5, f(2)=26, f(3)=677$. We add 1 at the end for the empty set. Hence our answer is $f(3)+1=678$.

Determine the total number of different selections possible for five donuts when choosing from four types of donuts (glazed, chocolate, powdered, and jelly), with the additional constraint of purchasing at least one jelly donut.

35

The medians $ A M $ and $ B E $ of triangle $ A B C $ intersect at point $ O $. Points $ O, M, E, C $ lie on the same circle. Find $ A B $ if $ B E = A M = 3 $.

2\sqrt{3}

Given that the power function $y=x^{m}$ is an even function and is a decreasing function when $x \in (0,+\infty)$, determine the possible value of the real number $m$.

-2

Let $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with chords $\overline{AC}$ and $\overline{BC}$. The largest possible value of $d$ can be written in the form $r-s\sqrt{t}$, where $r, s$ and $t$ are positive integers and $t$ is not divisible by the square of any prime. Find $r+s+t$.

14

By Pythagoras in $\triangle BMN,$ we get $BN=\dfrac{4}{5}.$ Since cross ratios are preserved upon projecting, note that $(M,Y;X,N)\stackrel{C}{=}(M,B;A,N).$ By definition of a cross ratio, this becomes \[\dfrac{XM}{NY}:\dfrac{NM}{NY}=\dfrac{AM}{AB}:\dfrac{MN}{NB}.\] Let $MY=a,YX=b,XN=c$ such that $a+b+c=1.$ We know that $XM=a+b,XY=b,NM=1,NY=b+c,$ so the LHS becomes $\dfrac{(a+b)(b+c)}{b}.$ In the RHS, we are given every value except for $AB.$ However, Ptolemy's Theorem on $MBAN$ gives $AB\cdot MN+AN\cdot BM=AM\cdot BN\implies AB+\dfrac{3}{5\sqrt{2}}=\dfrac{4}{5\sqrt{2}}\implies AB=\dfrac{1}{5\sqrt{2}}.$ Substituting, we get $\dfrac{(a+b)(b+c)}{b}=4\implies b(a+b+c)+ac=4b, b=\dfrac{ac}{3}$ where we use $a+b+c=1.$ Again using $a+b+c=1,$ we have $a+b+c=1\implies a+\dfrac{ac}{3}+c=1\implies a=3\dfrac{1-c}{c+3}.$ Then $b=\dfrac{ac}{3}=\dfrac{c-c^2}{c+3}.$ Since this is a function in $c,$ we differentiate WRT $c$ to find its maximum. By quotient rule, it suffices to solve \[(-2c+1)(c+3)-(c-c^2)=0 \implies c^2+6c-3,c=-3+2\sqrt{3}.\] Substituting back yields $b=7-4\sqrt{3},$ so $7+4+3=\boxed{014}$ is the answer. ~Generic_Username

$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$ . A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$ . Find the area of the square.

36

A regular octahedron has a side length of 1. What is the distance between two opposite faces?

\sqrt{6} / 3

Imagine orienting the octahedron so that the two opposite faces are horizontal. Project onto a horizontal plane; these two faces are congruent equilateral triangles which (when projected) have the same center and opposite orientations. Hence, the vertices of the octahedron project to the vertices of a regular hexagon $A B C D E F$. Let $O$ be the center of the hexagon and $M$ the midpoint of $A C$. Now $A B M$ is a 30-60-90 triangle, so $A B=A M /(\sqrt{3} / 2)=(1 / 2) /(\sqrt{3} / 2)=\sqrt{3} / 3$. If we let $d$ denote the desired vertical distance between the opposite faces (which project to $A C E$ and $B D F)$, then by the Pythagorean Theorem, $A B^{2}+d^{2}=1^{2}$, so $d=\sqrt{1-A B^{2}}=\sqrt{6} / 3$.

Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $4$ per gallon. How many miles can Margie drive on $20$ worth of gas?

160

1. **Calculate the number of gallons Margie can buy with $\textdollar 20$:** Given that the cost of one gallon of gas is $\textdollar 4$, the number of gallons Margie can buy with $\textdollar 20$ is calculated by dividing the total amount of money by the cost per gallon: \[ \text{Number of gallons} = \frac{\textdollar 20}{\textdollar 4 \text{ per gallon}} = 5 \text{ gallons} \] 2. **Calculate the total miles Margie can drive with 5 gallons of gas:** Since Margie's car can travel 32 miles per gallon, the total distance she can travel with 5 gallons is: \[ \text{Total miles} = 32 \text{ miles per gallon} \times 5 \text{ gallons} = 160 \text{ miles} \] 3. **Conclusion:** Margie can drive a total of 160 miles with $\textdollar 20$ worth of gas. Thus, the correct answer is $\boxed{\textbf{(C)}~160}$.

