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 |
|---|---|---|---|---|---|---|
A certain unit has 160 young employees. The number of middle-aged employees is twice the number of elderly employees. The total number of elderly, middle-aged, and young employees is 430. In order to understand the physical condition of the employees, a stratified sampling method is used for the survey. In a sample of ... | 18 | 0 | 5,154.3125 | -1 | 5,154.3125 | |
A nine-joint bamboo tube has rice capacities of 4.5 *Sheng* in the lower three joints and 3.8 *Sheng* in the upper four joints. Find the capacity of the middle two joints. | 2.5 | 0 | 6,496.8125 | -1 | 6,496.8125 | |
Find the value of $c$ if the roots of the quadratic $9x^2 - 5x + c$ are $\frac{-5\pm i\sqrt{415}}{18}$. | \frac{110}{9} | 0.5 | 6,909.875 | 5,627.75 | 8,192 | |
Given $\frac{e}{f}=\frac{3}{4}$ and $\sqrt{e^{2}+f^{2}}=15$, find $ef$. | 108 | We know that $\frac{e}{f}=\frac{3}{4}$ and $\sqrt{e^{2}+f^{2}}=15$. Solving for $e$ and $f$, we find that $e^{2}+f^{2}=225$, so $16 e^{2}+16 f^{2}=3600$, so $(4 e)^{2}+(4 f)^{2}=3600$, so $(3 f)^{2}+(4 f)^{2}=3600$, so $f^{2}\left(3^{2}+4^{2}\right)=3600$, so $25 f^{2}=3600$, so $f^{2}=144$ and $f=12$. Thus, $e=\frac{3... | 1 | 1,980.1875 | 1,980.1875 | -1 |
In the manufacturing of a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=18
\end{array}\right.
$$
Find the length of the cable.
| 4 \pi \sqrt{\frac{23}{3}} | 0 | 6,930.4375 | -1 | 6,930.4375 | |
Given that $n \in \mathbb{N}^*$, the coefficient of the second term in the expansion of $(x+2)^n$ is $\frac{1}{5}$ of the coefficient of the third term.
(1) Find the value of $n$;
(2) Find the term with the maximum binomial coefficient in the expansion;
(3) If $(x+2)^n = a\_0 + a\_1(x+1) + a\_2(x+1)^2 + \dots + a\_n(x+... | 64 | 0.625 | 5,535 | 5,078.6 | 6,295.666667 | |
Given that when $(a+b+c+d+e+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly 2002 terms that include all five variables $a, b, c, d, e$, each to some positive power, find the value of $N$. | 16 | 0 | 4,867.125 | -1 | 4,867.125 | |
Find the minimum value of
\[(15 - x)(8 - x)(15 + x)(8 + x).\] | -6480.25 | 0.0625 | 7,168.9375 | 5,334 | 7,291.266667 | |
If a $5\times 5$ chess board exists, in how many ways can five distinct pawns be placed on the board such that each column and row contains no more than one pawn? | 14400 | 0.6875 | 4,714.125 | 3,937.727273 | 6,422.2 | |
Given $a \in \{0,1,2\}, b \in \{-1,1,3,5\}$, determine the probability that the function $f(x)=ax^{2}-2bx$ is increasing in the interval $(1,+\infty)$. | \dfrac{5}{12} | 0.5625 | 5,917.625 | 5,195.111111 | 6,846.571429 | |
Given the domain of the function $f(x)$ is $(4a-3,3-2a^{2})$, where $a\in \mathbb{R}$, and $y=f(2x-3)$ is an even function. If $B_{n}=1\times a^{1}+4\times a^{2}+7\times a^{3}+\cdots +(3n-2)a^{n}$, then $B_{50}=$ ? | 75 | 0.125 | 7,013.4375 | 5,067 | 7,291.5 | |
