FlyLee/deepscaler-randomsampled-10k

problem stringlengths 11 4.03k	answer stringlengths 0 157	solution stringlengths 0 11.1k
A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimeter of the largest rectangle.	\frac{5}{6}
Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.	25
If a number $N, N \ne 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is	R	1. Formulate the Equation: Given that a number $N$, diminished by four times its reciprocal, equals a real constant $R$, we can write the equation as: \[ N - \frac{4}{N} = R. \] 2. Manipulate the Equation: To eliminate the fraction, multiply through by $N$ (assuming $N \neq 0$): \[ N^2 - 4 = RN. \] Rearrange this to form a quadratic equation: \[ N^2 - RN - 4 = 0. \] 3. Apply Vieta's Formulas: For a quadratic equation of the form $ax^2 + bx + c = 0$, Vieta's formulas tell us that the sum of the roots, $x_1 + x_2$, is given by $-b/a$. In our equation, $a = 1$, $b = -R$, and $c = -4$. Therefore, the sum of the roots is: \[ -\frac{-R}{1} = R. \] 4. Conclusion: The sum of all possible values of $N$ for a given $R$ is $R$. Thus, the correct answer is $\boxed{\textbf{(B) } R}$.
Let $P$ be a point inside regular pentagon $A B C D E$ such that $\angle P A B=48^{\circ}$ and $\angle P D C=42^{\circ}$. Find $\angle B P C$, in degrees.	84^{\circ}	Since a regular pentagon has interior angles $108^{\circ}$, we can compute $\angle P D E=66^{\circ}, \angle P A E=60^{\circ}$, and $\angle A P D=360^{\circ}-\angle A E D-\angle P D E-\angle P A E=126^{\circ}$. Now observe that drawing $P E$ divides quadrilateral $P A E D$ into equilateral triangle $P A E$ and isosceles triangle $P E D$, where $\angle D P E=\angle E D P=66^{\circ}$. That is, we get $P A=P E=s$, where $s$ is the side length of the pentagon. Now triangles $P A B$ and $P E D$ are congruent (with angles $48^{\circ}-66^{\circ}-66^{\circ}$), so $P D=P B$ and $\angle P D C=\angle P B C=42^{\circ}$. This means that triangles $P D C$ and $P B C$ are congruent (side-angle-side), so $\angle B P C=\angle D P C$. Finally, we compute $\angle B P C+\angle D P C=2 \angle B P C=360^{\circ}-\angle A P B-\angle E P A-\angle D P E=168^{\circ}$, meaning $\angle B P C=84^{\circ}$.
Thirty people are divided into three groups (I, II, and III) with 10 people in each group. How many different compositions of groups can there be?	\frac{30!}{(10!)^3}
The distance from Goteborg to Jonkiping on a map is 88 cm. The scale on the map is 1 cm: 15 km. How far is it between the two city centers, in kilometers?	1320
Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is [asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7)); [/asy]	6	#### Solution 1: Using Pick's Theorem 1. Identify the number of interior and boundary points: - Interior points: $5$ - Boundary points: $4$ 2. Apply Pick's Theorem: - Pick's Theorem states that the area $A$ of a simple lattice polygon is given by: \[ A = I + \frac{B}{2} - 1 \] where $I$ is the number of interior points and $B$ is the number of boundary points. 3. Calculate the area: - Substitute the values into Pick's Theorem: \[ A = 5 + \frac{4}{2} - 1 = 5 + 2 - 1 = 6 \] 4. Conclusion: - The area of the quadrilateral is $\boxed{6}$. #### Solution 2: Using Geometric Decomposition 1. Draw and label the rectangle and points: - Consider the rectangle $ABCD$ with $AB = 4$ units and $BC = 3$ units. - Label the points $E, F, G, H$ on the rectangle as shown in the diagram. 2. Calculate the areas of triangles: - $[AEH] = \frac{1}{2} \times 1 \times 2 = 1$ - $[EBF] = \frac{1}{2} \times 3 \times 2 = 3$ - $[FCG] = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$ - $[GDH] = \frac{1}{2} \times 3 \times 1 = \frac{3}{2}$ 3. Calculate the area of rectangle $ABCD$: - $[ABCD] = 3 \times 4 = 12$ 4. Calculate the area of quadrilateral $EFGH$: - Subtract the areas of the triangles from the area of the rectangle: \[ [EFGH] = [ABCD] - ([AEH] + [EBF] + [FCG] + [GDH]) \] \[ [EFGH] = 12 - (1 + 3 + \frac{1}{2} + \frac{3}{2}) = 12 - 6 = 6 \] 5. Conclusion: - The area of the quadrilateral is $\boxed{6}$.
