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Given a rhombus \(ABCD\) with diagonals equal to 3 cm and 4 cm. From the vertex of the obtuse angle \(B\), draw the altitudes \(BE\) and \(BF\). Calculate the area of the quadrilateral \(BFDE\).
|
4.32
| |
Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
|
k = \left\lfloor \sqrt{n - 1}\right\rfloor
|
Let \( n \geq 2 \) be an integer, and consider an \( n \times n \) chessboard. We place \( n \) rooks on this board such that each row and each column contains exactly one rook. This is defined as a peaceful configuration of rooks. The objective is to find the greatest positive integer \( k \) such that, in every possible peaceful configuration of \( n \) rooks, there exists a \( k \times k \) sub-square on the chessboard that is completely empty of any rooks.
### Step-by-step Solution
1. **Understanding the Problem:**
- In a peaceful configuration, since there is exactly one rook per row and one per column, it ensures all \( n \) rooks are placed in unique row-column intersections across the \( n \times n \) board.
2. **Identifying Empty Squares:**
- We need to ensure every configuration allows for a square sub-board of size \( k \times k \) which is void of rooks.
3. **Calculation of Maximum \( k \):**
- If we realize a peaceful configuration where rooks are distributed such that they occupy maximum area of the available board, each row and column combination will optimally cover the board minimally.
- The goal is maximizing \( k \), ensuring the largest empty \( k \times k \) square still forms on any part of the board in spite of any rook configuration.
4. **Using Combinatorial and Geometric Argument:**
- Let’s consider placing \( n-1 \) rooks. In this optimal configuration, potentially every position leading to \( n-1 \) coverages leaves a square potentially of size up to
\(\sqrt{n - 1} \times \sqrt{n - 1}\) that is free.
- For all \( n \) positions to be filled, this sub-square will obviously be smaller in the maximal empty form.
5. **Conclusion:**
- Upon deriving these options and observance that the largest \( k \times k \) square exists, due to \(\lceil\frac{n}{k}\rceil\) fraction of remaining free subset, we form:
- The greatest \( k \) ensuring a \( k \times k \) rupe-free square is presented by the integer part:
\[
k = \left\lfloor \sqrt{n - 1} \right\rfloor.
\]
Thus, the greatest positive integer \( k \) such that for any peaceful configuration, there exists a \( k \times k \) sub-square devoid of rooks, is:
\[
\boxed{\left\lfloor \sqrt{n - 1} \right\rfloor}.
\]
|
Ted is solving the equation by completing the square: $$64x^2+48x-36 = 0.$$ He aims to write the equation in a form: $$(ax + b)^2 = c,$$ with \(a\), \(b\), and \(c\) as integers and \(a > 0\). Determine the value of \(a + b + c\).
|
56
| |
Let \( m \in \mathbf{N}^{*} \), and let \( F(m) \) represent the integer part of \( \log_{2} m \). Determine the value of \( F(1) + F(2) + \cdots + F(1024) \).
|
8204
| |
The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a$.
[asy]
size(140);
Label f;
f.p=fontsize(4);
xaxis(-3,3,Ticks(f, 1.0));
yaxis(-4,4,Ticks(f, 1.0));
real f(real x)
{
return -2x^2+4x+1;
}
draw(graph(f,-.7,2.7),linewidth(1),Arrows(6));
[/asy]
|
-2
| |
If an integer \( n \) that is greater than 8 is a solution to the equation \( x^2 - ax + b = 0 \), and the representation of \( a \) in base \( n \) is 18, then the representation of \( b \) in base \( n \) is:
|
80
| |
In the diagram, the square has a perimeter of $48$ and the triangle has a height of $48.$ If the square and the triangle have the same area, what is the value of $x?$ [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((3,0)--(6,0)--(6,5)--cycle);
draw((5.8,0)--(5.8,.2)--(6,.2));
label("$x$",(4.5,0),S);
label("48",(6,2.5),E);
[/asy]
|
6
| |
A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\frac{\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Find the particle's position after $150$ moves.
|
(-5 \sqrt{2}, 5 + 5 \sqrt{2})
| |
What is the sum of the greatest common divisor of $50$ and $5005$ and the least common multiple of $50$ and $5005$?
|
50055
| |
Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 98.$ Find $\frac n{100}.$
|
196
|
Define $x_i = 2y_i - 1$. Then $2\left(\sum_{i = 1}^4 y_i\right) - 4 = 98$, so $\sum_{i = 1}^4 y_i = 51$.