Given a woman was x years old in the year $x^2$, determine her birth year.

1980

Let $x_1,$ $x_2,$ $x_3,$ $x_4,$ $x_5$ be the roots of the polynomial $f(x) = x^5 + x^2 + 1,$ and let $g(x) = x^2 - 2.$ Find \[g(x_1) g(x_2) g(x_3) g(x_4) g(x_5).\]

-23

Which triplet of numbers has a sum NOT equal to 1?

1.1 + (-2.1) + 1.0

To find which triplet of numbers has a sum NOT equal to 1, we will add the numbers in each triplet and check the result. 1. **Triplet (A):** $(1/2, 1/3, 1/6)$ \[ \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1 \] 2. **Triplet (B):** $(2, -2, 1)$ \[ 2 + (-2) + 1 = 0 + 1 = 1 \] 3. **Triplet (C):** $(0.1, 0.3, 0.6)$ \[ 0.1 + 0.3 + 0.6 = 1.0 \] 4. **Triplet (D):** $(1.1, -2.1, 1.0)$ \[ 1.1 + (-2.1) + 1.0 = -1.0 + 1.0 = 0 \] 5. **Triplet (E):** $(-3/2, -5/2, 5)$ \[ -\frac{3}{2} - \frac{5}{2} + 5 = -\frac{8}{2} + 5 = -4 + 5 = 1 \] From the calculations above, we see that all triplets except for **Triplet (D)** sum to 1. Therefore, the triplet that does not sum to 1 is: \[ \boxed{(D)} \]

In the drawing, 5 lines intersect at a single point. One of the resulting angles is $34^\circ$. What is the sum of the four angles shaded in gray, in degrees?

146

Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$

37

Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac {p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p$ + $q$ + $r$. Diagram [asy] size(120); pathpen = linewidth(0.7); pointpen = black; pen f = fontsize(10); pair B=(0,0), A=(5,0), C=(0,13), E=(-5,0), O = incenter(E,C,A), D=IP(A -- A+3*(O-A),E--C); D(A--B--C--cycle); D(A--D--C); D(D--E--B, linetype("4 4")); MP("5", (A+B)/2, f); MP("13", (A+C)/2, NE,f); MP("A",D(A),f); MP("B",D(B),f); MP("C",D(C),N,f); MP("A'",D(E),f); MP("D",D(D),NW,f); D(rightanglemark(C,B,A,20)); D(anglemark(D,A,E,35));D(anglemark(C,A,D,30)); [/asy] -asjpz

96

Let $O$ be the intersection of $BC$ and $AD$. By the Angle Bisector Theorem, $\frac {5}{BO}$ = $\frac {13}{CO}$, so $BO$ = $5x$ and $CO$ = $13x$, and $BO$ + $OC$ = $BC$ = $12$, so $x$ = $\frac {2}{3}$, and $OC$ = $\frac {26}{3}$. Let $P$ be the foot of the altitude from $D$ to $OC$. It can be seen that triangle $DOP$ is similar to triangle $AOB$, and triangle $DPC$ is similar to triangle $ABC$. If $DP$ = $15y$, then $CP$ = $36y$, $OP$ = $10y$, and $OD$ = $5y\sqrt {13}$. Since $OP$ + $CP$ = $46y$ = $\frac {26}{3}$, $y$ = $\frac {13}{69}$, and $AD$ = $\frac {60\sqrt{13}}{23}$ (by the pythagorean theorem on triangle $ABO$ we sum $AO$ and $OD$). The answer is $60$ + $13$ + $23$ = $\boxed{096}$.

For his birthday, Piglet baked a big cake weighing 10 kg and invited 100 guests. Among them was Winnie-the-Pooh, who has a weakness for sweets. The birthday celebrant announced the cake-cutting rule: the first guest cuts themselves a piece of cake equal to $1\%$ of the remaining cake, the second guest cuts themselves a piece of cake equal to $2\%$ of the remaining cake, the third guest cuts themselves a piece of cake equal to $3\%$ of the remaining cake, and so on. Which position in the queue should Winnie-the-Pooh take to get the largest piece of cake?

10

Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$ .