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with magnitudes $\|\overrightarrow{a}\| = 1$ and $\|\overrightarrow{b}\| = 2$, if for any unit vector $\overrightarrow{e}$, the inequality $\|\overrightarrow{a} \cdot \overrightarrow{e}\| + \|\overrightarrow{b} \cdot \overrightarrow{e}\| \leq \sqrt{6}$ holds, find the maximum value of $\overrightarrow{a} \cdot \overrightarrow{b}$.	\frac{1}{2}
Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.	90^\circ	Consider triangle \(\triangle ABC\) with altitudes \(AD\), \(BF\), and \(CE\). The orthocenter of the triangle is denoted by \(H\). A line through \(D\) that is parallel to \(AB\) intersects line \(EF\) at point \(G\). To find the angle \(\angle CGH\), follow these steps: 1. Identify the orthocenter \(H\): Since \(AD\), \(BF\), and \(CE\) are altitudes of \(\triangle ABC\), they meet at the orthocenter \(H\) of the triangle. 2. Analyze parallelism: The line passing through \(D\) and parallel to \(AB\), when intersecting \(EF\) at \(G\), means that \(DG \parallel AB\). 3. Use properties of cyclic quadrilaterals: The points \(A\), \(B\), \(C\), and \(H\) are concyclic in the circumcircle. The key insight is noticing the properties of angles formed by such a configuration: - Since \(AB \parallel DG\), the angles \(\angle DAG = \angle ABC\) are equal. - Consider the quadrilateral \(BFDG\): since it is cyclic, the opposite angles are supplementary. 4. Calculate \(\angle CGH\): - Since \(H\) lies on the altitude \(AD\), \(\angle CDH = 90^\circ\). - Now observe the angles formed at \(G\): - Since \(DG \parallel AB\) and both are perpendicular to \(CE\), we have \(\angle CGH = 90^\circ\). 5. Conclusion: This analysis ensures that \(\angle CGH\) is a right angle since \(G\) is where the line parallel to \(AB\) meets \(EF\) and forms perpendicularity with \(CE\). Thus, the angle \(\angle CGH\) is: \[ \boxed{90^\circ} \]
Given the relationship $P={P}_{0}{e}^{-kt}$ between the concentration of toxic and harmful substances $P$ (unit: $mg/L$) in the exhaust gas and time $t$ (unit: $h$) during the filtration process, determine the percentage of the original toxic and harmful substances that will remain after $5$ hours, given that $20\%$ of the substances are eliminated after $2$ hours.	57\%
How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there?	27
Given complex numbers \( z \) and \( \omega \) that satisfy the following two conditions: 1. \( z + \omega + 3 = 0 \); 2. \( \| z \|, 2, \| \omega \| \) form an arithmetic sequence. Does \( \cos (\arg z - \arg \omega) \) have a maximum value? If it does, find the maximum value.	\frac{1}{8}
In a certain circle, the chord of a $d$-degree arc is $22$ centimeters long, and the chord of a $2d$-degree arc is $20$ centimeters longer than the chord of a $3d$-degree arc, where $d < 120.$ The length of the chord of a $3d$-degree arc is $- m + \sqrt {n}$ centimeters, where $m$ and $n$ are positive integers. Find $m + n.$	174	Let $z=\frac{d}{2}$, $R$ be the circumradius, and $a$ be the length of 3d degree chord. Using the extended sine law, we obtain: \[22=2R\sin(z)\] \[20+a=2R\sin(2z)\] \[a=2R\sin(3z)\] Dividing the second from the first we get $\cos(z)=\frac{20+a}{44}$ By the triple angle formula we can manipulate the third equation as follows: $a=2R\times \sin(3z)=\frac{22}{\sin(z)} \times (3\sin(z)-4\sin^3(z)) = 22(3-4\sin^2(z))=22(4\cos^2(z)-1)=\frac{(20+a)^2}{22}-22$ Solving the quadratic equation gives the answer to be $\boxed{174}$.
What is the value of $K$ in the equation $16^3\times8^3=2^K$?	21