So we want to find four natural numbers that sum up to 51; we can imagine this as trying to split up 51 on the number line into 4 ranges. This is equivalent to trying to place 3 markers on the numbers 1 through 50; thus the answer is $n = {50\choose3} = \frac{50 * 49 * 48}{3 * 2} = 19600$, and $\frac n{100} = \boxed{196}$.
|
For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$.
|
504
| |
How many positive integers \( n \) are there such that \( n \) is a multiple of 4, and the least common multiple of \( 4! \) and \( n \) equals 4 times the greatest common divisor of \( 8! \) and \( n \)?
|
12
| |
What is the greatest possible value of $x$ for the equation $$\left(\frac{4x-16}{3x-4}\right)^2+\left(\frac{4x-16}{3x-4}\right)=12?$$
|
2
| |
Let \(n = 2^{20}3^{25}\). How many positive integer divisors of \(n^2\) are less than \(n\) but do not divide \(n\)?
|
499
| |
Compute
\[ e^{2 \pi i/17} + e^{4 \pi i/17} + e^{6 \pi i/17} + \dots + e^{32 \pi i/17}. \]
|
-1
| |
A pen and its ink refill together cost $\;\$1.10$. The pen costs $\;\$1$ more than the ink refill. What is the cost of the pen in dollars?
|
1.05
| |
Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?
|
\frac{1}{2}
|
1. **Calculate the initial lap time**:
When Elisa started swimming, she completed 10 laps in 25 minutes. To find the time it took for one lap, we divide the total time by the number of laps:
\[
\text{Initial lap time} = \frac{25 \text{ minutes}}{10 \text{ laps}} = 2.5 \text{ minutes per lap}
\]
2. **Calculate the current lap time**:
Now, Elisa can finish 12 laps in 24 minutes. Similarly, we find the time for one lap by dividing the total time by the number of laps:
\[
\text{Current lap time} = \frac{24 \text{ minutes}}{12 \text{ laps}} = 2 \text{ minutes per lap}
\]
3. **Determine the improvement in lap time**:
To find out by how many minutes she has improved her lap time, we subtract the current lap time from the initial lap time:
\[
\text{Improvement} = 2.5 \text{ minutes per lap} - 2 \text{ minutes per lap} = 0.5 \text{ minutes per lap}
\]
4. **Conclusion**:
Elisa has improved her lap time by 0.5 minutes per lap, which can also be expressed as $\frac{1}{2}$ minute per lap.
\[
\boxed{\textbf{(A)}\ \frac{1}{2}}
\]
|
In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$ .
|
23
| |
Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$ .
*Proposed by Michael Tang*
|
423
| |
How many distinct arrangements of the letters in the word "basic'' are there?
|
120
| |
Let $n^{}_{}$ be the smallest positive integer that is a multiple of $75_{}^{}$ and has exactly $75_{}^{}$ positive integral divisors, including $1_{}^{}$ and itself. Find $\frac{n}{75}$.
|
432
|
The prime factorization of $75 = 3^15^2 = (2+1)(4+1)(4+1)$. For $n$ to have exactly $75$ integral divisors, we need to have $n = p_1^{e_1-1}p_2^{e_2-1}\cdots$ such that $e_1e_2 \cdots = 75$. Since $75|n$, two of the prime factors must be $3$ and $5$. To minimize $n$, we can introduce a third prime factor, $2$. Also to minimize $n$, we want $5$, the greatest of all the factors, to be raised to the least power. Therefore, $n = 2^43^45^2$ and $\frac{n}{75} = \frac{2^43^45^2}{3 \cdot 5^2} = 16 \cdot 27 = \boxed{432}$.
|
Compute $\begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}^6.$
|
\begin{pmatrix} -64 & 0 \\ 0 & -64 \end{pmatrix}
| |
A circle with equation $x^{2}+y^{2}=1$ passes through point $P(1, \sqrt {3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$.
|
\sqrt {3}
| |
Let \(b = 8\) and \(S_n\) be the sum of the reciprocals of the non-zero digits of the integers from \(1\) to \(8^n\) inclusive. Find the smallest positive integer \(n\) for which \(S_n\) is an integer.
|
105
| |
In tetrahedron $ABCD,$
\[\angle ADB = \angle ADC = \angle BDC = 90^\circ.\]Also, $x = \sin \angle CAD$ and $y = \sin \angle CBD.$ Express $\cos \angle ACB$ in terms of $x$ and $y.$
|
xy
| |
In the diagram, $P$ is on $RS$ so that $QP$ bisects $\angle SQR$. Also, $PQ=PR$, $\angle RSQ=2y^\circ$, and $\angle RPQ=3y^\circ$. What is the measure, in degrees, of $\angle RPQ$? [asy]
// C14
import olympiad;
size(7cm);
real x = 50; real y = 20;
pair q = (1, 0);
pair r = (0, 0);
pair p = intersectionpoints((10 * dir(x))--r, q--(shift(q) * 10 * dir(180 - x)))[0];
pair s = intersectionpoints(r--(r + 10 * (p - r)), 10 * dir(180 - 2 * x)--q)[0];
// Draw lines
draw(p--s--q--p--r--q);
// Label points
label("$R$", r, SW);
label("$Q$", q, SE);
label("$S$", s, N);
label("$P$", p, NW);
// Label angles
label("$x^\circ$", q, 2 * W + 2 * NW);
label("$x^\circ$", q, 4 * N + 2 * NW);
label("$2y^\circ$", s, 5 * S + 4 * SW);
label("$3y^\circ$", p, 4 * S);
// Tick marks
add(pathticks(r--p, 2, spacing=0.6, s=2));
add(pathticks(p--q, 2, spacing=0.6, s=2));
[/asy]
|
108
| |
Find all pairs of integer solutions $(n, m)$ to $2^{3^{n}}=3^{2^{m}}-1$.