Let $p(x) = 2x - 7$ and $q(x) = 3x - b$. If $p(q(4)) = 7$, what is $b$? | 5 | 1 | 1,357.3125 | 1,357.3125 | -1 | |
Divide the product of the first five positive composite integers by the product of the next five composite integers. Express your answer as a common fraction. | \frac{1}{42} | 0.875 | 5,001.6875 | 4,914.214286 | 5,614 | |
Calculate $7 \cdot 9\frac{2}{5}$. | 65\frac{4}{5} | 0.625 | 1,655.3125 | 1,974.4 | 1,123.5 | |
What is the coefficient of $x^3$ in the product of the polynomials $$x^4 - 2x^3 + 3x^2 - 4x + 5$$ and $$3x^3 - 4x^2 + x + 6$$ after combining like terms? | 22 | 0.75 | 4,655.75 | 3,968.166667 | 6,718.5 | |
Let $a$ and $b$ be real numbers such that $a + b = 4.$ Find the maximum value of
\[a^4 b + a^3 b + a^2 b + ab + ab^2 + ab^3 + ab^4.\] | \frac{7225}{56} | 0 | 8,192 | -1 | 8,192 | |
Given a sequence $1$, $1$, $3$, $1$, $3$, $5$, $1$, $3$, $5$, $7$, $1$, $3$, $5$, $7$, $9$, $\ldots$, where the first term is $1$, the next two terms are $1$, $3$, and the next three terms are $1$, $3$, $5$, and so on. Let $S_{n}$ denote the sum of the first $n$ terms of this sequence. Find the smallest positive intege... | 59 | 0.5 | 7,245.5625 | 6,453 | 8,038.125 | |
Given real numbers $a$ and $b$ satisfying $a^{2}-4\ln a-b=0$, find the minimum value of $\left(a-c\right)^{2}+\left(b+2c\right)^{2}$. | \frac{9}{5} | 0.625 | 6,953.875 | 6,211 | 8,192 | |
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i] | f(x) = 2 - x \text{ and } f(x) = x |
To solve the functional equation:
\[
f(x + f(x+y)) + f(xy) = x + f(x+y) + yf(x)
\]
for all \( x, y \in \mathbb{R} \), we start by considering particular values for \( x \) and \( y \) to simplify the equation and gain insight into the form of the function \( f \).
### Step 1: Substitute \( y = 0 \)
Let \( y = 0 \)... | 0 | 8,089.875 | -1 | 8,089.875 |
Let \( T \) be the set of positive real numbers. Let \( g : T \to \mathbb{R} \) be a function such that
\[ g(x) g(y) = g(xy) + 2006 \left( \frac{1}{x} + \frac{1}{y} + 2005 \right) \] for all \( x, y > 0 \).
Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of... | \frac{6019}{3} | 0.0625 | 8,192 | 8,192 | 8,192 | |
In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 2\angle D$, and $\angle C = 3\angle B$. Find $\angle B$. | 45^\circ | 0.75 | 3,816.9375 | 2,720.083333 | 7,107.5 | |
Given $|m|=3$, $|n|=2$, and $m<n$, find the value of $m^2+mn+n^2$. | 19 | 0.4375 | 3,028.625 | 3,861.571429 | 2,380.777778 | |
Let the function \( f(x) = 4x^3 + bx + 1 \) with \( b \in \mathbb{R} \). For any \( x \in [-1, 1] \), \( f(x) \geq 0 \). Find the range of the real number \( b \). | -3 | 0.125 | 7,748 | 8,051 | 7,704.714286 | |
A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is | 30\pi - 9\sqrt3 | 1. **Identify the dimensions of the plot and path**: The circular grass plot has a diameter of 12 feet, so its radius \( r \) is \( \frac{12}{2} = 6 \) feet. The gravel path is 3 feet wide.
