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 cone is formed from a 270-degree sector of a circle of radius 18 by aligning the two straight sides. What is the result when the volume of the cone is divided by $\pi$? | 60.75\sqrt{141.75} | 0 | 5,873.9375 | -1 | 5,873.9375 | |
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 - x - 1 = 0$, find the value of $\frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma}$. | -7 | 0.75 | 6,421.9375 | 5,831.916667 | 8,192 | |
Equilateral $\triangle ABC$ has side length $2$, and shapes $ABDE$, $BCHT$, $CAFG$ are formed outside the triangle such that $ABDE$ and $CAFG$ are squares, and $BCHT$ is an equilateral triangle. What is the area of the geometric shape formed by $DEFGHT$?
A) $3\sqrt{3} - 1$
B) $3\sqrt{3} - 2$
C) $3\sqrt{3} + 2$
D) $4\sq... | 3\sqrt{3} - 2 | 0 | 8,192 | -1 | 8,192 | |
5 people are standing in a row for a photo, among them one person must stand in the middle. There are ways to arrange them. | 24 | 0.75 | 4,666 | 3,794.833333 | 7,279.5 | |
Given sets $A=\{1,2,3,4,5\}$, $B=\{0,1,2,3,4\}$, and a point $P$ with coordinates $(m,n)$, where $m\in A$ and $n\in B$, find the probability that point $P$ lies below the line $x+y=5$. | \dfrac{2}{5} | 1 | 4,050.4375 | 4,050.4375 | -1 | |
How many integers between 100 and 300 are multiples of both 5 and 7, but are not multiples of 10? | 3 | The integers that are multiples of both 5 and 7 are the integers that are multiples of 35. The smallest multiple of 35 greater than 100 is $3 imes 35=105$. Starting at 105 and counting by 35s, we obtain 105, 140, 175, 210, 245, 280, 315. The integers in this list that are between 100 and 300 and are not multiples of 1... | 0.8125 | 4,317.25 | 3,492.538462 | 7,891 |
From a group of $3$ orthopedic surgeons, $4$ neurosurgeons, and $5$ internists, a medical disaster relief team of $5$ people is to be formed. How many different ways can the team be selected such that there is at least one person from each specialty? | 590 | 0.5625 | 5,883.4375 | 4,087.888889 | 8,192 | |
In the figure, $PA$ is tangent to semicircle $SAR$, $PB$ is tangent to semicircle $RBT$, and $SRT$ is a straight line. If arc $AS$ is $58^\circ$ and arc $BT$ is $37^\circ$, then find $\angle APB$, in degrees.
[asy]
import graph;
unitsize(1.5 cm);
pair A, B, P, R, S, T;
pair[] O;
real[] r;
r[1] = 1;
r[2] = 0.8;
S ... | 95^\circ | 0.0625 | 7,882 | 7,637 | 7,898.333333 | |
An 8.5-meter ladder is leaning against a vertical wall. How many meters is its base from the wall if the ladder reaches 7.5 meters up the wall? | 4 | 1 | 1,750.1875 | 1,750.1875 | -1 | |
Simplify the expression $(-\frac{1}{343})^{-2/3}$. | 49 | 1 | 3,732.875 | 3,732.875 | -1 | |
Given the function $f(x)=\frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))$ where $n \in \mathbb{N}^{*}$, vector $\overrightarrow{a}=(0,1)$, and $\theta_{n}$ is the angle between vector $\overrightarrow{OA}_{n}$ and $\overrightarrow{a}$. Compute the value of $\frac{\cos \theta_{1}}{\sin \theta_{1... | \frac{2016}{2017} | 0.75 | 3,986.5 | 2,843.166667 | 7,416.5 | |
Evaluate the expression \[ \frac{a+3}{a+1} \cdot \frac{b-2}{b-3} \cdot \frac{c + 9}{c+7} , \] given that $c = b-11$, $b = a+3$, $a = 5$, and none of the denominators are zero. | \frac{1}{3} | 0 | 3,198.6875 | -1 | 3,198.6875 | |
Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n$, where $n=1,2,\cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: | 1 | To solve for $S_n = 1 - 2 + 3 - 4 + \cdots + (-1)^{n-1}n$, we need to consider the cases when $n$ is even and when $n$ is odd separately.