Let $P$ be a point selected uniformly at random in the cube $[0,1]^{3}$. The plane parallel to $x+y+z=0$ passing through $P$ intersects the cube in a two-dimensional region $\mathcal{R}$. Let $t$ be the expected value of the perimeter of $\mathcal{R}$. If $t^{2}$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.	12108	We can divide the cube into 3 regions based on the value of $x+y+z$ which defines the plane: $x+y+z<1,1 \leq x+y+z \leq 2$, and $x+y+z>2$. The two regions on the ends create tetrahedra, each of which has volume $1 / 6$. The middle region is a triangular antiprism with volume $2 / 3$. If our point $P$ lies in the middle region, we can see that we will always get the same value $3 \sqrt{2}$ for the perimeter of $\mathcal{R}$. Now let us compute the expected perimeter given that we pick a point $P$ in the first region $x+y+z<1$. If $x+y+z=a$, then the perimeter of $\mathcal{R}$ will just be $3 \sqrt{2} a$, so it is sufficient to find the expected value of $a$. $a$ is bounded between 0 and 1, and forms a continuous probability distribution with value proportional to $a^{2}$, so we can see with a bit of calculus that its expected value is $3 / 4$. The region $x+y+z>2$ is identical to the region $x+y+z<1$, so we get the same expected perimeter. Thus we have a $2 / 3$ of a guaranteed $3 \sqrt{2}$ perimeter, and a $1 / 3$ of having an expected $\frac{9}{4} \sqrt{2}$ perimeter, which gives an expected perimeter of $\frac{2}{3} \cdot 3 \sqrt{2}+\frac{1}{3} \cdot \frac{9}{4} \sqrt{2}=\frac{11 \sqrt{2}}{4}$. The square of this is $\frac{121}{8}$, giving an extraction of 12108.
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.	8.8
Given that $\{a_n\}$ is an arithmetic sequence, if $a_3 + a_5 + a_{12} - a_2 = 12$, calculate the value of $a_7 + a_{11}$.	12
We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules: 1. $g(b)=3b+1$ if $b$ is odd 2. $g(b)=\frac{b}{2}$ if $b$ is even. What is the smallest possible number of integers in the domain of $g$?	23
A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?	11475
There are 4 different points \( A, B, C, D \) on two non-perpendicular skew lines \( a \) and \( b \), where \( A \in a \), \( B \in a \), \( C \in b \), and \( D \in b \). Consider the following two propositions: (1) Line \( AC \) and line \( BD \) are always skew lines. (2) Points \( A, B, C, D \) can never be the four vertices of a regular tetrahedron. Which of the following is correct?	(1)(2)
Given a sequence $\{a_{n}\}$ such that $a_{2}=2a_{1}=4$, and $a_{n+1}-b_{n}=2a_{n}$, where $\{b_{n}\}$ is an arithmetic sequence with a common difference of $-1$. $(1)$ Investigation: Determine whether the sequence $\{a_{n}-n\}$ is an arithmetic sequence or a geometric sequence, and provide a justification. $(2)$ Find the smallest positive integer value of $n$ that satisfies $a_{1}+a_{2}+\ldots +a_{n} \gt 2200$.	12
Completely factor the following expression: \[(6a^3+92a^2-7)-(-7a^3+a^2-7)\]	13a^2(a+7)
Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form?	-30
Points $A, B, C$, and $D$ are on a line in that order. The distance from $A$ to $D$ is 24. The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$. Point $C$ is halfway between $B$ and $D$. What is the distance from $A$ to $C$?	15	Since $B$ is between $A$ and $D$ and $B D=3 A B$, then $B$ splits $A D$ in the ratio $1: 3$. Since $A D=24$, then $A B=6$ and $B D=18$. Since $C$ is halfway between $B$ and $D$, then $B C=rac{1}{2} B D=9$. Thus, $A C=A B+B C=6+9=15$.
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$?	3
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{1}{2}$, a circle $\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\sqrt{6}=0$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ A line passing through the fixed point $Q(1,0)$ with slope $k$ intersects the ellipse $C$ at points $M$ and $N$. If $\overrightarrow{OM}•\overrightarrow{ON}=-2$, find the value of the real number $k$ and the area of $\triangle MON$.	\frac{6\sqrt{6}}{11}
Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$ . A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$ , compute $100m+n$ . Proposed by Eugene Chen	106