|
(0,0) \text{ and } (1,1)
|
We find all solutions of $2^{x}=3^{y}-1$ for positive integers $x$ and $y$. If $x=1$, we obtain the solution $x=1, y=1$, which corresponds to $(n, m)=(0,0)$ in the original problem. If $x>1$, consider the equation modulo 4. The left hand side is 0, and the right hand side is $(-1)^{y}-1$, so $y$ is even. Thus we can write $y=2 z$ for some positive integer $z$, and so $2^{x}=(3^{z}-1)(3^{z}+1)$. Thus each of $3^{z}-1$ and $3^{z}+1$ is a power of 2, but they differ by 2, so they must equal 2 and 4 respectively. Therefore, the only other solution is $x=3$ and $y=2$, which corresponds to $(n, m)=(1,1)$ in the original problem.
|
Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, he caught no more fish than the previous day. In total, he caught exactly 100 fish over the five days. What is the minimum total number of fish he could have caught on Monday, Wednesday, and Friday?
|
50
| |
In the regular quadrangular pyramid \(P-ABCD\), \(M\) and \(N\) are the midpoints of \(PA\) and \(PB\) respectively. If the tangent of the dihedral angle between a side face and the base is \(\sqrt{2}\), find the cosine of the angle between skew lines \(DM\) and \(AN\).
|
1/6
| |
\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\]
|
-1
| |
Find the maximum value of
\[
\frac{\sin^4 x + \cos^4 x + 2}{\sin^6 x + \cos^6 x + 2}
\]
over all real values \(x\).
|
\frac{10}{9}
| |
Given $\sin\left( \frac{\pi}{3} + a \right) = \frac{5}{13}$, and $a \in \left( \frac{\pi}{6}, \frac{2\pi}{3} \right)$, find the value of $\sin\left( \frac{\pi}{12} + a \right)$.
|
\frac{17\sqrt{2}}{26}
| |
In right triangle $MNO$, $\tan{M}=\frac{5}{4}$, $OM=8$, and $\angle O = 90^\circ$. Find $MN$. Express your answer in simplest radical form.
|
2\sqrt{41}
| |
Given $f(x) = 4\cos x\sin \left(x+ \frac{\pi}{6}\right)-1$.
(Ⅰ) Determine the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\left[- \frac{\pi}{6}, \frac{\pi}{4}\right]$.
|
-1
| |
Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?
|
5
|
To find the minimum number of packs needed to buy exactly 90 cans of soda, we consider the sizes of the packs available: 6, 12, and 24 cans. We aim to use the largest packs first to minimize the total number of packs.
1. **Using the largest pack (24 cans)**:
- Calculate how many 24-can packs can be used without exceeding 90 cans.
- Since $24 \times 4 = 96$ exceeds 90, we can use at most 3 packs of 24 cans.
- Total cans covered by three 24-packs: $3 \times 24 = 72$ cans.
- Remaining cans needed: $90 - 72 = 18$ cans.
2. **Using the next largest pack (12 cans)**:
- Calculate how many 12-can packs can be used to cover some or all of the remaining 18 cans.
- Since $12 \times 2 = 24$ exceeds 18, we can use at most 1 pack of 12 cans.
- Total cans covered by one 12-pack: $12$ cans.
- Remaining cans needed after using one 12-pack: $18 - 12 = 6$ cans.
3. **Using the smallest pack (6 cans)**:
- Calculate how many 6-can packs are needed to cover the remaining 6 cans.
- Since $6 \times 1 = 6$ exactly covers the remaining cans, we use 1 pack of 6 cans.
4. **Total packs used**:
- Number of 24-can packs used: 3
- Number of 12-can packs used: 1
- Number of 6-can packs used: 1
- Total number of packs: $3 + 1 + 1 = 5$
Thus, the minimum number of packs needed to buy exactly 90 cans of soda is $\boxed{\textbf{(B)}\ 5}$.
|
In how many distinct ways can I arrange my six keys on a keychain, if my house key must be exactly opposite my car key and my office key should be adjacent to my house key? For arrangement purposes, two placements are identical if one can be obtained from the other through rotation or flipping the keychain.
|
12
| |
A cylindrical barrel with a radius of 5 feet and a height of 15 feet is full of water. A solid cube with side length 7 feet is set into the barrel so that one edge of the cube is vertical. Calculate the square of the volume of water displaced, $v^2$, when the cube is fully submerged.
|
117649
| |
Arrange the 7 numbers $39, 41, 44, 45, 47, 52, 55$ in a sequence such that the sum of any three consecutive numbers is a multiple of 3. What is the maximum value of the fourth number in all such arrangements?
|
47
| |
Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$ . Compute the area of region $R$ . Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$ .
|
4 - 2\sqrt{2}
| |
Find an $n$ such that $n!-(n-1)!+(n-2)!-(n-3)!+\cdots \pm 1$ ! is prime. Be prepared to justify your answer for $\left\{\begin{array}{c}n, \\ {\left[\frac{n+225}{10}\right],}\end{array} n \leq 25\right.$ points, where $[N]$ is the greatest integer less than $N$.
|
3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160
|
$3,4,5,6,7,8,10,15,19,41$ (26 points), 59, 61 (28 points), 105 (33 points), 160 (38 points) are the only ones less than or equal to 335. If anyone produces an answer larger than 335, then we ask for justification to call their bluff. It is not known whether or not there are infinitely many such $n$.
|
In my office, there are two digital 24-hour clocks. One clock gains one minute every hour and the other loses two minutes every hour. Yesterday, I set both of them to the same time, but when I looked at them today, I saw that the time shown on one was 11:00 and the time on the other was 12:00. What time was it when I set the two clocks?