1. **Case 1: $n$ is even.**
- Pair the terms: $(1-2) + (3-4) + \cdots + (n-1 - n)$.
- Each pair sums to $-1$.
- There are $n/2$ pairs.
- Therefore, $S_n = -1 \times (n/2... | 1 | 3,764 | 3,764 | -1 |
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her ... | 31 | 0.375 | 6,312.9375 | 5,982.666667 | 6,511.1 | |
A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock. | 12.925 | 0.6875 | 6,623.0625 | 6,458.090909 | 6,986 | |
Let $n \ge 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is $orderly$ if:
no matter how Rowan permutes the rows ... | \[ 2 \cdot n! + 2 \] | We focus on the leftmost column for simplicity. Let $m$ be the number of red squares in this column. We then have five cases:
1. $m=1$
When Rowan permutes the rows of the coloring, we consider only the first column, which by the above contains $m=1$ red colors, so there are ${n \choose 1}=n$ ways to permute the first... | 0 | 8,192 | -1 | 8,192 |
A scale drawing of a park shows that one inch represents 800 feet. A line segment in the drawing that is 4.75 inches long represents how many feet? | 3800 | 0.625 | 562.25 | 721.3 | 297.166667 | |
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$ | 1011 | 0.0625 | 8,056.75 | 8,192 | 8,047.733333 | |
Two numbers \( x \) and \( y \) satisfy the equation \( 26x^2 + 23xy - 3y^2 - 19 = 0 \) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | -3 | 0.1875 | 7,699.5625 | 6,290.666667 | 8,024.692308 | |
Suppose $a_{1}, a_{2}, \ldots, a_{100}$ are positive real numbers such that $$a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}$$ for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$. | 215 | If we cross multiply, we obtain $a_{n} a_{n-1}=n a_{n-1}+(n-1) a_{n}$, which we can rearrange and factor as $\left(a_{n}-n\right)\left(a_{n-1}-(n-1)\right)=n(n-1)$. Let $b_{n}=a_{n}-n$. Then, $b_{n} b_{n-1}=n(n-1)$. If we let $b_{1}=t$, then we have by induction that $b_{n}=n t$ if $n$ is odd and $b_{n}=n / t$ if $n$ i... | 0.375 | 6,395.0625 | 4,247.833333 | 7,683.4 |
The cross below is made up of five congruent squares. The perimeter of the cross is $72$ . Find its area.
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(2... | 180 | 0.5 | 6,988.1875 | 6,385.25 | 7,591.125 | |
In the trapezoid $ABCD$, $CD$ is three times the length of $AB$. Given that the area of the trapezoid is $30$ square units, determine the area of $\triangle ABC$.
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((1,4)--(0,0));
label("$A$",(1,4),NW);
label("$B$",(7,4),NE);
label("$C$",(12,0),E);
label("$D$",(0,0),W... | 7.5 | 0.0625 | 5,869.5 | 4,378 | 5,968.933333 | |
In Mrs. Marsh's algebra class, 30 of the 36 students took the Chapter 4 test, and their mean score was $72 \%$. The next day the other six students took the test and their mean score was $78 \%$. What is the new class mean? Express your answer as a percent. | 73\% | 1 | 1,836.625 | 1,836.625 | -1 | |
The surface of a clock is circular, and on its circumference, there are 12 equally spaced points representing the hours. Calculate the total number of rectangles that can have these points as vertices. | 15 | 0.0625 | 8,028.3125 | 5,573 | 8,192 | |