The number of ounces of water needed to reduce $9$ ounces of shaving lotion containing $50$ % alcohol to a lotion containing $30$ % alcohol is:	6	1. Identify the amount of alcohol in the original lotion: The original shaving lotion is 9 ounces with 50% alcohol. Therefore, the amount of alcohol in the lotion is: \[ \frac{50}{100} \times 9 = \frac{9}{2} \text{ ounces} \] 2. Set up the equation for the final alcohol concentration: Let $N$ be the number of ounces of water added. The total volume of the new solution (lotion plus water) becomes $9 + N$ ounces. We want the alcohol to constitute 30% of this new solution. Thus, the equation based on the concentration of alcohol is: \[ \frac{9}{2} = 0.3 \times (9 + N) \] Here, $\frac{9}{2}$ ounces is the amount of alcohol (which remains constant), and $0.3$ represents 30%. 3. Solve the equation: Start by expanding and simplifying the equation: \[ \frac{9}{2} = 0.3 \times (9 + N) \implies \frac{9}{2} = 2.7 + 0.3N \] Rearrange to solve for $N$: \[ \frac{9}{2} - 2.7 = 0.3N \implies 4.5 - 2.7 = 0.3N \implies 1.8 = 0.3N \implies N = \frac{1.8}{0.3} = 6 \] 4. Conclusion: The number of ounces of water needed to achieve the desired alcohol concentration is $\boxed{\textbf{(D) } 6}$.
What is the value of $x + y$ if the sequence $3, ~9, ~x, ~y, ~30$ is an arithmetic sequence?	36
Given that $\|\vec{a}\| = 2$, $\|\vec{b}\| = 1$, and $(2\vec{a} - 3\vec{b}) \cdot (2\vec{a} + \vec{b}) = 9$. (I) Find the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$; (II) Find $\|\vec{a} + \vec{b}\|$ and the projection of vector $\vec{a}$ in the direction of $\vec{a} + \vec{b}$.	\frac{5\sqrt{7}}{7}
If three different natural numbers $a$, $b$ and $c$ each have exactly four natural-number factors, how many factors does $a^3b^4c^5$ have?	2080
Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is ( ) A 36 B 48 C 52 D 54	48
Abby, Bart, Cindy and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, and Cindy and Damon weigh 270 pounds. How many pounds do Abby and Damon weigh together?	285
The highest temperatures from April 1st to April 6th in a certain region were 28℃, 21℃, 22℃, 26℃, 28℃, and 25℃, respectively. Calculate the variance of the highest temperature data for these six days.	\frac{22}{3}
Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that \[ A + A^{\times} = I_{2020} \] \[ A \cdot A^{\times} = I_{2020} \] where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating the row \(j\) and the column \(i\). Find the maximum number of matrices verifying these conditions such that any two of them are not similar.	673
Given that the determinant of a $2 \times 2$ matrix $A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is 3, find the values of $\begin{vmatrix} 3a & 3b \\ 3c & 3d \end{vmatrix}$ and $\begin{vmatrix} 4a & 2b \\ 4c & 2d \end{vmatrix}$.	24
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $$2\sin^{2} \frac {A+B}{2}+\cos2C=1$$ (1) Find the magnitude of angle $C$; (2) If vector $$\overrightarrow {m}=(3a，b)$$ and vector $$\overrightarrow {n}=(a，- \frac {b}{3})$$, with $$\overrightarrow {m} \perp \overrightarrow {n}$$ and $$( \overrightarrow {m}+ \overrightarrow {n})(- \overrightarrow {m}+ \overrightarrow {n})=-16$$, find the values of $a$, $b$, and $c$.	\sqrt {7}
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 3.$ Find the minimum value of \[\frac{a + b}{abc}.\]	\frac{16}{9}
The projection of $\begin{pmatrix} 0 \\ 1 \\ 4 \end{pmatrix}$ onto a certain vector $\mathbf{w}$ is $\begin{pmatrix} 1 \\ -1/2 \\ 1/2 \end{pmatrix}.$ Find the projection of $\begin{pmatrix} 3 \\ 3 \\ -2 \end{pmatrix}$ onto $\mathbf{w}.$	\begin{pmatrix} 1/3 \\ -1/6 \\ 1/6 \end{pmatrix}
The New Year's gala has a total of 8 programs, 3 of which are non-singing programs. When arranging the program list, it is stipulated that the non-singing programs are not adjacent, and the first and last programs are singing programs. How many different ways are there to arrange the program list?	720