A) 23:00
B) 19:40
C) 15:40
D) 14:00
E) 11:20
|
15:40
| |
How many digits does the number \(2^{100}\) have? What are its last three digits? (Give the answers without calculating the power directly or using logarithms!) If necessary, how could the power be quickly calculated?
|
376
| |
Let $A$ be a point on the parabola $y = x^2 - 9x + 25,$ and let $B$ be a point on the line $y = x - 8.$ Find the shortest possible distance $AB.$
|
4 \sqrt{2}
| |
Let \( x \) be a real number such that \( x + \frac{1}{x} = 5 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_6 \).
|
12098
| |
Xiaoting's average score for five math tests is 85, the median is 86, and the mode is 88. What is the sum of the scores of the two lowest tests?
|
163
| |
Solve the inequality
\[\dfrac{x+1}{x+2}>\dfrac{3x+4}{2x+9}.\]
|
\left( -\frac{9}{2} , -2 \right) \cup \left( \frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2} \right)
| |
Let \( N_{0} \) be the set of non-negative integers, and \( f: N_{0} \rightarrow N_{0} \) be a function such that \( f(0)=0 \) and for any \( n \in N_{0} \), \( [f(2n+1)]^{2} - [f(2n)]^{2} = 6f(n) + 1 \) and \( f(2n) > f(n) \). Determine how many elements in \( f(N_{0}) \) are less than 2004.
|
128
| |
The sum of three numbers \(x, y,\) and \(z\) is 120. If we decrease \(x\) by 10, we get the value \(M\). If we increase \(y\) by 10, we also get the value \(M\). If we multiply \(z\) by 10, we also get the value \(M\). What is the value of \(M\)?
|
\frac{400}{7}
| |
For each pair of distinct natural numbers \(a\) and \(b\), not exceeding 20, Petya drew the line \( y = ax + b \) on the board. That is, he drew the lines \( y = x + 2, y = x + 3, \ldots, y = x + 20, y = 2x + 1, y = 2x + 3, \ldots, y = 2x + 20, \ldots, y = 3x + 1, y = 3x + 2, y = 3x + 4, \ldots, y = 3x + 20, \ldots, y = 20x + 1, \ldots, y = 20x + 19 \). Vasia drew a circle of radius 1 with center at the origin on the same board. How many of Petya’s lines intersect Vasia’s circle?
|
190
| |
Given in the Cartesian coordinate system $xOy$, a line $l$ passing through a fixed point $P$ with an inclination angle of $\alpha$ has the parametric equation: $$\begin{cases} x=t\cos\alpha \\ y=-2+t\sin\alpha \end{cases}$$ (where $t$ is the parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinates of the center of the circle are $(3, \frac{\pi}{2})$, and the circle $C$ with a radius of 3 intersects the line $l$ at points $A$ and $B$. Then, $|PA|\cdot|PB|=$ \_\_\_\_\_.
|
16
| |
Inside the square \(ABCD\) with side length 5, there is a point \(X\). The areas of triangles \(AXB\), \(BXC\), and \(CXD\) are in the ratio \(1:5:9\). Find the sum of the squares of the distances from point \(X\) to the sides of the square.
|
33
| |
A racer departs from point \( A \) along the highway, maintaining a constant speed of \( a \) km/h. After 30 minutes, a second racer starts from the same point with a constant speed of \( 1.25a \) km/h. How many minutes after the start of the first racer was a third racer sent from the same point, given that the third racer developed a speed of \( 1.5a \) km/h and simultaneously with the second racer caught up with the first racer?
|
50
| |
A point $Q$ is randomly placed in the interior of the right triangle $XYZ$ with $XY = 10$ units and $XZ = 6$ units. What is the probability that the area of triangle $QYZ$ is less than one-third of the area of triangle $XYZ$?
|
\frac{1}{3}
| |
There are 19 candy boxes arranged in a row, with the middle box containing $a$ candies. Moving to the right, each box contains $m$ more candies than the previous one; moving to the left, each box contains $n$ more candies than the previous one ($a$, $m$, and $n$ are all positive integers). If the total number of candies is 2010, then the sum of all possible values of $a$ is.
|
105
| |
Solve for $y$: $3y+7y = 282-8(y-3)$.
|
17
| |
The side length of square \(ABCD\) is 4. Point \(E\) is the midpoint of \(AB\), and point \(F\) is a moving point on side \(BC\). Triangles \(\triangle ADE\) and \(\triangle DCF\) are folded up along \(DE\) and \(DF\) respectively, making points \(A\) and \(C\) coincide at point \(A'\). Find the maximum distance from point \(A'\) to plane \(DEF\).