Four families visit a tourist spot that has four different routes available for exploration. Calculate the number of scenarios in which exactly one route is not visited by any of the four families. | 144 | 0.625 | 6,720.625 | 5,837.8 | 8,192 | |
Given in the polar coordinate system, circle $C$: $p=2\cos (\theta+ \frac {\pi}{2})$ and line $l$: $\rho\sin (\theta+ \frac {\pi}{4})= \sqrt {2}$, point $M$ is a moving point on circle $C$. Find the maximum distance from point $M$ to line $l$. | \frac {3 \sqrt {2}}{2}+1 | 0 | 6,565.3125 | -1 | 6,565.3125 | |
Let $u$ and $v$ be real numbers satisfying the inequalities $2u + 3v \le 10$ and $4u + v \le 9.$ Find the largest possible value of $u + 2v$. | 6.1 | 0 | 6,986.6875 | -1 | 6,986.6875 | |
What is the area of the quadrilateral formed by the points of intersection of the circle \(x^2 + y^2 = 16\) and the ellipse \((x-3)^2 + 4y^2 = 36\). | 14 | 0 | 8,192 | -1 | 8,192 | |
Let $f(x)=\frac{3x^2+5x+8}{x^2-x+4}$ and $g(x)=x-1$. Find $f(g(x))+g(f(x))$, evaluated when $x=1$. | 5 | 0.9375 | 4,511.75 | 4,531.466667 | 4,216 | |
Nine balls numbered $1, 2, \cdots, 9$ are placed in a bag. These balls differ only in their numbers. Person A draws a ball from the bag, the number on the ball is $a$, and after returning it to the bag, person B draws another ball, the number on this ball is $b$. The probability that the inequality $a - 2b + 10 > 0$ ho... | 61/81 | 0.8125 | 5,910.5625 | 5,384.076923 | 8,192 | |
In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\), calculate the value of \(DE + DF\). | 24 | 0.125 | 7,885.0625 | 5,736.5 | 8,192 | |
Find the number of cubic centimeters in the volume of the cylinder formed by rotating a rectangle with side lengths 8 cm and 16 cm about its longer side. Express your answer in terms of \(\pi\). | 256\pi | 0 | 1,009.3125 | -1 | 1,009.3125 | |
Below is a portion of the graph of a function, $y=f(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=5.25,ymin=-3.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy... | 2 | 0.875 | 4,225 | 3,658.285714 | 8,192 | |
For how many values of $c$ in the interval $[0, 1000]$ does the equation \[7 \lfloor x \rfloor + 2 \lceil x \rceil = c\]have a solution for $x$? | 223 | 0.8125 | 6,437.1875 | 6,032.230769 | 8,192 | |
Given the sequence $\{a_k\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,
$a_k=\begin{cases}
(0.\underbrace{501\cdots 01}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd,} \\
(0.\underbrace{501\cdots 011}_{k+1\text{ digits}})... | 30 | 0 | 7,829.8125 | -1 | 7,829.8125 | |
In right triangle $ABC$, we have $\sin A = \frac{3}{5}$ and $\sin B = 1$. Find $\sin C$. | \frac{4}{5} | 1 | 2,418.875 | 2,418.875 | -1 | |
The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations:
\[x=x^2+y^2\] \[y=2xy\]
is | 4 | We are given the system of equations:
1. \(x = x^2 + y^2\)
2. \(y = 2xy\)
We will analyze this system by considering different cases.