2. **Calculate the area of the entire circle**: The area \( A \) of a circle is given by the formula \( A = \pi r^2 \). Substitut... | 0.1875 | 7,534.3125 | 7,185.333333 | 7,614.846154 |
In triangle \( \triangle ABC \), \( AB = BC = 2 \) and \( AC = 3 \). Let \( O \) be the incenter of \( \triangle ABC \). If \( \overrightarrow{AO} = p \overrightarrow{AB} + q \overrightarrow{AC} \), find the value of \( \frac{p}{q} \). | 2/3 | 0 | 4,195.8125 | -1 | 4,195.8125 | |
We inscribe spheres with a radius of \(\frac{1}{2}\) around the vertices of a cube with edge length 1. There are two spheres that touch each of these eight spheres. Calculate the difference in volume between these two spheres. | \frac{10}{3} \pi | 0.0625 | 7,889.5 | 3,352 | 8,192 | |
There was a bonus fund in a certain institution. It was planned to distribute the fund such that each employee of the institution would receive $50. However, it turned out that the last employee on the list would receive only $45. Then, in order to ensure fairness, it was decided to give each employee $45, leaving $95 ... | 950 | 0.125 | 4,335.75 | 3,648 | 4,434 | |
Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\). | \frac{3\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
The numbers \( x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3 \) are equal to the numbers \( 1, 2, 3, \ldots, 9 \) in some order. Find the smallest possible value of
\[ x_1 x_2 x_3 + y_1 y_2 y_3 + z_1 z_2 z_3. \] | 214 | 0.125 | 8,192 | 8,192 | 8,192 | |
An escalator has \( n \) visible steps and descends at a constant speed. Two boys, \( A \) and \( Z \), walk down the moving escalator at a steady pace. Boy \( A \) walks twice as many steps per minute as boy \( Z \). \( A \) reaches the bottom after walking 27 steps, and \( Z \) reaches the bottom after walking 18 ste... | 54 | 0.8125 | 4,205.6875 | 3,285.769231 | 8,192 | |
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved tw... | 2 | To solve this problem, we need to analyze the movements of each friend and determine the original seating arrangement based on the given movements.
1. **Initial Setup**: There are 5 seats, and each friend occupies one seat. Ada leaves, creating one empty seat.
2. **Movements**:
- **Bea** moves two seats to the rig... | 0.0625 | 7,973.8125 | 4,701 | 8,192 |
Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score? | 171 | 1 | 1,385.75 | 1,385.75 | -1 | |
Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line ... | 2x-3y=0 | 1. **Identify the given information:**
- Line $\ell$ has the equation $5x - y = 0$.
- Point $P$ has coordinates $(-1, 4)$.
- Point $P''$ has coordinates $(4, 1)$.
- Lines $\ell$ and $m$ intersect at the origin $O$.
2. **Understand the geometric transformations:**
- $P$ is reflected about line $\ell$ to ... | 0.1875 | 7,112.4375 | 6,394.666667 | 7,278.076923 |
Given $S$, $P$ (not the origin) are two different points on the parabola $y=x^{2}$, the tangent line at point $P$ intersects the $x$ and $y$ axes at $Q$ and $R$, respectively.
(Ⅰ) If $\overrightarrow{PQ}=\lambda \overrightarrow{PR}$, find the value of $\lambda$;
(Ⅱ) If $\overrightarrow{SP} \perp \overrightarrow{PR}$,... | \frac{4\sqrt{3}}{9} | 0 | 6,730.0625 | -1 | 6,730.0625 | |
In the Cartesian coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $P$ are $(3, \frac{\pi}{4})$. The parametric equation of curve $C$ is $\rho=2\cos (\theta- \frac{\pi}{4})$ (with $\the... | \frac{\sqrt{10}-1}{2} | 0 | 7,065.875 | -1 | 7,065.875 | |
The area of triangle \( ABC \) is 1. On the rays \( AB \), \( BC \), and \( CA \), points \( B' \), \( C' \), and \( A' \) are marked respectively, such that:
\[ BB' = AB, \quad CC' = 2BC, \quad AA' = 3CA \]
Calculate the area of triangle \( A'B'C' \). | 18 | 0.375 | 7,784.125 | 7,104.333333 | 8,192 | |
Let $f(x)=|x-p|+|x-15|+|x-p-15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \leq x\leq15$. | 15 | It is best to get rid of the absolute values first.
Under the given circumstances, we notice that $|x-p|=x-p$, $|x-15|=15-x$, and $|x-p-15|=15+p-x$.