|
\frac{4\sqrt{5}}{5}
| |
Given that \( f(x) \) is an odd function defined on \( \mathbf{R} \), and for any \( x \in \mathbf{R} \), the following holds:
$$
f(2+x) + f(2-x) = 0.
$$
When \( x \in [-1, 0) \), it is given that
$$
f(x) = \log_{2}(1-x).
$$
Find \( f(1) + f(2) + \cdots + f(2021) \).
|
-1
| |
The pages of a book are numbered $1_{}^{}$ through $n_{}^{}$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of $1986_{}^{}$. What was the number of the page that was added twice?
|
33
|
Denote the page number as $x$, with $x < n$. The sum formula shows that $\frac{n(n + 1)}{2} + x = 1986$. Since $x$ cannot be very large, disregard it for now and solve $\frac{n(n+1)}{2} = 1986$. The positive root for $n \approx \sqrt{3972} \approx 63$. Quickly testing, we find that $63$ is too large, but if we plug in $62$ we find that our answer is $\frac{62(63)}{2} + x = 1986 \Longrightarrow x = \boxed{033}$.
|
Find the simplest method to solve the system of equations using substitution $$\begin{cases} x=2y\textcircled{1} \\ 2x-y=5\textcircled{2} \end{cases}$$
|
y = \frac{5}{3}
| |
How many pairs of integers $(a, b)$, with $1 \leq a \leq b \leq 60$, have the property that $b$ is divisible by $a$ and $b+1$ is divisible by $a+1$?
|
106
|
The divisibility condition is equivalent to $b-a$ being divisible by both $a$ and $a+1$, or, equivalently (since these are relatively prime), by $a(a+1)$. Any $b$ satisfying the condition is automatically $\geq a$, so it suffices to count the number of values $b-a \in$ $\{1-a, 2-a, \ldots, 60-a\}$ that are divisible by $a(a+1)$ and sum over all $a$. The number of such values will be precisely $60 /[a(a+1)]$ whenever this quantity is an integer, which fortunately happens for every $a \leq 5$; we count: $a=1$ gives 30 values of $b ;$ $a=2$ gives 10 values of $b ;$ $a=3$ gives 5 values of $b$; $a=4$ gives 3 values of $b$; $a=5$ gives 2 values of $b$; $a=6$ gives 2 values ($b=6$ or 48); any $a \geq 7$ gives only one value, namely $b=a$, since $b>a$ implies $b \geq a+a(a+1)>60$. Adding these up, we get a total of 106 pairs.
|
All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown. Continuing the pattern, in what column will the integer 800 be written?
[asy]
label("A",(0,0),N);
label("B",(10,0),N);
label("C",(20,0),N);
label("D",(30,0),N);
label("E",(40,0),N);
label("Row 1",(-10,-7),W);
label("2",(10,-12),N);
label("3",(20,-12),N);
label("4",(30,-12),N);
label("5",(40,-12),N);
label("Row 2",(-10,-24),W);
label("9",(0,-29),N);
label("8",(10,-29),N);
label("7",(20,-29),N);
label("6",(30,-29),N);
label("Row 3",(-10,-41),W);
label("10",(10,-46),N);
label("11",(20,-46),N);
label("12",(30,-46),N);
label("13",(40,-46),N);
label("Row 4",(-10,-58),W);
label("17",(0,-63),N);
label("16",(10,-63),N);
label("15",(20,-63),N);
label("14",(30,-63),N);
label("Row 5",(-10,-75),W);
label("18",(10,-80),N);
label("19",(20,-80),N);
label("20",(30,-80),N);
label("21",(40,-80),N);
[/asy]
|
\text{B}
| |
In the Cartesian coordinate system xOy, the equation of line l is given as x+1=0, and curve C is a parabola with the coordinate origin O as the vertex and line l as the axis. Establish a polar coordinate system with the coordinate origin O as the pole and the non-negative semi-axis of the x-axis as the polar axis.
1. Find the polar coordinate equations for line l and curve C respectively.
2. Point A is a moving point on curve C in the first quadrant, and point B is a moving point on line l in the second quadrant. If ∠AOB=$$\frac {π}{4}$$, find the maximum value of $$\frac {|OA|}{|OB|}$$.
|
\frac { \sqrt {2}}{2}
| |
Let \( D \) be a point inside \( \triangle ABC \) such that \( \angle BAD = \angle BCD \) and \( \angle BDC = 90^\circ \). If \( AB = 5 \), \( BC = 6 \), and \( M \) is the midpoint of \( AC \), find the length of \( DM \).
|
\frac{\sqrt{11}}{2}
| |
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \frac{1}{2}$?
|
10
|
1. **Identify the expression and substitute $a$:**
Given the expression $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$, we need to evaluate it at $a = \frac{1}{2}$.
2. **Calculate $a^{-1}$:**
Since $a^{-1}$ is the reciprocal of $a$, when $a = \frac{1}{2}$, we have:
\[
a^{-1} = \left(\frac{1}{2}\right)^{-1} = 2.
