problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
The national security agency's wiretap recorded a conversation between two spies and found that on a 30-minute tape, starting from the 30-second mark, there was a 10-second segment of conversation containing information about the spies' criminal activities. Later, it was discovered that part of this conversation was er... | \frac{1}{45} | 0.3125 | 6,106 | 4,508.2 | 6,832.272727 | |
On the number line, points $M$ and $N$ divide $L P$ into three equal parts. What is the value at $M$? | \frac{1}{9} | The difference between $\frac{1}{6}$ and $\frac{1}{12}$ is $\frac{1}{6}-\frac{1}{12}=\frac{2}{12}-\frac{1}{12}=\frac{1}{12}$, so $L P=\frac{1}{12}$. Since $L P$ is divided into three equal parts, then this distance is divided into three equal parts, each equal to $\frac{1}{12} \div 3=\frac{1}{12} \times \frac{1}{3}=\fr... | 0 | 645.4375 | -1 | 645.4375 |
The sides of a triangle are $30$, $70$, and $80$ units. If an altitude is dropped upon the side of length $80$, the larger segment cut off on this side is: | 65 | 1. **Assign Variables:**
Let the shorter segment of the side of length $80$ be $x$, and let the altitude dropped to this side be $y$. The larger segment is then $80 - x$.
2. **Apply the Pythagorean Theorem:**
Since the altitude divides the triangle into two right triangles, we can apply the Pythagorean theorem t... | 0.875 | 4,466.875 | 3,934.714286 | 8,192 |
Find the shortest distance from the line $3 x+4 y=25$ to the circle $x^{2}+y^{2}=6 x-8 y$. | 7 / 5 | The circle is $(x-3)^{2}+(y+4)^{2}=5^{2}$. The center $(3,-4)$ is a distance of $$ \frac{|3 \cdot 3+4 \cdot-4-25|}{\sqrt{3^{2}+4^{2}}}=\frac{32}{5} $$ from the line, so we subtract 5 for the radius of the circle and get $7 / 5$. | 1 | 3,336.25 | 3,336.25 | -1 |
Let \(\{a, b, c, d\}\) be a subset of \(\{1, 2, \ldots, 17\}\). If 17 divides \(a - b + c - d\), then \(\{a, b, c, d\}\) is called a "good subset." Find the number of good subsets. | 476 | 0 | 8,192 | -1 | 8,192 | |
If three coins are tossed at the same time, what is the probability of getting two tails and one head? Express your answer as a common fraction. | \frac{3}{8} | 0.9375 | 2,361.9375 | 1,973.266667 | 8,192 | |
In the diagram, the circle has center \( O \) and square \( OPQR \) has vertex \( Q \) on the circle. If the area of the circle is \( 72 \pi \), the area of the square is: | 36 | 0.3125 | 5,289.4375 | 3,436.2 | 6,131.818182 | |
Ten points are spaced around at intervals of one unit around a modified $2 \times 2$ square such that each vertex of the square and midpoints on each side of the square are included, along with two additional points, each located midway on a diagonal extension from opposite corners of the square. Two of the 10 points a... | \frac{14}{45} | 0 | 8,192 | -1 | 8,192 | |
Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7. | 16687 | 0 | 8,142.6875 | -1 | 8,142.6875 | |
Given the differences between the scores of 14 students in a group and the class average score of 85 are 2, 3, -3, -5, 12, 12, 8, 2, -1, 4, -10, -2, 5, 5, find the average score of this group. | 87.29 | 0 | 4,990.4375 | -1 | 4,990.4375 | |
(The 2018 Anqing City, Anhui Province, China, High School Second Mock Exam) Given that the focus of the parabola $x^{2}=4y$ is $F$, points $A$ and $B$ are on the parabola, and satisfy $\overrightarrow{AF}=λ\overrightarrow{FB}$. If $|\overrightarrow{AF}|=\frac{3}{2}$, then the value of $\lambda$ is ____. | \frac{1}{2} | 0.6875 | 6,144.5 | 5,213.818182 | 8,192 | |