Adding these together, we find that the sum is equal to $30-x$, which attains its minimum value (on the given interval $p \leq x \leq 15$) when $x=15$, giving a minimum ... | 0.9375 | 5,026.0625 | 4,815 | 8,192 |
Given the general term formula of the sequence $\{a\_n\}$, where $a\_n=n\cos \frac {nπ}{2}$, and the sum of the first $n$ terms is represented by $S\_n$, find the value of $S\_{2016}$. | 1008 | 0.5625 | 6,151.4375 | 5,265.555556 | 7,290.428571 | |
Let $f_1(x)=x^2-1$ , and for each positive integer $n \geq 2$ define $f_n(x) = f_{n-1}(f_1(x))$ . How many distinct real roots does the polynomial $f_{2004}$ have? | 2005 | 0 | 8,192 | -1 | 8,192 | |
What is the smallest positive value of $x$ such that $x + 5678$ results in a palindrome? | 97 | 0.4375 | 7,195.8125 | 6,106 | 8,043.444444 | |
Let $\mathbf{a} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$ | 9 | 1 | 1,989.875 | 1,989.875 | -1 | |
A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result o... | 16 | 0.0625 | 8,124.1875 | 7,107 | 8,192 | |
Given that $x$ varies directly as the square of $y$, and $y$ varies directly as the cube root of $z$, determine the power $n$ such that $x$ varies as $x^n$. | \frac{2}{3} | 0.9375 | 3,313.375 | 3,150.2 | 5,761 | |
Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00. Each stays for 15 minutes, then leaves. What is the probability that Allen and Bethany see each other at the party? | \frac{7}{16} | 0.875 | 5,014.8125 | 4,560.928571 | 8,192 | |
Car X is traveling at a constant speed of 90 km/h and has a length of 5 meters, while Car Y is traveling at a constant speed of 91 km/h and has a length of 6 meters. Given that Car Y starts behind Car X and eventually passes Car X, calculate the length of time between the instant when the front of Car Y is lined up wit... | 39.6 | 0.8125 | 4,519.3125 | 3,772.230769 | 7,756.666667 | |
Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge... | \frac{25}{32} | To solve this problem, we will calculate the probability that Frieda reaches a corner square within four hops, starting from the center of a $3 \times 3$ grid. We will use a state-based approach to model Frieda's possible positions and transitions.
#### Definitions:
- **State**: Represents Frieda's position on the gri... | 0 | 7,983 | -1 | 7,983 |
Let $T$ be the set of all complex numbers $z$ where $z = w - \frac{1}{w}$ for some complex number $w$ of absolute value $2$. Determine the area inside the curve formed by $T$ in the complex plane. | \frac{9}{4} \pi | 0 | 4,386.9375 | -1 | 4,386.9375 | |
For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? | 231 | 0 | 6,917.4375 | -1 | 6,917.4375 | |
For what value of $k$ does the equation $x^2+10x+y^2+6y-k=0$ represent a circle of radius 6? | 2 | 1 | 1,482.9375 | 1,482.9375 | -1 | |
The apex of a regular pyramid with a square base $ABCD$ of unit side length is $E$. Point $P$ lies on the base edge $AB$ and point $Q$ lies on the lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Additionally, we know that $AP : PB = 6 : 1$. What are the lengths of the lateral edges? | \sqrt{2} | 0.875 | 3,848.375 | 3,620.285714 | 5,445 | |
Shuxin begins with 10 red candies, 7 yellow candies, and 3 blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the smallest possible number of candies that Shuxin ate? | 11 | For there to be equal numbers of each colour of candy, there must be at most 3 red candies and at most 3 yellow candies, since there are 3 blue candies to start. Thus, Shuxin ate at least 7 red candies and at least 4 yellow candies. This means that Shuxin ate at least $7+4=11$ candies. We note that if Shuxin eats 7 red... | 0.8125 | 4,476.9375 | 3,619.615385 | 8,192 |
Find the constant $b$ such that
$$\left(5x^2-3x+\frac{7}{3}\right)(ax^2+bx+c) = 15x^4 - 14x^3 + 20x^2 - \frac{25}{3}x + \frac{14}{3}$$ | -1 | 1 | 2,891 | 2,891 | -1 | |