A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimeter of the largest rectangle.

\frac{5}{6}

Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.

25

If a number $N, N \ne 0$, diminished by four times its reciprocal, equals a given real constant $R$, then, for this given $R$, the sum of all such possible values of $N$ is

R

1. **Formulate the Equation**: Given that a number $N$, diminished by four times its reciprocal, equals a real constant $R$, we can write the equation as: \[ N - \frac{4}{N} = R. \] 2. **Manipulate the Equation**: To eliminate the fraction, multiply through by $N$ (assuming $N \neq 0$): \[ N^2 - 4 = RN. \] Rearrange this to form a quadratic equation: \[ N^2 - RN - 4 = 0. \] 3. **Apply Vieta's Formulas**: For a quadratic equation of the form $ax^2 + bx + c = 0$, Vieta's formulas tell us that the sum of the roots, $x_1 + x_2$, is given by $-b/a$. In our equation, $a = 1$, $b = -R$, and $c = -4$. Therefore, the sum of the roots is: \[ -\frac{-R}{1} = R. \] 4. **Conclusion**: The sum of all possible values of $N$ for a given $R$ is $R$. Thus, the correct answer is $\boxed{\textbf{(B) } R}$.

Let $P$ be a point inside regular pentagon $A B C D E$ such that $\angle P A B=48^{\circ}$ and $\angle P D C=42^{\circ}$. Find $\angle B P C$, in degrees.

84^{\circ}

Since a regular pentagon has interior angles $108^{\circ}$, we can compute $\angle P D E=66^{\circ}, \angle P A E=60^{\circ}$, and $\angle A P D=360^{\circ}-\angle A E D-\angle P D E-\angle P A E=126^{\circ}$. Now observe that drawing $P E$ divides quadrilateral $P A E D$ into equilateral triangle $P A E$ and isosceles triangle $P E D$, where $\angle D P E=\angle E D P=66^{\circ}$. That is, we get $P A=P E=s$, where $s$ is the side length of the pentagon. Now triangles $P A B$ and $P E D$ are congruent (with angles $48^{\circ}-66^{\circ}-66^{\circ}$), so $P D=P B$ and $\angle P D C=\angle P B C=42^{\circ}$. This means that triangles $P D C$ and $P B C$ are congruent (side-angle-side), so $\angle B P C=\angle D P C$. Finally, we compute $\angle B P C+\angle D P C=2 \angle B P C=360^{\circ}-\angle A P B-\angle E P A-\angle D P E=168^{\circ}$, meaning $\angle B P C=84^{\circ}$.

Thirty people are divided into three groups (I, II, and III) with 10 people in each group. How many different compositions of groups can there be?

\frac{30!}{(10!)^3}

The distance from Goteborg to Jonkiping on a map is 88 cm. The scale on the map is 1 cm: 15 km. How far is it between the two city centers, in kilometers?

1320

Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is [asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7)); [/asy]

6

#### Solution 1: Using Pick's Theorem 1. **Identify the number of interior and boundary points:** - Interior points: $5$ - Boundary points: $4$ 2. **Apply Pick's Theorem:** - Pick's Theorem states that the area $A$ of a simple lattice polygon is given by: \[ A = I + \frac{B}{2} - 1 \] where $I$ is the number of interior points and $B$ is the number of boundary points. 3. **Calculate the area:** - Substitute the values into Pick's Theorem: \[ A = 5 + \frac{4}{2} - 1 = 5 + 2 - 1 = 6 \] 4. **Conclusion:** - The area of the quadrilateral is $\boxed{6}$. #### Solution 2: Using Geometric Decomposition 1. **Draw and label the rectangle and points:** - Consider the rectangle $ABCD$ with $AB = 4$ units and $BC = 3$ units. - Label the points $E, F, G, H$ on the rectangle as shown in the diagram. 2. **Calculate the areas of triangles:** - $[AEH] = \frac{1}{2} \times 1 \times 2 = 1$ - $[EBF] = \frac{1}{2} \times 3 \times 2 = 3$ - $[FCG] = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$ - $[GDH] = \frac{1}{2} \times 3 \times 1 = \frac{3}{2}$ 3. **Calculate the area of rectangle $ABCD$:** - $[ABCD] = 3 \times 4 = 12$ 4. **Calculate the area of quadrilateral $EFGH$:** - Subtract the areas of the triangles from the area of the rectangle: \[ [EFGH] = [ABCD] - ([AEH] + [EBF] + [FCG] + [GDH]) \] \[ [EFGH] = 12 - (1 + 3 + \frac{1}{2} + \frac{3}{2}) = 12 - 6 = 6 \] 5. **Conclusion:** - The area of the quadrilateral is $\boxed{6}$.

Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with magnitudes $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, if for any unit vector $\overrightarrow{e}$, the inequality $|\overrightarrow{a} \cdot \overrightarrow{e}| + |\overrightarrow{b} \cdot \overrightarrow{e}| \leq \sqrt{6}$ holds, find the maximum value of $\overrightarrow{a} \cdot \overrightarrow{b}$.

\frac{1}{2}

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

90^\circ

Consider triangle $\triangle ABC$ with altitudes $AD$, $BF$, and $CE$. The orthocenter of the triangle is denoted by $H$. A line through $D$ that is parallel to $AB$ intersects line $EF$ at point $G$. To find the angle $\angle CGH$, follow these steps: 1. **Identify the orthocenter $H$:** Since $AD$, $BF$, and $CE$ are altitudes of $\triangle ABC$, they meet at the orthocenter $H$ of the triangle. 2. **Analyze parallelism:** The line passing through $D$ and parallel to $AB$, when intersecting $EF$ at $G$, means that $DG \parallel AB$. 3. **Use properties of cyclic quadrilaterals:** The points $A$, $B$, $C$, and $H$ are concyclic in the circumcircle. The key insight is noticing the properties of angles formed by such a configuration: - Since $AB \parallel DG$, the angles $\angle DAG = \angle ABC$ are equal. - Consider the quadrilateral $BFDG$: since it is cyclic, the opposite angles are supplementary. 4. **Calculate $\angle CGH$:** - Since $H$ lies on the altitude $AD$, $\angle CDH = 90^\circ$. - Now observe the angles formed at $G$: - Since $DG \parallel AB$ and both are perpendicular to $CE$, we have $\angle CGH = 90^\circ$. 5. **Conclusion:** This analysis ensures that $\angle CGH$ is a right angle since $G$ is where the line parallel to $AB$ meets $EF$ and forms perpendicularity with $CE$. Thus, the angle $\angle CGH$ is: \[ \boxed{90^\circ} \]

Given the relationship $P={P}_{0}{e}^{-kt}$ between the concentration of toxic and harmful substances $P$ (unit: $mg/L$) in the exhaust gas and time $t$ (unit: $h$) during the filtration process, determine the percentage of the original toxic and harmful substances that will remain after $5$ hours, given that $20\%$ of the substances are eliminated after $2$ hours.

57\%

How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there?

27

Given complex numbers $ z $ and $ \omega $ that satisfy the following two conditions: 1. $ z + \omega + 3 = 0 $; 2. $ | z |, 2, | \omega | $ form an arithmetic sequence. Does $ \cos (\arg z - \arg \omega) $ have a maximum value? If it does, find the maximum value.

\frac{1}{8}

In a certain circle, the chord of a $d$-degree arc is $22$ centimeters long, and the chord of a $2d$-degree arc is $20$ centimeters longer than the chord of a $3d$-degree arc, where $d < 120.$ The length of the chord of a $3d$-degree arc is $- m + \sqrt {n}$ centimeters, where $m$ and $n$ are positive integers. Find $m + n.$

174

Let $z=\frac{d}{2}$, $R$ be the circumradius, and $a$ be the length of 3d degree chord. Using the extended sine law, we obtain: \[22=2R\sin(z)\] \[20+a=2R\sin(2z)\] \[a=2R\sin(3z)\] Dividing the second from the first we get $\cos(z)=\frac{20+a}{44}$ By the triple angle formula we can manipulate the third equation as follows: $a=2R\times \sin(3z)=\frac{22}{\sin(z)} \times (3\sin(z)-4\sin^3(z)) = 22(3-4\sin^2(z))=22(4\cos^2(z)-1)=\frac{(20+a)^2}{22}-22$ Solving the quadratic equation gives the answer to be $\boxed{174}$.

What is the value of $K$ in the equation $16^3\times8^3=2^K$?

21

Let $P$ be a point selected uniformly at random in the cube $[0,1]^{3}$. The plane parallel to $x+y+z=0$ passing through $P$ intersects the cube in a two-dimensional region $\mathcal{R}$. Let $t$ be the expected value of the perimeter of $\mathcal{R}$. If $t^{2}$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.