Given the lines $l_1: ax+2y-1=0$ and $l_2: x+by-3=0$, where the angle of inclination of $l_1$ is $\frac{\pi}{4}$, find the value of $a$. If $l_1$ is perpendicular to $l_2$, find the value of $b$. If $l_1$ is parallel to $l_2$, find the distance between the two lines. | \frac{7\sqrt{2}}{4} | 0 | 5,147.5625 | -1 | 5,147.5625 | |
A certain regular tetrahedron has three of its vertices at the points $(0,1,2),$ $(4,2,1),$ and $(3,1,5).$ Find the coordinates of the fourth vertex, given that they are also all integers. | (3,-2,2) | 0.625 | 6,121.5625 | 4,879.3 | 8,192 | |
A sphere is inscribed in a cube, and the cube has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? | 8 | 1 | 1,900.125 | 1,900.125 | -1 | |
Given two boxes, each containing the chips numbered $1$, $2$, $4$, $5$, a chip is drawn randomly from each box. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. | \frac{1}{2} | 0.25 | 7,960.5625 | 7,266.25 | 8,192 | |
Let $W$ be the hypercube $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid 0 \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 1\right\}$. The intersection of $W$ and a hyperplane parallel to $x_{1}+x_{2}+x_{3}+x_{4}=0$ is a non-degenerate 3-dimensional polyhedron. What is the maximum number of faces of this polyhedron? | 8 | The number of faces in the polyhedron is equal to the number of distinct cells (3-dimensional faces) of the hypercube whose interior the hyperplane intersects. However, it is possible to arrange the hyperplane such that it intersects all 8 cells. Namely, $x_{1}+x_{2}+x_{3}+x_{4}=\frac{3}{2}$ intersects all 8 cells beca... | 0.0625 | 7,999.8125 | 5,117 | 8,192 |
Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the n... | 13 | 1. **Define Variables:**
Let $c$ be the number of consecutive sheets Hiram’s roommate borrows, and let $b$ be the number of sheets preceding the $c$ borrowed sheets. For example, if the roommate borrows sheets $3$, $4$, and $5$, then $c=3$ and $b=2$.
2. **Calculate the Sum of Page Numbers:**
The sum of the page ... | 0.4375 | 6,772.1875 | 4,946.714286 | 8,192 |
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class's mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all... | 76 | 1. **Identify the given information:**
- Mean score of the morning class, $M = 84$.
- Mean score of the afternoon class, $A = 70$.
- Ratio of the number of students in the morning class to the afternoon class, $\frac{m}{a} = \frac{3}{4}$.
2. **Express the number of students in the morning class in terms of th... | 1 | 1,559.0625 | 1,559.0625 | -1 |
Let $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$ | \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} | 0.75 | 4,956.875 | 3,882.5 | 8,180 | |
Given a square \( PQRS \) with an area of \( 120 \, \text{cm}^2 \). Point \( T \) is the midpoint of \( PQ \). The ratios are given as \( QU: UR = 2:1 \), \( RV: VS = 3:1 \), and \( SW: WP = 4:1 \).
Find the area, in \(\text{cm}^2\), of quadrilateral \( TUVW \). | 67 | 0.5 | 6,575.5 | 5,928.625 | 7,222.375 | |
Given the function $f(x)=\sin(2x+\varphi)$ where $(0 < \varphi < \pi)$ satisfies $f(x) \leq |f(\frac{\pi}{6})|$, and $f(x_{1}) = f(x_{2}) = -\frac{3}{5}$, calculate the value of $\sin(x_{2}-x_{1})$. | \frac{4}{5} | 0.5 | 6,696.25 | 5,200.5 | 8,192 | |
Determine the maximum possible value of the expression
$$
27abc + a\sqrt{a^2 + 2bc} + b\sqrt{b^2 + 2ca} + c\sqrt{c^2 + 2ab}
$$
where \(a, b, c\) are positive real numbers such that \(a + b + c = \frac{1}{\sqrt{3}}\). | \frac{2}{3 \sqrt{3}} | 0 | 8,182.875 | -1 | 8,182.875 | |
In the polar coordinate system, the equation of curve C is $\rho^2\cos2\theta=9$. Point P is $(2\sqrt{3}, \frac{\pi}{6})$. Establish a Cartesian coordinate system with the pole O as the origin and the positive half-axis of the x-axis as the polar axis.