If $8 \tan \theta = 3 \cos \theta$ and $0 < \theta < \pi,$ then determine the value of $\sin \theta.$ | \frac{1}{3} | 1 | 3,799.75 | 3,799.75 | -1 | |
Denote by \( f(n) \) the integer obtained by reversing the digits of a positive integer \( n \). Find the greatest integer that is certain to divide \( n^{4} - f(n)^{4} \) regardless of the choice of \( n \). | 99 | 0.0625 | 7,957.125 | 5,258 | 8,137.066667 | |
Given the function $f(x)=-\frac{1}{3}x^{3}+bx^{2}+cx+bc$ has an extreme value of $-\frac{4}{3}$ at $x=1$, find the value of $b$. | -1 | 0.8125 | 4,986.625 | 4,347.615385 | 7,755.666667 | |
Let $f(x)=x^{2}-2 x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c))))=3$ ? | 9 | We see the size of the set $f^{-1}\left(f^{-1}\left(f^{-1}\left(f^{-1}(3)\right)\right)\right)$. Note that $f(x)=(x-1)^{2}-1=3$ has two solutions: $x=3$ and $x=-1$, and that the fixed points $f(x)=x$ are $x=3$ and $x=0$. Therefore, the number of real solutions is equal to the number of distinct real numbers $c$ such th... | 0.125 | 7,784.8125 | 4,934.5 | 8,192 |
On each of the first three days of January, there is a $\frac{1}{3}$ chance that it will snow where Bob lives. On each of the next four days, there is a $\frac{1}{4}$ chance that it will snow. What is the probability that it snows at least once during the first week of January? | \frac{29}{32} | 0.9375 | 3,249.375 | 2,919.866667 | 8,192 | |
There are 2 dimes of Chinese currency, how many ways can they be exchanged into coins (1 cent, 2 cents, and 5 cents)? | 28 | 0 | 5,224.75 | -1 | 5,224.75 | |
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 ... | 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 poss... | 0 | 8,192 | -1 | 8,192 |
$ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$ , respectively. Find the largest possible value of $AB$ .
*Team #6* | 58\sqrt{3} | 0 | 8,189.875 | -1 | 8,189.875 | |
Points $E$ and $F$ lie on $\overline{GH}$. The length of $\overline{GE}$ is $3$ times the length of $\overline{EH}$, and the length of $\overline{GF}$ is $5$ times the length of $\overline{FH}$. Determine the length of $\overline{EF}$ as a fraction of the length of $\overline{GH}$.
A) $\frac{1}{10}$
B) $\frac{1}{12}$... | \frac{1}{12} | 0 | 3,247.8125 | -1 | 3,247.8125 | |
George has an unfair six-sided die. The probability that it rolls a 6 is $\frac{1}{2}$, and the probability that it rolls any other number is $\frac{1}{10}$. What is the expected value of the number shown when this die is rolled? Express your answer as a decimal. | 4.5 | 1 | 2,907.6875 | 2,907.6875 | -1 | |
The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape? | 14 | 0.0625 | 7,133.375 | 7,177 | 7,130.466667 | |
Calculate the value of $\text{rem} \left(\frac{5}{7}, \frac{3}{4}\right)$ and then multiply the result by $-2$. | -\frac{10}{7} | 0.625 | 2,946.5625 | 4,146.7 | 946.333333 | |
One hundred bricks, each measuring $3''\times 8''\times 15''$, are stacked to form a tower. Each brick can contribute $3''$, $8''$, or $15''$ to the height of the tower. How many different tower heights can be achieved using all one hundred bricks? | 1201 | 0 | 8,003.75 | -1 | 8,003.75 | |
Given a $4 \times 4$ square grid partitioned into $16$ unit squares, each of which is painted white or black with a probability of $\frac{1}{2}$, determine the probability that the grid is entirely black after a $90^{\circ}$ clockwise rotation and any white square landing in a position previously occupied by a black sq... | \frac{1}{65536} | 0.0625 | 7,424.6875 | 5,050 | 7,583 | |
On hypotenuse $AB$ of a right triangle $ABC$ a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $BC=1$, $AC=b$, and $AD=2$, then $BD$ equals: | \sqrt{b^2-3} | 1. **Identify the triangles and their properties**: We have two right triangles, $ABC$ and $ABD$, sharing the hypotenuse $AB$. Triangle $ABC$ has legs $\overline{BC} = 1$ and $\overline{AC} = b$. Triangle $ABD$ has one leg $\overline{AD} = 2$.