#### Case 1: \(y = 0\)
Substituting \(y = 0\) into the first equation:
\[ x = x^2 + 0^2 \]
\[ x = x^2 \]
\[ x^2 - x = 0 \]
\[ x(x - 1) = 0 \]
This gives us \(x = 0\) or \(x = 1\). Both... | 1 | 3,584.5625 | 3,584.5625 | -1 |
Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly, in a ring. Determine the probability that no two red marbles are adjacent. | \frac{7}{33} | Select any blue marble and consider the remaining eleven marbles, arranged in a line. The proportion of arrangement for which no two red marbles are adjacent will be the same as for the original twelve marbles, arranged in a ring. The total number of ways of arranging 4 red marbles out of 11 is $\binom{11}{4}=330$. To ... | 0.125 | 7,479.1875 | 5,877.5 | 7,708 |
Experts and Viewers play "What? Where? When?" until one side wins six rounds—the first to win six rounds wins the game. The probability of the Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of $3:4$. Find the probability that the Experts will still win. | 0.4752 | 0 | 7,772.125 | -1 | 7,772.125 | |
A four-digit number $2\Box\Box5$ is divisible by $45$. How many such four-digit numbers are there? | 11 | 0.9375 | 4,067.0625 | 3,792.066667 | 8,192 | |
Find the smallest integer $n \geq 5$ for which there exists a set of $n$ distinct pairs $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ of positive integers with $1 \leq x_{i}, y_{i} \leq 4$ for $i=1,2, \ldots, n$, such that for any indices $r, s \in\{1,2, \ldots, n\}$ (not necessarily distinct), there ex... | 8 | In other words, we have a set $S$ of $n$ pairs in $(\mathbb{Z} / 4 \mathbb{Z})^{2}$ closed under addition. Since $1+1+1+1 \equiv 0(\bmod 4)$ and $1+1+1 \equiv-1(\bmod 4),(0,0) \in S$ and $S$ is closed under (additive) inverses. Thus $S$ forms a group under addition (a subgroup of $(\mathbb{Z} / 4 \mathbb{Z})^{2}$ ). By... | 0 | 8,192 | -1 | 8,192 |
Javier is going to Disneyland during spring break. He plans on visiting four particular attractions all before lunch. In how many orders could he visit all four attractions once? | 24 | 1 | 1,110.4375 | 1,110.4375 | -1 | |
Find the focus of the parabola $y = -3x^2 - 6x.$ | \left( -1, \frac{35}{12} \right) | 1 | 3,972.3125 | 3,972.3125 | -1 | |
The operation $\otimes$ is defined for all nonzero numbers by $a \otimes b = \frac{a^{2}}{b}$. Determine $[(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]$. | -\frac{2}{3} | 1. **Apply the operation $\otimes$ to the innermost parentheses first:**
\[
(1 \otimes 2) \otimes 3 = \left(\frac{1^2}{2}\right) \otimes 3 = \frac{1}{2} \otimes 3
\]
\[
1 \otimes (2 \otimes 3) = 1 \otimes \left(\frac{2^2}{3}\right) = 1 \otimes \frac{4}{3}
\]
2. **Calculate each part using the definit... | 1 | 2,341.9375 | 2,341.9375 | -1 |
How many non-congruent triangles with perimeter 7 have integer side lengths? | 2 | 1 | 4,665.1875 | 4,665.1875 | -1 | |
Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of $x$ such that the area of this new triang... | 22 | 0 | 8,192 | -1 | 8,192 | |
Simplify the expression $20(x+y)-19(y+x)$ for all values of $x$ and $y$. | x+y | Simplifying, we see that $20(x+y)-19(y+x)=20x+20y-19y-19x=x+y$ for all values of $x$ and $y$. | 1 | 1,430.625 | 1,430.625 | -1 |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2\sqrt{2}$, and passing through the point $A(\frac{3}{2}, -\frac{1}{2})$.
(1) Find the equation of the ellipse;
(2) Find the coordinates of a point $P$ on the ellipse $C$ such that its distance to the line $l$: $x+y+... | \sqrt{2} | 0.625 | 6,666.8125 | 5,963.7 | 7,838.666667 | |
We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree. | 2500 | 0 | 7,919.625 | -1 | 7,919.625 | |
Let $M = 39 \cdot 48 \cdot 77 \cdot 150$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | \frac{1}{62} | 0.5 | 6,337.8125 | 5,413.625 | 7,262 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 | 0.875 | 5,201.5625 | 4,774.357143 | 8,192 | |
Calculate the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, $(4,-2)$, and $(3,4)$. | \frac{11}{2} | 0.4375 | 7,155.6875 | 6,145.857143 | 7,941.111111 | |
Given $|a|=1$, $|b|=2$, and $a+b=(1, \sqrt{2})$, the angle between vectors $a$ and $b$ is _______. | \frac{2\pi}{3} | 0.0625 | 4,335.25 | 3,391 | 4,398.2 | |
Let $g$ be defined by \[g(x) = \left\{
\begin{array}{cl}
x+3 & \text{ if } x \leq 2, \\
x^2 - 4x + 5 & \text{ if } x > 2.