831

A trapezoid $ABCD$ has bases $AD$ and $BC$. If $BC = 60$ units, and altitudes from $B$ and $C$ to line $AD$ divide it into segments of lengths $AP = 20$ units and $DQ = 19$ units, with the length of the altitude itself being $30$ units, what is the perimeter of trapezoid $ABCD$? **A)** $\sqrt{1300} + 159$ **B)** $\sqrt{1261} + 159$ **C)** $\sqrt{1300} + \sqrt{1261} + 159$ **D)** $259$ **E)** $\sqrt{1300} + 60 + \sqrt{1261}$

\sqrt{1300} + \sqrt{1261} + 159

Given $x$, $y$, and $a \in R^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Then, calculate the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$.

6\sqrt{3}

Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person gets more heads than the first person.

1/2

Determine the numerical value of $k$ such that \[\frac{12}{x + z} = \frac{k}{z - y} = \frac{5}{y - x}.\]

17

The Gauss family has three boys aged $7,$ a girl aged $14,$ and a boy aged $15.$ What is the mean (average) of the ages of the children?

10

In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$.

145

A five-digit number $abcde$ satisfies: \[ a < b, \, b > c > d, \, d < e, \, \text{and} \, a > d, \, b > e. \] For example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the five-digit number is said to follow the "sine rule." Find the total number of five-digit numbers that follow the sine rule. Note: Please disregard any references or examples provided within the original problem if they are not part of the actual problem statement.

2892

Two students are having a pie eating contest. The first student eats $\frac{6}{7}$ of one pie. The second student eats $\frac{3}{4}$ of one pie. How much more pie did the first student finish than the second student? Express your answer as a fraction of one pie, reduced to simplest form.

\frac{3}{28}

In triangle $XYZ$, the sides are in the ratio $3:4:5$. If segment $XM$ bisects the largest angle at $X$ and divides side $YZ$ into two segments, find the length of the shorter segment given that the length of side $YZ$ is $12$ inches.

\frac{9}{2}

The scientific notation for 0.000048 is $4.8\times 10^{-5}$.

4.8 \times 10^{-5}

The value of $ a $ is chosen such that the number of roots of the first equation $ 4^{x} - 4^{-x} = 2 \cos a x $ is 2007. How many roots does the second equation $ 4^{x} + 4^{-x} = 2 \cos a x + 4 $ have for the same $ a $?

4014

How many integers between 100 and 300 have both 11 and 8 as factors?

2

Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$?

1009

Given the set $\{1,2,3,5,8,13,21,34,55\}$, calculate the number of integers between 3 and 89 that cannot be written as the sum of exactly two elements of the set.

53

Given a right circular cone with three mutually perpendicular side edges, each with a length of $\sqrt{3}$, determine the surface area of the circumscribed sphere.

9\pi

Max bought a new dirt bike and paid $10\%$ of the cost upfront, which was $\$150$. What was the price of the bike?

\$ 1500

How many zeroes does $10!$ end with, when $10!$ is written in base 9?

2

Given $\tan \theta =2$, find $\cos 2\theta =$____ and $\tan (\theta -\frac{π}{4})=$____.

\frac{1}{3}

In a trapezoid $ABCD$ with bases $AD=12$ and $BC=8$, circles constructed on the sides $AB$, $BC$, and $CD$ as diameters intersect at one point. The length of diagonal $AC$ is 12. Find the length of $BD$.

16

Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions.

\dfrac{19}{36}

Given that $40\%$ of the birds were pigeons, $20\%$ were sparrows, $15\%$ were crows, and the remaining were parakeets, calculate the percent of the birds that were not sparrows and were crows.

18.75\%

What is the greatest product obtainable from two integers whose sum is 246?

15129

Find the range of the function $f(x) = \arcsin x + \arccos x + \arctan x.$ All functions are in radians.

\left[ \frac{\pi}{4}, \frac{3 \pi}{4} \right]

Given a regular tetrahedron A-BCD with an edge length of 1, and $\overrightarrow{AE} = 2\overrightarrow{EB}$, $\overrightarrow{AF} = 2\overrightarrow{FD}$, calculate $\overrightarrow{EF} \cdot \overrightarrow{DC}$.

-\frac{1}{3}

Consider the sequence $\left\{a_{n}\right\}$ defined by $a_{0}=\frac{1}{2}$ and $a_{n+1}=a_{n}+\frac{a_{n}^{2}}{2023}$ for $n = 0, 1, \ldots$. Find the integer $k$ such that $a_{k} < 1 < a_{k+1}$.

2023

Five positive integers from $1$ to $15$ are chosen without replacement. What is the probability that their sum is divisible by $3$ ?

1/3