12108

We can divide the cube into 3 regions based on the value of $x+y+z$ which defines the plane: $x+y+z<1,1 \leq x+y+z \leq 2$, and $x+y+z>2$. The two regions on the ends create tetrahedra, each of which has volume $1 / 6$. The middle region is a triangular antiprism with volume $2 / 3$. If our point $P$ lies in the middle region, we can see that we will always get the same value $3 \sqrt{2}$ for the perimeter of $\mathcal{R}$. Now let us compute the expected perimeter given that we pick a point $P$ in the first region $x+y+z<1$. If $x+y+z=a$, then the perimeter of $\mathcal{R}$ will just be $3 \sqrt{2} a$, so it is sufficient to find the expected value of $a$. $a$ is bounded between 0 and 1, and forms a continuous probability distribution with value proportional to $a^{2}$, so we can see with a bit of calculus that its expected value is $3 / 4$. The region $x+y+z>2$ is identical to the region $x+y+z<1$, so we get the same expected perimeter. Thus we have a $2 / 3$ of a guaranteed $3 \sqrt{2}$ perimeter, and a $1 / 3$ of having an expected $\frac{9}{4} \sqrt{2}$ perimeter, which gives an expected perimeter of $\frac{2}{3} \cdot 3 \sqrt{2}+\frac{1}{3} \cdot \frac{9}{4} \sqrt{2}=\frac{11 \sqrt{2}}{4}$. The square of this is $\frac{121}{8}$, giving an extraction of 12108.

A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.

8.8

Given that $\{a_n\}$ is an arithmetic sequence, if $a_3 + a_5 + a_{12} - a_2 = 12$, calculate the value of $a_7 + a_{11}$.

12

We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules: 1. $g(b)=3b+1$ if $b$ is odd 2. $g(b)=\frac{b}{2}$ if $b$ is even. What is the smallest possible number of integers in the domain of $g$?

23

A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?

11475

There are 4 different points $ A, B, C, D $ on two non-perpendicular skew lines $ a $ and $ b $, where $ A \in a $, $ B \in a $, $ C \in b $, and $ D \in b $. Consider the following two propositions: (1) Line $ AC $ and line $ BD $ are always skew lines. (2) Points $ A, B, C, D $ can never be the four vertices of a regular tetrahedron. Which of the following is correct?

(1)(2)

Given a sequence $\{a_{n}\}$ such that $a_{2}=2a_{1}=4$, and $a_{n+1}-b_{n}=2a_{n}$, where $\{b_{n}\}$ is an arithmetic sequence with a common difference of $-1$. $(1)$ Investigation: Determine whether the sequence $\{a_{n}-n\}$ is an arithmetic sequence or a geometric sequence, and provide a justification. $(2)$ Find the smallest positive integer value of $n$ that satisfies $a_{1}+a_{2}+\ldots +a_{n} \gt 2200$.

12

Completely factor the following expression: \[(6a^3+92a^2-7)-(-7a^3+a^2-7)\]

13a^2(a+7)

Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form?

-30

Points $A, B, C$, and $D$ are on a line in that order. The distance from $A$ to $D$ is 24. The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$. Point $C$ is halfway between $B$ and $D$. What is the distance from $A$ to $C$?

15

Since $B$ is between $A$ and $D$ and $B D=3 A B$, then $B$ splits $A D$ in the ratio $1: 3$. Since $A D=24$, then $A B=6$ and $B D=18$. Since $C$ is halfway between $B$ and $D$, then $B C=rac{1}{2} B D=9$. Thus, $A C=A B+B C=6+9=15$.

Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$?

3

Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{1}{2}$, a circle $\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\sqrt{6}=0$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ A line passing through the fixed point $Q(1,0)$ with slope $k$ intersects the ellipse $C$ at points $M$ and $N$. If $\overrightarrow{OM}•\overrightarrow{ON}=-2$, find the value of the real number $k$ and the area of $\triangle MON$.