\]
3. **Substitute $a^{-1}$ into the expression:**
Replace $a^{-1}$ with $2$ in the expression:
\[
\frac{2 \cdot 2 + \frac{2}{2}}{\frac{1}{2}} = \frac{4 + 1}{\frac{1}{2}} = \frac{5}{\frac{1}{2}}.
\]
4. **Simplify the expression:**
To simplify $\frac{5}{\frac{1}{2}}$, we multiply by the reciprocal of the denominator:
\[
\frac{5}{\frac{1}{2}} = 5 \cdot 2 = 10.
\]
5. **Conclude with the final answer:**
Thus, the value of the expression when $a = \frac{1}{2}$ is $\boxed{\textbf{(D)}\ 10}$.
|
Given that $$α∈(0， \frac {π}{3})$$ and vectors $$a=( \sqrt {6}sinα， \sqrt {2})$$, $$b=(1，cosα- \frac { \sqrt {6}}{2})$$ are orthogonal,
(1) Find the value of $$tan(α+ \frac {π}{6})$$;
(2) Find the value of $$cos(2α+ \frac {7π}{12})$$.
|
\frac { \sqrt {2}- \sqrt {30}}{8}
| |
Let \( n \) be the positive integer such that
\[ \frac{1}{9 \sqrt{11} + 11 \sqrt{9}} + \frac{1}{11 \sqrt{13} + 13 \sqrt{11}} + \frac{1}{13 \sqrt{15} + 15 \sqrt{13}} + \cdots + \frac{1}{n \sqrt{n+2} + (n+2) \sqrt{n}} = \frac{1}{9} . \]
Find the value of \( n \).
|
79
| |
If set $A=\{x\in N\left|\right.-1 \lt x\leqslant 2\}$, $B=\{x\left|\right.x=ab,a,b\in A\}$, then the number of non-empty proper subsets of set $B$ is ______.
|
14
| |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a King?
|
\dfrac{4}{663}
| |
On a table, there are 2020 boxes. Some of them contain candies, while others are empty. The first box has a label that reads: "All boxes are empty." The second box reads: "At least 2019 boxes are empty." The third box reads: "At least 2018 boxes are empty," and so on, up to the 2020th box, which reads: "At least one box is empty." It is known that the labels on the empty boxes are false, and the labels on the boxes with candies are true. Determine how many boxes contain candies. Justify your answer.
|
1010
| |
For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots?
|
625
| |
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
|
20
|
1. **Determine individual rates**:
- Cagney's rate of frosting cupcakes is 1 cupcake per 20 seconds.
- Lacey's rate of frosting cupcakes is 1 cupcake per 30 seconds.
2. **Calculate combined rate**:
- The combined rate of frosting cupcakes when Cagney and Lacey work together can be calculated using the formula for the combined work rate of two people working together:
\[
\text{Combined rate} = \frac{1}{\text{Time taken by Cagney}} + \frac{1}{\text{Time taken by Lacey}} = \frac{1}{20} + \frac{1}{30}
\]
- To add these fractions, find a common denominator (which is 60 in this case):
\[
\frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60}
\]
- Therefore, the combined rate is:
\[
\frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \text{ cupcakes per second}
\]
3. **Convert the combined rate to a single cupcake time**:
- The time taken to frost one cupcake together is the reciprocal of the combined rate:
\[
\text{Time for one cupcake} = \frac{1}{\frac{1}{12}} = 12 \text{ seconds}
\]
4. **Calculate total cupcakes in 5 minutes**:
- Convert 5 minutes to seconds:
\[
5 \text{ minutes} = 5 \times 60 = 300 \text{ seconds}
\]
- The number of cupcakes frosted in 300 seconds is:
\[
\frac{300 \text{ seconds}}{12 \text{ seconds per cupcake}} = 25 \text{ cupcakes}
\]
5. **Conclusion**:
- Cagney and Lacey can frost 25 cupcakes in 5 minutes.
Thus, the answer is $\boxed{\textbf{(D)}\ 25}$.
|
Given that $2x + 5y = 20$ and $5x + 2y = 26$, find $20x^2 + 60xy + 50y^2$.
|
\frac{59600}{49}
| |
Evaluate the sum of $1001101_2$ and $111000_2$, and then add the decimal equivalent of $1010_2$. Write your final answer in base $10$.
|
143
| |
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$?
|
16.67\%
| |
Jim borrows $1500$ dollars from Sarah, who charges an interest rate of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Jim will owe more than twice as much as he borrowed?
|
12
| |
A phone number \( d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7} \) is called "legal" if the number \( d_{1} d_{2} d_{3} \) is equal to \( d_{4} d_{5} d_{6} \) or to \( d_{5} d_{6} d_{7} \). For example, \( 234-2347 \) is a legal phone number. Assume each \( d_{i} \) can be any digit from 0 to 9. How many legal phone numbers exist?
|
19990
| |
In a right triangle JKL, where angle J is the right angle, KL measures 20 units, and JL measures 12 units. Calculate $\tan K$.
|
\frac{4}{3}
| |
For lines $l_1: x + ay + 3 = 0$ and $l_2: (a-2)x + 3y + a = 0$ to be parallel, determine the values of $a$.