(1) Find the parametric equation of line OP and the Cartesian equ... | \sqrt{2} | 0.9375 | 5,164.75 | 4,962.933333 | 8,192 | |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $a = b\cos C + c\sin B$.
(1) Find angle $B$.
(2) If $b = 4$, find the maximum area of triangle $ABC$. | 4\sqrt{2} + 4 | 0 | 7,133 | -1 | 7,133 | |
Given that \( \mathrm{a}, \mathrm{b}, \mathrm{c} \) are three natural numbers, and the least common multiple (LCM) of \( \mathrm{a} \) and \( \mathrm{b} \) is 60, and the LCM of \( \mathrm{a} \) and \( \mathrm{c} \) is 270, find the LCM of \( \mathrm{b} \) and \( \mathrm{c} \). | 540 | 0.4375 | 7,133.1875 | 5,771.857143 | 8,192 | |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy: $|\overrightarrow {a}| = \sqrt {2}$, $|\overrightarrow {b}| = 4$, and $\overrightarrow {a} \cdot (\overrightarrow {b} - \overrightarrow {a}) = 2$.
1. Find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
2. Find the m... | 2\sqrt{2} | 0 | 2,819.75 | -1 | 2,819.75 | |
A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction. | \frac{9}{2} | 0.75 | 5,998.375 | 5,267.166667 | 8,192 | |
Find the coefficient of the $x^3$ term in the expansion of the product $$(3x^3 + 2x^2 + 4x + 5)(4x^2 + 5x + 6).$$ | 44 | 0.8125 | 3,735.3125 | 2,706.846154 | 8,192 | |
Given the curve $C$: $\begin{cases}x=2\cos \alpha \\ y= \sqrt{3}\sin \alpha\end{cases}$ ($\alpha$ is a parameter) and the fixed point $A(0, \sqrt{3})$, $F_1$ and $F_2$ are the left and right foci of this curve, respectively. Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis ... | \frac{12\sqrt{3}}{13} | 0 | 7,408.5 | -1 | 7,408.5 | |
Given that points $\mathbf{A}$ and $\mathbf{B}$ lie on the curves $C_{1}: x^{2} - y + 1 = 0$ and $C_{2}: y^{2} - x + 1 = 0$ respectively, determine the minimum value of $|AB|$. | \frac{3 \sqrt{2}}{4} | 0 | 7,465 | -1 | 7,465 | |
A store owner bought 2000 markers at $0.20 each. To make a minimum profit of $200, if he sells the markers for $0.50 each, calculate the number of markers he must sell at least to achieve or exceed this profit. | 1200 | 0.8125 | 846.375 | 902.769231 | 602 | |
Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has 28 solutions in positive integers $x$, $y$, and $z$, then $n$ must be either | 17 or 18 | 1. **Rewrite the equation**: Start by rewriting the given equation $2x + 2y + z = n$ in a form that is easier to analyze for solutions in positive integers. We can express $z$ as $z = n - 2x - 2y$. Since $x, y, z$ are positive integers, we have $x \geq 1$, $y \geq 1$, and $z \geq 1$. Thus, $n - 2x - 2y \geq 1$.