2. **Apply the Pythagorean Theorem to triangle $ABC$**:
\[
AB^2 = AC^... | 0.625 | 7,274.625 | 6,724.2 | 8,192 |
The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. | 44 | 1 | 1,986.3125 | 1,986.3125 | -1 | |
A new solid $T$ is defined by the set of all points $(x, y, z)$ in space such that $|x| + |y| + |z| \leq 2$. Find the volume of solid $T$. | \frac{32}{3} | 0.875 | 4,618.5 | 4,108 | 8,192 | |
If $a = \log 8$ and $b = \log 25,$ compute
\[5^{a/b} + 2^{b/a}.\] | 2 \sqrt{2} + 5^{2/3} | 0.0625 | 7,333.8125 | 7,308 | 7,335.533333 | |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 3000$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 3000... | 54 | 0.0625 | 8,078.0625 | 8,192 | 8,070.466667 | |
On the refrigerator, MATHCOUNTS is spelled out with 10 magnets, one letter per magnet. Two vowels and three consonants fall off and are put away in a bag. If the Ts are indistinguishable, how many distinct possible collections of letters could be put in the bag? | 75 | 0.4375 | 7,201.25 | 6,483.142857 | 7,759.777778 | |
What is $\left(\frac{6}{7}\right)^2 \cdot \left(\frac{1}{2}\right)^2$? | \frac{9}{49} | 0.9375 | 2,618.625 | 2,247.066667 | 8,192 | |
Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? | 25 | 1. **Convert the work day into minutes**:
Makarla's work day is $9$ hours long. Since there are $60$ minutes in an hour, the total number of minutes in her work day is:
\[
9 \times 60 = 540 \text{ minutes}
\]
2. **Calculate the total time spent in meetings**:
- The duration of the first meeting is $45$... | 1 | 1,404.5625 | 1,404.5625 | -1 |
If $y=f(x)$ has an inverse function $y=f^{-1}(x)$, and $y=f(x+2)$ and $y=f^{-1}(x-1)$ are inverse functions of each other, then $f^{-1}(2007)-f^{-1}(1)=$ . | 4012 | 0.6875 | 6,108.3125 | 5,161.181818 | 8,192 | |
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{36 - 4\sqrt{5}} \sin x + 2) \cdot (3 + 2\sqrt{10 - \sqrt{5}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number. | -27 | 0.0625 | 8,159.5625 | 7,673 | 8,192 | |
Given that a new kitchen mixer is listed in a store for $\textdollar 129.99$ and an online advertisement offers the same mixer for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling fee of $\textdollar 19.99$, calculate how many cents are saved by purchasing the mixer through the online adve... | 996 | 0.875 | 487.4375 | 473.357143 | 586 | |
A shooter has a probability of hitting the target of $0.8$ each time. Now, using the method of random simulation to estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots: first, use a calculator to generate random integers between $0$ and $9$, where $0$, $1$ represent missing the... | 0.75 | 0.125 | 6,325.6875 | 6,073.5 | 6,361.714286 | |
Given vectors $\overrightarrow{a}=(\cos α,\sin α)$ and $\overrightarrow{b}=(-2,2)$.