|
-1
| |
In a square array of 25 dots arranged in a 5x5 grid, what is the probability that five randomly chosen dots will be collinear? Express your answer as a common fraction.
|
\frac{2}{8855}
| |
Let $r=H_{1}$ be the answer to this problem. Given that $r$ is a nonzero real number, what is the value of $r^{4}+4 r^{3}+6 r^{2}+4 r ?$
|
-1
|
Since $H_{1}$ is the answer, we know $r^{4}+4 r^{3}+6 r^{2}+4 r=r \Rightarrow(r+1)^{4}=r+1$. Either $r+1=0$, or $(r+1)^{3}=1 \Rightarrow r=0$. Since $r$ is nonzero, $r=-1$.
|
The segments connecting the feet of the altitudes of an acute-angled triangle form a right triangle with a hypotenuse of 10. Find the radius of the circumcircle of the original triangle.
|
10
| |
Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1+b_2+ \cdots +b_{150}=150 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{300}=450.
$$What is the value of $b_2 - b_1$? Express your answer as a common fraction.
|
\frac{1}{75}
| |
Calculate the definite integral:
$$
\int_{0}^{2} e^{\sqrt{(2-x) /(2+x)}} \cdot \frac{d x}{(2+x) \sqrt{4-x^{2}}}
$$
|
\frac{e-1}{2}
| |
Let \( P \) be a point inside regular pentagon \( ABCDE \) such that \( \angle PAB = 48^\circ \) and \( \angle PDC = 42^\circ \). Find \( \angle BPC \), in degrees.
|
84
| |
Given an ellipse with the equation \\(\\dfrac{x^{2}}{a^{2}}+\\dfrac{y^{2}}{b^{2}}=1(a > b > 0)\\) and an eccentricity of \\(\\dfrac{\\sqrt{3}}{2}\\). A line $l$ is drawn through one of the foci of the ellipse, perpendicular to the $x$-axis, and intersects the ellipse at points $M$ and $N$, with $|MN|=1$. Point $P$ is located at $(-b,0)$. Point $A$ is any point on the circle $O:x^{2}+y^{2}=b^{2}$ that is different from point $P$. A line is drawn through point $P$ perpendicular to $PA$ and intersects the circle $x^{2}+y^{2}=a^{2}$ at points $B$ and $C$.
(1) Find the standard equation of the ellipse;
(2) Determine whether $|BC|^{2}+|CA|^{2}+|AB|^{2}$ is a constant value. If it is, find that value; if not, explain why.
|
26
| |
On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$.
|
810
|
Drawing the square and examining the given lengths, [asy] size(2inch, 2inch); currentpen = fontsize(8pt); pair A = (0, 0); dot(A); label("$A$", A, plain.SW); pair B = (3, 0); dot(B); label("$B$", B, plain.SE); pair C = (3, 3); dot(C); label("$C$", C, plain.NE); pair D = (0, 3); dot(D); label("$D$", D, plain.NW); pair E = (0, 1); dot(E); label("$E$", E, plain.W); pair F = (3, 2); dot(F); label("$F$", F, plain.E); label("$\frac x3$", E--A); label("$\frac x3$", F--C); label("$x$", A--B); label("$x$", C--D); label("$\frac {2x}3$", B--F); label("$\frac {2x}3$", D--E); label("$30$", B--E); label("$30$", F--E); label("$30$", F--D); draw(B--C--D--F--E--B--A--D); [/asy] you find that the three segments cut the square into three equal horizontal sections. Therefore, ($x$ being the side length), $\sqrt{x^2+(x/3)^2}=30$, or $x^2+(x/3)^2=900$. Solving for $x$, we get $x=9\sqrt{10}$, and $x^2=810.$
Area of the square is $\boxed{810}$.
|
In the center of a circular field, there is a geologist's cabin. From it extend 6 straight roads, dividing the field into 6 equal sectors. Two geologists start a journey from their cabin at a speed of 4 km/h each on a randomly chosen road. Determine the probability that the distance between them will be at least 6 km after one hour.
|
0.5
| |
A circle with a radius of 15 is tangent to two adjacent sides \( AB \) and \( AD \) of square \( ABCD \). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from vertex \( B \) to the point of tangency.
|
12
| |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\sin A= \sqrt {7}a$.
(I) Find the value of $\sin B$;
(II) If $a$, $b$, and $c$ form an arithmetic sequence with a common difference greater than $0$, find the value of $\cos A-\cos C$.
|
\frac { \sqrt {7}}{2}
| |
Let $a = \pi/2008$. Find the smallest positive integer $n$ such that \[2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]\] is an integer.