2. **S... | 0 | 7,722.125 | -1 | 7,722.125 |
Consider the arithmetic sequence $1$, $4$, $7$, $10$, $13$, $\ldots$. Find the $15^{\text{th}}$ term in the sequence. | 43 | 1 | 2,040.625 | 2,040.625 | -1 | |
The hare and the tortoise had a race over 100 meters, in which both maintained constant speeds. When the hare reached the finish line, it was 75 meters in front of the tortoise. The hare immediately turned around and ran back towards the start line. How far from the finish line did the hare and the tortoise meet? | 60 | 0.5 | 5,733.875 | 5,548.25 | 5,919.5 | |
Given \( P \) is the product of \( 3,659,893,456,789,325,678 \) and \( 342,973,489,379,256 \), find the number of digits of \( P \). | 34 | 0.4375 | 6,973.125 | 6,578.428571 | 7,280.111111 | |
The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP = 3$, $PB = 4$, $AC... | 227 | Let the incircle of $ABC$ be tangent to $AB$ and $AC$ at $M$ and $N$. By Brianchon's theorem on tangential hexagons $QNCBMP$ and $PYQCXB$, we know that $MN,CP,BQ$ and $XY$ are concurrent at a point $O$. Let $PQ \cap BC = Z$. Then by La Hire's $A$ lies on the polar of $Z$ so $Z$ lies on the polar of $A$. Therefore, $MN$... | 0 | 8,192 | -1 | 8,192 |
Masha has three identical dice, each face of which has one of six different prime numbers with a total sum of 87.
Masha rolled all three dice twice. The first time, the sum of the numbers rolled was 10, and the second time, the sum of the numbers rolled was 62.
Exactly one of the six numbers never appeared. What numb... | 17 | 0 | 8,192 | -1 | 8,192 | |
Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$. The diagonals have length $10\sqrt {21}$, and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitud... | 32 | 0.0625 | 7,891.8125 | 4,818 | 8,096.733333 | |
How many multiples of 4 are there between 200 and 500? | 74 | 0.5625 | 3,380.75 | 4,546.444444 | 1,882 | |
Compute the unique positive integer $n$ such that
\[2 \cdot 2^2 + 3 \cdot 2^3 + 4 \cdot 2^4 + \dots + n \cdot 2^n = 2^{n + 10}.\] | 513 | 0.75 | 4,733.125 | 3,580.166667 | 8,192 | |
Given $\sin 2α - 2 = 2\cos 2α$, find the value of $\sin^{2}α + \sin 2α$. | \frac{8}{5} | 0.5 | 7,276.8125 | 6,361.625 | 8,192 | |
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 $\frac {(a+b)^{2}-c^{2}}{3ab}=1$.
$(1)$ Find $\angle C$;
$(2)$ If $c= \sqrt {3}$ and $b= \sqrt {2}$, find $\angle B$ and the area of $\triangle ABC$. | \frac {3+ \sqrt {3}}{4} | 0 | 5,341.4375 | -1 | 5,341.4375 | |
How many three-digit whole numbers have at least one 7 or at least one 9 as digits? | 452 | 0.125 | 7,846.1875 | 5,425.5 | 8,192 | |
At lunch, $60\%$ of the students selected soda while $20\%$ selected milk. If 72 students selected soda, how many students selected milk? | 24 | 1 | 1,383 | 1,383 | -1 | |
Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$. | 4 | 1 | 1,746.4375 | 1,746.4375 | -1 | |
The length of the longer side of rectangle $R$ is $10$ percent more than the length of a side of square $S.$ The length of the shorter side of rectangle $R$ is $10$ percent less than the length of a side of square $S.$ What is the ratio of the area of rectangle $R$ to the area of square $S?$ Express your answer as a co... | \frac{99}{100} | 1 | 1,458.875 | 1,458.875 | -1 | |
Given that points \( B \) and \( C \) are in the fourth and first quadrants respectively, and both lie on the parabola \( y^2 = 2px \) where \( p > 0 \). Let \( O \) be the origin, and \(\angle OBC = 30^\circ\) and \(\angle BOC = 60^\circ\). If \( k \) is the slope of line \( OC \), find the value of \( k^3 + 2k \). | \sqrt{3} | 0 | 8,149.4375 | -1 | 8,149.4375 | |
In an arithmetic sequence $\{a_n\}$, if $a_3 - a_2 = -2$ and $a_7 = -2$, find the value of $a_9$. | -6 | 1 | 1,420.1875 | 1,420.1875 | -1 | |
Suppose $a$, $b$, $c$, and $d$ are positive integers satisfying $a + b + c + d = 3000$. Calculate $a!b!c!d! = m \cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by 10. What is the smallest possible value of $n$?