\frac{6\sqrt{6}}{11}

Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$ . A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$ , compute $100m+n$ . *Proposed by Eugene Chen*

106

The number of ounces of water needed to reduce $9$ ounces of shaving lotion containing $50$ % alcohol to a lotion containing $30$ % alcohol is:

6

1. **Identify the amount of alcohol in the original lotion**: The original shaving lotion is 9 ounces with 50% alcohol. Therefore, the amount of alcohol in the lotion is: \[ \frac{50}{100} \times 9 = \frac{9}{2} \text{ ounces} \] 2. **Set up the equation for the final alcohol concentration**: Let $N$ be the number of ounces of water added. The total volume of the new solution (lotion plus water) becomes $9 + N$ ounces. We want the alcohol to constitute 30% of this new solution. Thus, the equation based on the concentration of alcohol is: \[ \frac{9}{2} = 0.3 \times (9 + N) \] Here, $\frac{9}{2}$ ounces is the amount of alcohol (which remains constant), and $0.3$ represents 30%. 3. **Solve the equation**: Start by expanding and simplifying the equation: \[ \frac{9}{2} = 0.3 \times (9 + N) \implies \frac{9}{2} = 2.7 + 0.3N \] Rearrange to solve for $N$: \[ \frac{9}{2} - 2.7 = 0.3N \implies 4.5 - 2.7 = 0.3N \implies 1.8 = 0.3N \implies N = \frac{1.8}{0.3} = 6 \] 4. **Conclusion**: The number of ounces of water needed to achieve the desired alcohol concentration is $\boxed{\textbf{(D) } 6}$.

What is the value of $x + y$ if the sequence $3, ~9, ~x, ~y, ~30$ is an arithmetic sequence?

36

Given that $|\vec{a}| = 2$, $|\vec{b}| = 1$, and $(2\vec{a} - 3\vec{b}) \cdot (2\vec{a} + \vec{b}) = 9$. (I) Find the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$; (II) Find $|\vec{a} + \vec{b}|$ and the projection of vector $\vec{a}$ in the direction of $\vec{a} + \vec{b}$.

\frac{5\sqrt{7}}{7}

If three different natural numbers $a$, $b$ and $c$ each have exactly four natural-number factors, how many factors does $a^3b^4c^5$ have?

2080

Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is ( ) A 36 B 48 C 52 D 54

48

Abby, Bart, Cindy and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, and Cindy and Damon weigh 270 pounds. How many pounds do Abby and Damon weigh together?

285

The highest temperatures from April 1st to April 6th in a certain region were 28℃, 21℃, 22℃, 26℃, 28℃, and 25℃, respectively. Calculate the variance of the highest temperature data for these six days.

\frac{22}{3}

Consider $A \in \mathcal{M}_{2020}(\mathbb{C})$ such that \[ A + A^{\times} = I_{2020} \] \[ A \cdot A^{\times} = I_{2020} \] where $A^{\times}$ is the adjugate matrix of $A$, i.e., the matrix whose elements are $a_{ij} = (-1)^{i+j} d_{ji}$, where $d_{ji}$ is the determinant obtained from $A$, eliminating the row $j$ and the column $i$. Find the maximum number of matrices verifying these conditions such that any two of them are not similar.

673

Given that the determinant of a $2 \times 2$ matrix $A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is 3, find the values of $\begin{vmatrix} 3a & 3b \\ 3c & 3d \end{vmatrix}$ and $\begin{vmatrix} 4a & 2b \\ 4c & 2d \end{vmatrix}$.

24

In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $$2\sin^{2} \frac {A+B}{2}+\cos2C=1$$ (1) Find the magnitude of angle $C$; (2) If vector $$\overrightarrow {m}=(3a，b)$$ and vector $$\overrightarrow {n}=(a，- \frac {b}{3})$$, with $$\overrightarrow {m} \perp \overrightarrow {n}$$ and $$( \overrightarrow {m}+ \overrightarrow {n})(- \overrightarrow {m}+ \overrightarrow {n})=-16$$, find the values of $a$, $b$, and $c$.

\sqrt {7}

Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 3.$ Find the minimum value of \[\frac{a + b}{abc}.\]

\frac{16}{9}

The projection of $\begin{pmatrix} 0 \\ 1 \\ 4 \end{pmatrix}$ onto a certain vector $\mathbf{w}$ is $\begin{pmatrix} 1 \\ -1/2 \\ 1/2 \end{pmatrix}.$ Find the projection of $\begin{pmatrix} 3 \\ 3 \\ -2 \end{pmatrix}$ onto $\mathbf{w}.$

\begin{pmatrix} 1/3 \\ -1/6 \\ 1/6 \end{pmatrix}

The New Year's gala has a total of 8 programs, 3 of which are non-singing programs. When arranging the program list, it is stipulated that the non-singing programs are not adjacent, and the first and last programs are singing programs. How many different ways are there to arrange the program list?

720