(1) If $\overrightarrow{a}\cdot \overrightarrow{b}= \frac {14}{5}$, find the value of $(\sin α+\cos α)^{2}$;
(2) If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\sin (π-α)\cdot\sin ( \frac {π}{2}+α)$. | -\frac{1}{2} | 0.8125 | 4,386.5625 | 3,899.769231 | 6,496 | |
In rectangle \(ABCD\), \(AB = 2\) and \(AD = 1\), point \(P\) is a moving point on side \(DC\) (including points \(D\) and \(C\)), and point \(Q\) is a moving point on the extension line of \(CB\) (including point \(B\)), such that \(|\overrightarrow{DP}| = |\overrightarrow{BQ}|\). Determine the minimum value of the do... | 3/4 | 0.4375 | 4,667.25 | 3,815.714286 | 5,329.555556 | |
A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of $ABCD$. | 4(36 + \sqrt{113}) | 1. **Area and Diagonal Relationship**:
Given that the area of convex quadrilateral $ABCD$ is $2002$, we can use the inequality for the area of a convex quadrilateral split by its diagonals:
\[
[ABCD] \leq \frac{1}{2} (AC \cdot BD)
\]
This inequality holds because the area of a triangle is given by $\fr... | 0 | 7,810.625 | -1 | 7,810.625 |
The four zeros of the polynomial \(x^4 + px^2 + qx - 144\) are distinct real numbers in arithmetic progression. Compute the value of \(p.\) | -40 | 0 | 8,192 | -1 | 8,192 | |
Let the polynomial be defined as $$Q(x) = \left(\frac{x^{20} - 1}{x-1}\right)^2 - x^{20}.$$ Calculate the sum of the first five distinct $\alpha_k$ values where each zero of $Q(x)$ can be expressed in the complex form $z_k = r_k [\cos(2\pi \alpha_k) + i\sin(2\pi \alpha_k)]$, with $\alpha_k \in (0, 1)$ and $r_k > 0$. | \frac{3}{4} | 0 | 8,192 | -1 | 8,192 | |
A certain brand of computers has a warranty period of $1$ year. Based on a large amount of repair record data, the maximum number of repairs for this brand of computers within one year is $3$ times, with $15\%$ needing $1$ repair, $6\%$ needing $2$ repairs, and $4\%$ needing $3$ repairs. <br/>$(1)$ If a person buys $1$... | 0.9 | 0.125 | 6,021.8125 | 4,308 | 6,266.642857 | |
If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area 6, then $ab=$ | 3 | 1. **Identify the intercepts**:
The equation of the line is given by $ax + by = 6$. To find the $x$-intercept, set $y = 0$:
\[
ax = 6 \implies x = \frac{6}{a}.
\]
Similarly, to find the $y$-intercept, set $x = 0$:
\[
by = 6 \implies y = \frac{6}{b}.
\]
2. **Calculate the area of the triangle**... | 1 | 1,577.5 | 1,577.5 | -1 |
Regular hexagon $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6}$ has side length 2. For $1 \leq i \leq 6$, let $C_{i}$ be a unit circle centered at $P_{i}$ and $\ell_{i}$ be one of the internal common tangents of $C_{i}$ and $C_{i+2}$, where $C_{7}=C_{1}$ and $C_{8}=C_{2}$. Assume that the lines $\{\ell_{1}, \ell_{2}, \ell_{3}, \... | 1603 | The only way for the lines $\ell_{i}$ to bound a regular hexagon $H$ is if they are rotationally symmetric around the center $O$ of the original hexagon. (A quick way to see this is to note that the angle between the two internal common tangents of $C_{i}$ and $C_{i+2}$ cannot be a multiple of $60^{\circ}$.) Thus all w... | 0 | 8,192 | -1 | 8,192 |
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $50^{\mbox{th}}$ term of this sequence. | 327 | 0.6875 | 5,651.0625 | 4,889.090909 | 7,327.4 | |
Define the sequence $\{x_{i}\}_{i \geq 0}$ by $x_{0}=2009$ and $x_{n}=-\frac{2009}{n} \sum_{k=0}^{n-1} x_{k}$ for all $n \geq 1$. Compute the value of $\sum_{n=0}^{2009} 2^{n} x_{n}$ | 2009 | We have $-\frac{n x_{n}}{2009}=x_{n-1}+x_{n-2}+\ldots+x_{0}=x_{n-1}+\frac{(n-1) x_{n-1}}{2009}$, which yields the recursion $x_{n}=\frac{n-2010}{n} x_{n-1}$. Unwinding this recursion, we find $x_{n}=(-1)^{n} \cdot 2009$. $\binom{2008}{n}$. Thus $\sum_{k=0}^{2009} 2^{n} x_{n} =\sum_{k=0}^{2009}(-2)^{n} \cdot 2009 \cdot\... | 0 | 8,192 | -1 | 8,192 |
Place the numbers $1, 2, 3, \cdots, 2001$ in a clockwise direction on a circle. First, eliminate the number 2. Then proceed to eliminate every second number in a clockwise direction until only one number remains. What is the last remaining number? | 1955 | 0.3125 | 7,180.375 | 4,954.8 | 8,192 | |
A circle with radius 4 cm is tangent to three sides of a rectangle, as shown. The area of the rectangle is twice the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$.