A) 745
B) 748
C) 751
D) 754
E) 757 | 748 | 0 | 8,192 | -1 | 8,192 | |
The numbers 60, 221, and 229 are the legs and hypotenuse of a right triangle. Find the multiplicative inverse to 450 modulo 3599. (Express your answer as an integer $n$ with $0\leq n<3599$.) | 8 | 0.875 | 4,326.4375 | 3,774.214286 | 8,192 | |
In a school, there are 30 students who are enrolled in at least one of the offered foreign language classes: German or Italian. The information available indicates that 22 students are taking German and 26 students are taking Italian. Sarah, who is writing an article for the school magazine, needs to interview two stud... | \frac{401}{435} | 0.0625 | 6,037.625 | 4,974 | 6,108.533333 | |
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} | 0.75 | 6,108.1875 | 5,413.583333 | 8,192 | |
At an exchange point, there are two types of transactions:
1) Give 2 euros - receive 3 dollars and a candy as a gift.
2) Give 5 dollars - receive 3 euros and a candy as a gift.
When the wealthy Buratino came to the exchange point, he only had dollars. When he left, he had fewer dollars, he did not get any euros, but h... | 10 | 0 | 6,278.5 | -1 | 6,278.5 | |
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square? | 12.5 | 1. **Assigning Dimensions Based on Given Ratios:**
- Let the side length of the square be $s$.
- According to the problem, the width of the rectangle is twice the side of the square. Therefore, the width of the rectangle is $2s$.
- The ratio of the rectangle's length to its width is $2:1$. Hence, if the width ... | 1 | 3,249.9375 | 3,249.9375 | -1 |
Given that the function $f(x)$ is monotonic on $(-1, +\infty)$, and the graph of the function $y = f(x - 2)$ is symmetrical about the line $x = 1$, if the sequence $\{a_n\}$ is an arithmetic sequence with a nonzero common difference and $f(a_{50}) = f(a_{51})$, determine the sum of the first 100 terms of $\{a_n\}$. | -100 | 0.5625 | 5,712.3125 | 5,331.444444 | 6,202 | |
Let $\phi$ be the smallest acute angle for which $\cos \phi,$ $\cos 2 \phi,$ $\cos 3 \phi$ form an arithmetic progression, in some order. Find $\sin \phi.$ | \frac{\sqrt{3}}{2} | 0 | 7,556.6875 | -1 | 7,556.6875 | |
Find the smallest natural number that begins with the digit five, which is reduced by four times if this five is removed from the beginning of its decimal representation and appended to its end. | 512820 | 0.625 | 6,667 | 5,752 | 8,192 | |
Given that in $\triangle ABC$, $BD:DC = 3:2$ and $AE:EC = 3:4$, and the area of $\triangle ABC$ is 1, find the area of $\triangle BMD$. | \frac{4}{15} | 0.0625 | 8,115.875 | 7,573 | 8,152.066667 | |
In a bag, there are several balls, including red, black, yellow, and white balls. The probability of drawing a red ball is $\frac{1}{3}$, the probability of drawing either a black or a yellow ball is $\frac{5}{12}$, and the probability of drawing either a yellow or a white ball is $\frac{5}{12}$. The probability of dra... | \frac{1}{4} | 1 | 3,196.75 | 3,196.75 | -1 | |
For which $x$ and $y$ is the number $x x y y$ a square of a natural number? | 7744 | 0.625 | 6,001.5625 | 5,500.5 | 6,836.666667 | |
Place four balls numbered 1, 2, 3, and 4 into three boxes labeled A, B, and C.