|
251
|
By the product-to-sum identities, we have that $2\cos a \sin b = \sin (a+b) - \sin (a-b)$. Therefore, this reduces to a telescoping series: \begin{align*} \sum_{k=1}^{n} 2\cos(k^2a)\sin(ka) &= \sum_{k=1}^{n} [\sin(k(k+1)a) - \sin((k-1)ka)]\\ &= -\sin(0) + \sin(2a)- \sin(2a) + \sin(6a) - \cdots - \sin((n-1)na) + \sin(n(n+1)a)\\ &= -\sin(0) + \sin(n(n+1)a) = \sin(n(n+1)a) \end{align*}
Thus, we need $\sin \left(\frac{n(n+1)\pi}{2008}\right)$ to be an integer; this can be only $\{-1,0,1\}$, which occur when $2 \cdot \frac{n(n+1)}{2008}$ is an integer. Thus $1004 = 2^2 \cdot 251 | n(n+1) \Longrightarrow 251 | n, n+1$. It easily follows that $n = \boxed{251}$ is the smallest such integer.
|
In a photograph, Aristotle, David, Flora, Munirah, and Pedro are seated in a random order in a row of 5 chairs. If David is seated in the middle of the row, what is the probability that Pedro is seated beside him?
|
\frac{1}{2}
|
After David is seated, there are 4 seats in which Pedro can be seated, of which 2 are next to David. Thus, the probability that Pedro is next to David is $ rac{2}{4}$ or $ rac{1}{2}$.
|
Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.
Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.
Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n$-th attempt or earlier, regardless of the locations of the monsters.
[i]
|
3
|
To solve this problem, we will analyze the board's structure and derive a strategy for Turbo to ensure he reaches the last row in a guaranteed number of attempts. We'll consider the distribution of monsters and Turbo's possible paths.
Given:
- The board has 2024 rows and 2023 columns.
- There is exactly one monster in each row except the first and last, totaling 2022 monsters.
- Each column contains at most one monster.
**Objective:**
Determine the minimum number \( n \) of attempts Turbo requires to guarantee reaching the last row, regardless of monster placement.
### Analysis
1. **Board Configuration:**
- In total, 2022 monsters are distributed such that each row (except the first and last) contains exactly one monster.
- Since each column has at most one monster, not all columns have a monster.
2. **Turbo's Strategy:**
- Turbo needs to explore the board in a manner that efficiently identifies safe columns and rows without encountering a monster multiple times unnecessarily.
- Turbo can determine whether a column is safe (contains no monsters) by exploring strategic positions across breadth and depth on the board.
3. **Strategy Application:**
- **First Attempt:** Turbo starts by exploring a single path down a column from the first row to the last row.
- If no monster is encountered, Turbo completes the game in the first attempt.
- If a monster is encountered, Turbo records the dangerous columns.
- **Second Attempt:** Turbo tries an adjacent column next to the previously explored path.
- In this attempt, he checks whether this path leads to a monster-free path.
- **Third Attempt:** Combining information from the first and second attempts, Turbo systematically explores remaining unchecked paths.
With a systematic exploration strategy, Turbo uses at most three different attempts because:
- **Attempt 1:** It eliminates either the path as safe or identifies monsters, removing knowledge uncertainties.
- **Attempt 2:** Validates adjacent safe paths based on new or old information.
- **Attempt 3:** Finishes off ensuring any unclear pathways are confirmed.
Considering the constraints (2024 rows but only one monster per row, and each column has at most one monster), and considering that Turbo can remember the unsafe paths and adjust his route, the minimum number of guaranteed attempts is 3:
\[
\boxed{3}
\]
This ensures that Turbo utilizes a strategic exploration pattern, minimizing redundant moves while guaranteeing reaching the destination row.
|
How many integers between $100$ and $150$ have three different digits in increasing order? One such integer is $129$.
|
18
| |
Given the sequence $\{a_k\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,
$a_k=\begin{cases}
(0.\underbrace{501\cdots 01}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd,} \\
(0.\underbrace{501\cdots 011}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is even.}
\end{cases}$
Rearrange the numbers in the sequence $\{a_k\}_{k=1}^{11}$ in decreasing order to produce a new sequence $\{b_k\}_{k=1}^{11}$. Find the sum of all integers $k$, $1\le k \le 11$, such that $a_k = b_k$.
|
30
| |
In the equation "Xiwangbei jiushi hao $\times$ 8 = Jiushihao Xiwangbei $\times$ 5", different Chinese characters represent different digits. The six-digit even number represented by "Xiwangbei jiushi hao" is ____.
|
256410
| |
A building has three different staircases, all starting at the base of the building and ending at the top. One staircase has 104 steps, another has 117 steps, and the other has 156 steps. Whenever the steps of the three staircases are at the same height, there is a floor. How many floors does the building have?
|
13
| |
If the function \( f(x) = (x^2 - 1)(x^2 + ax + b) \) satisfies \( f(x) = f(4 - x) \) for any \( x \in \mathbb{R} \), what is the minimum value of \( f(x) \)?
|
-16
| |
Find the number of ways to pave a $1 \times 10$ block with tiles of sizes $1 \times 1, 1 \times 2$ and $1 \times 4$, assuming tiles of the same size are indistinguishable. It is not necessary to use all the three kinds of tiles.
|
169
| |
A department dispatches 4 researchers to 3 schools to investigate the current status of the senior year review and preparation for exams, requiring at least one researcher to be sent to each school. Calculate the number of different distribution schemes.
|
36
| |
$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$ . What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$ ?
*2018 CCA Math Bonanza Lightning Round #1.3*
|
45
|
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