[asy]
import graph;
draw((0,0)--(30,0)--(30,20)--(0,20)--cycle);
draw(C... | 4\pi | 0.4375 | 6,942.1875 | 5,512.285714 | 8,054.333333 | |
Given a seminar recording of 495 minutes that needs to be divided into multiple USB sticks, each capable of holding up to 65 minutes of audio, and the minimum number of USB sticks is used, calculate the length of audio that each USB stick will contain. | 61.875 | 0.1875 | 545 | 494 | 556.769231 | |
Sector $OAB$ is a quarter of a circle with radius 5 cm. Inside this sector, a circle is inscribed, tangent at three points. Find the radius of the inscribed circle in simplest radical form. | 5\sqrt{2} - 5 | 0.0625 | 4,276.875 | 6,279 | 4,143.4 | |
Translate the graph of $y= \sqrt {2}\sin (2x+ \frac {\pi}{3})$ to the right by $\phi(0 < \phi < \pi)$ units to obtain the graph of the function $y=2\sin x(\sin x-\cos x)-1$. Then, $\phi=$ ______. | \frac {13\pi}{24} | 0.75 | 6,451 | 5,870.666667 | 8,192 | |
Outstanding Brazilian footballer Ronaldinho Gaúcho will be $X$ years old in the year $X^{2}$. How old will he be in 2018, when the World Cup is held in Russia? | 38 | 0.875 | 5,545.8125 | 5,167.785714 | 8,192 | |
If $\sqrt2 \sin 10^\circ$ can be written as $\cos \theta - \sin\theta$ for some acute angle $\theta,$ what is $\theta?$ (Give your answer in degrees, not radians.) | 35^\circ | 0.9375 | 3,791.4375 | 3,498.066667 | 8,192 | |
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (... | \frac{5}{8} | We are given a problem where a frog jumps randomly in one of the four cardinal directions from a point $(1,2)$ within a square bounded by $(0,0), (0,4), (4,4),$ and $(4,0)$. We need to find the probability that the frog first reaches a vertical side of the square.
Let $P_{(x,y)}$ denote the probability that the frog's... | 0 | 8,192 | -1 | 8,192 |
Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares. | $4(n-k)$ |
Let \( n \) and \( k \) be positive integers such that \( \frac{1}{2}n < k \leq \frac{2}{3}n \). Our goal is to find the least number \( m \) for which it is possible to place \( m \) pawns on an \( n \times n \) chessboard such that no column or row contains a block of \( k \) adjacent unoccupied squares.
### Analys... | 0 | 8,192 | -1 | 8,192 |
Find the ratio of the volume of the cone to the volume of the cylinder. Express your answer as a common fraction.
[asy]
import solids; size(150); import three; defaultpen(linewidth(0.8)); currentprojection = orthographic(5,0,3);
revolution c = cylinder((0,0,0), 1, 3);
revolution c2 = cone((0,0,0), 1,1.5);
draw(c,black... | \frac{1}{6} | 0.6875 | 3,718.1875 | 2,684.545455 | 5,992.2 | |
At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but carefully observes its comfort level. If the co... | \frac{1}{2} | 0 | 8,053.375 | -1 | 8,053.375 | |
The Screamers are coached by Coach Yellsalot. The Screamers have 12 players, but two of them, Bob and Yogi, refuse to play together. How many starting lineups (of 5 players) can Coach Yellsalot make, if the starting lineup can't contain both Bob and Yogi? (The order of the 5 players in the lineup does not matter; th... | 672 | 0.8125 | 3,880.9375 | 3,086.076923 | 7,325.333333 |
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