(1) If none of the boxes are empty and ball number 3 must be in box B, how many different arrangements are there?
(2) If ball number 1 cannot be in box A and ball number 2 cannot be in box B, how many different arrangements are there? | 36 | 0.25 | 7,602.4375 | 6,701.75 | 7,902.666667 | |
Our water polo team has 15 members. I want to choose a starting team consisting of 7 players, one of whom will be the goalie (the other six positions are interchangeable, so the order in which they are chosen doesn't matter). In how many ways can I choose my starting team? | 45,\!045 | 0 | 3,997.875 | -1 | 3,997.875 | |
Jerry's favorite number is $97$ . He knows all kinds of interesting facts about $97$ :
- $97$ is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which $97$ can be written as a difference of two perfect squares.
- There is only one way in whic... | 96 | 0.875 | 4,077.9375 | 4,043.357143 | 4,320 | |
Given that $\tan \alpha = -2$, find the value of the following expressions:
$(1) \frac{\sin \alpha - 3 \cos \alpha}{\sin \alpha + \cos \alpha}$
$(2) \frac{1}{\sin \alpha \cdot \cos \alpha}$ | -\frac{5}{2} | 1 | 2,806.8125 | 2,806.8125 | -1 | |
The Greater Fourteen Basketball League has two divisions, each containing seven teams. Each team plays each of the other teams in its own division twice and every team in the other division twice. How many league games are scheduled? | 182 | 0.5 | 5,958.6875 | 3,725.375 | 8,192 | |
How many positive integers less than $201$ are multiples of either $4$ or $9$, but not both at once? | 62 | 0.9375 | 3,188.5 | 3,202 | 2,986 | |
When \( n \) is a positive integer, the function \( f \) satisfies
\[ f(n+3) = \frac{f(n) - 1}{f(n) + 1}, \]
with \( f(1) \neq 0 \), and \( f(1) \neq \pm 1 \).
Compute \( f(1) f(2023) \). | -1 | 0.75 | 5,349.875 | 4,402.5 | 8,192 | |
Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \_\_\_\_\_\_. | \sqrt{2} | 0.4375 | 3,931.3125 | 3,143.714286 | 4,543.888889 | |
The diagram shows a regular octagon and a square formed by drawing four diagonals of the octagon. The edges of the square have length 1. What is the area of the octagon?
A) \(\frac{\sqrt{6}}{2}\)
B) \(\frac{4}{3}\)
C) \(\frac{7}{5}\)
D) \(\sqrt{2}\)
E) \(\frac{3}{2}\) | \sqrt{2} | 0 | 7,738.8125 | -1 | 7,738.8125 | |
Simplify this expression to a common fraction: $\frac{1}{\frac{1}{(\frac{1}{2})^{1}}+\frac{1}{(\frac{1}{2})^{2}}+\frac{1}{(\frac{1}{2})^{3}}}$ | \frac{1}{14} | 1 | 1,941.6875 | 1,941.6875 | -1 | |
Among the natural numbers not exceeding 10,000, calculate the number of odd numbers with distinct digits. | 2605 | 0 | 7,635.375 | -1 | 7,635.375 | |
The numbers \( a, b, c, d \) belong to the interval \([-7, 7]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 210 | 0.1875 | 7,819.6875 | 6,206.333333 | 8,192 | |
In a trapezoid, the two non parallel sides and a base have length $1$ , while the other base and both the diagonals have length $a$ . Find the value of $a$ . | \frac{\sqrt{5} + 1}{2} | 0 | 5,788.75 | -1 | 5,788.75 | |
Two mutually perpendicular chords \( AB \) and \( CD \) are drawn in a circle. Determine the distance between the midpoint of segment \( AD \) and the line \( BC \), given that \( BD = 6 \), \( AC = 12 \), and \( BC = 10 \). If necessary, round your answer to two decimal places. | 2.5 | 0 | 8,171.0625 | -1 | 8,171.0625 | |
Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1. | 11111111100000000 | 0 | 8,192 | -1 | 8,192 | |
Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AL) + \log_{10} (AM) &= 2, \\
\log_{10} (ML) + \log_{10} (MR) &= 3, \\
\log_{10} (RA) + \log_{10} (RL) &= 4.
\end{align*}Compute the value of the product $ARML.$ | 1000 | 0.8125 | 5,486.25 | 4,972.615385 | 7,712 | |
A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is $12\sqrt{3}$ inches. A sphere is placed inside the cone so that it is tangent to the sides of the cone and sits on the table. What is the volume... | 288\pi | 0.75 | 6,246.75 | 5,598.333333 | 8,192 | |
Of the students attending a school party, $60\%$ of the students are girls, and $40\%$ of the students like to dance. After these students are joined by $20$ more boy students, all of whom like to dance, the party is now $58\%$ girls. How many students now at the party like to dance? | 252 | Let $p$ denote the total number of people at the party. Then, because we know the proportions of boys to $p$ both before and after 20 boys arrived, we can create the following equation: \[0.4p+20 = 0.42(p+20)\] Solving for p gives us $p=580$, so the solution is $0.4p+20 = \boxed{252}$ | 0.8125 | 3,934.125 | 3,100.153846 | 7,548 |
Lil writes one of the letters \( \text{P}, \text{Q}, \text{R}, \text{S} \) in each cell of a \( 2 \times 4 \) table. She does this in such a way that, in each row and in each \( 2 \times 2 \) square, all four letters appear. In how many ways can she do this? | 24 | 0 | 8,192 | -1 | 8,192 | |
Let $a,$ $b,$ $c$ be complex numbers such that
\[a + b + c = ab + ac + bc = abc = 1.\]Enter the values $a,$ $b,$ $c,$ separated by commas, in any order. | 1,i,-i | 0.1875 | 1,286.1875 | 1,506.666667 | 1,235.307692 | |
A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105? | 16 | 0 | 8,192 | -1 | 8,192 | |
Find the smallest positive integer $b$ such that $1111_{b}$ ( 1111 in base $b$) is a perfect square. If no such $b$ exists, write "No solution". | 7 | We have $1111_{b}=b^{3}+b^{2}+b+1=\left(b^{2}+1\right)(b+1)$. Note that $\operatorname{gcd}\left(b^{2}+1, b+1\right)=\operatorname{gcd}\left(b^{2}+1-(b+1)(b-1), b+1\right)=\operatorname{gcd}(2, b+1)$, which is either 1 or 2 . If the gcd is 1 , then there is no solution as this implies $b^{2}+1$ is a perfect square, whi... | 0.6875 | 5,352.5 | 4,092.545455 | 8,124.4 |
Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head. | 48 | 0 | 6,563.1875 | -1 | 6,563.1875 | |
Given that the odd function $f(x)$ is an increasing function defined on $\mathbb{R}$, and the sequence $x_n$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, then the value of $x_{2011}$ is equal to. | 4003 | 1 | 4,008.9375 | 4,008.9375 | -1 | |
All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2}{3} | 0.25 | 7,580.0625 | 5,744.25 | 8,192 | |
Find the maximum value of the following expression:
$$
|\cdots|\left|x_{1}-x_{2}\right|-x_{3}\left|-\cdots-x_{1990}\right|,
$$
where \( x_{1}, x_{2}, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990. | 1989 | 0 | 8,121.8125 | -1 | 8,121.8125 | |
In the dihedral angle $\alpha - E F - \beta $, $AE \subset \alpha, BF \subset \beta$, and $AE \perp EF, BF \perp EF$. Given $EF = 1$, $AE = 2$, and $AB = \sqrt{2}$, find the maximum volume of the tetrahedron $ABEF$. | \frac{1}{3} | 0 | 8,173.75 | -1 | 8,173.75 | |
Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is: | \frac{4}{3} | 1. **Identify the product of the roots using Vieta's formulas**:
For a quadratic equation of the form $ax^2 + bx + c = 0$, the product of the roots can be given by Vieta's formulas as $\frac{c}{a}$.
Here, $a = 3$, $b = -4$, and $c = k$. Therefore, the product of the roots is:
\[
\frac{k}{3}
\]
2. **... | 1 | 2,431.8125 | 2,431.8125 | -1 |
The number of games won by five cricket teams is displayed in a chart, but the team names are missing. Use the clues below to determine how many games the Hawks won:
1. The Hawks won fewer games than the Falcons.
2. The Raiders won more games than the Wolves, but fewer games than the Falcons.
3. The Wolves won more th... | 20 | 0 | 1,366.125 | -1 | 1,366.125 | |
The fraction $\frac{1}{5}$ is written as an infinite binary fraction. How many ones are there among the first 2022 digits after the binary point in this representation? (12 points) | 1010 | 0.6875 | 4,646 | 3,034.181818 | 8,192 | |
Find the number of strictly increasing sequences of nonnegative integers with the following properties: - The first term is 0 and the last term is 12. In particular, the sequence has at least two terms. - Among any two consecutive terms, exactly one of them is even. | 144 | For a natural number $n$, let $A_{n}$ be a set containing all sequences which satisfy the problem conditions but which 12 is replaced by $n$. Also, let $a_{n}$ be the size of $A_{n}$. We first consider $a_{1}$ and $a_{2}$. We get $a_{1}=1$, as the only sequence satisfying the problem conditions is 0,1. We also get $a_{... | 0.1875 | 8,025.375 | 7,413.666667 | 8,166.538462 |
Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having side length 1. Find the difference between the sum of the areas of the squares and the sum of the areas of the circles. | 2 - \frac{\pi}{2} | The ratio of the area of each square and the circle immediately inside it is $\frac{4}{\pi}$. The total sum of the areas of the squares is $1+\frac{1}{2}+\frac{1}{4}+\ldots=2$. Difference in area is then $2-2 \cdot \frac{4}{\pi} = 2 - \frac{\pi}{2}$. | 0.6875 | 5,234.625 | 4,535.909091 | 6,771.8 |
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 s... | 15:40 | 0 | 8,192 | -1 | 8,192 | |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge. | \frac{\sqrt{3}}{4} | 0 | 8,192 | -1 | 8,192 | |
Given the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and the line $l: 4x - 5y + 40 = 0$. Is there a point on the ellipse for which the distance to line $l$ is minimal? If so, what is the minimal distance? | \frac{15}{\sqrt{41}} | 0 | 6,611.6875 | -1 | 6,611.6875 | |
Given that the golden ratio $m = \frac{{\sqrt{5}-1}}{2}$, calculate the value of $\frac{{\sin{42}°+m}}{{\cos{42}°}}$. | \sqrt{3} | 0.4375 | 7,040.6875 | 5,560.428571 | 8,192 | |
Auston is 60 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is Auston in centimeters? Express your answer as a decimal to the nearest tenth. | 152.4 | 0.875 | 1,112.6875 | 1,228.857143 | 299.5 | |
Given a geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. Given that $a_1 + a_2 + a_3 = 3$ and $a_4 + a_5 + a_6 = 6$, calculate the value of $S_{12}$. | 45 | 0.75 | 5,206.6875 | 4,211.583333 | 8,192 | |
In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$
| 112 | 0 | 8,149.875 | -1 | 8,149.875 | |
Find the smallest natural number \( n \) that satisfies the following conditions:
1. The units digit of \( n \) is 6.
2. If the units digit 6 is moved to the front of the number, the new number is 4 times \( n \). | 153846 | 0.8125 | 5,389.875 | 4,743.230769 | 8,192 |
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