\end{array}
\right.\]Calculate $g^{-1}(1)+g^{-1}(6)+g^{-1}(11)$. | 2 + \sqrt{5} + \sqrt{10} | 0.9375 | 4,311.125 | 4,052.4 | 8,192 | |
The 12 numbers from 1 to 12 on a clock face divide the circumference into 12 equal parts. Using any 4 of these division points as vertices to form a quadrilateral, find the total number of rectangles that can be formed. | 15 | 0.0625 | 7,863.25 | 5,286 | 8,035.066667 | |
Given that there are 25 cities in the County of Maplewood, and the average population per city lies between $6,200$ and $6,800$, estimate the total population of all the cities in the County of Maplewood. | 162,500 | 0 | 2,317.8125 | -1 | 2,317.8125 | |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 40 | 0 | 7,775.75 | -1 | 7,775.75 | |
Triangle $ABC$ has an area 1. Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$. | \frac{7 - 3 \sqrt{5}}{4} | Choose $r,s,t$ so that $EC = rBC, FA = sCA, GB = tCB$, and let $[XYZ]$ denote the area of triangle $XYZ$. Then $[ABE] = [AFE]$ since the triangles have the same altitude and base. Also $[ABE] = (BE/BC) [ABC] = 1-r$, and $[ECF] = (EC/BC)(CF/CA)[ABC] = r(1-s)$ (e.g., by the law of sines). Adding this all up yields
\begin... | 0 | 8,192 | -1 | 8,192 |
What is the sum of the number of faces, edges and vertices of a triangular prism?
[asy]
draw((0,0)--(10,0)--(5,8.7)--cycle);
draw((0,0)--(20,20),dashed);
draw((10,0)--(30,20));
draw((5,8.7)--(25,28.7));
draw((25,28.7)--(30,20)--(20,20)--cycle,dashed);
draw((25,28.7)--(30,20));
[/asy] | 20 | 0.875 | 1,774.8125 | 1,851.214286 | 1,240 | |
A car's clock is running at a constant speed but is inaccurate. One day, when the driver begins shopping, he notices both the car clock and his wristwatch (which is accurate) show 12:00 noon. After shopping, the wristwatch reads 12:30, and the car clock reads 12:35. Later that day, he loses his wristwatch and looks at ... | 6:00 | 0.4375 | 4,372.5625 | 3,697.285714 | 4,897.777778 | |
Toner Drum and Celery Hilton are both running for president. A total of $129$ million people cast their vote in a random order, with exactly $63$ million and $66$ million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front p... | 1/43 | 0.25 | 7,036.5625 | 5,280.5 | 7,621.916667 | |
Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? | 31 | 1. **Calculate the cost of sandwiches**: Each sandwich costs $3$ dollars. Therefore, the cost for $5$ sandwiches is calculated by multiplying the number of sandwiches by the cost per sandwich:
\[
5 \times 3 = 15 \text{ dollars}
\]
2. **Calculate the cost of sodas**: Each soda costs $2$ dollars. Therefore, the... | 1 | 1,106.8125 | 1,106.8125 | -1 |
A running track is the ring formed by two concentric circles. If the circumferences of the two circles differ by $10\pi $ feet, how wide is the track in feet?
[asy]size(100); path g=scale(2)*unitcircle;
filldraw(unitcircle^^g,evenodd+grey,black);
[/asy] | 5 | 0.6875 | 769.5625 | 848 | 597 | |
A bag contains four balls, each labeled with one of the characters "美", "丽", "惠", "州". Balls are drawn with replacement until both "惠" and "州" are drawn, at which point the drawing stops. Use a random simulation method to estimate the probability that the drawing stops exactly on the third draw. Use a computer to rando... | \frac{1}{8} | 0 | 7,752.375 | -1 | 7,752.375 | |
Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$, where $x$ is in $\mathcal{S}$. In other words, $\mathcal{T}$ is the set of numbers that result when the last three digits of each number ... | 170 | The condition implies $x^2\equiv 256 \pmod{1000}$. Rearranging and factoring, \[(x-16)(x+16)\equiv 0\pmod {1000}.\] This can be expressed with the system of congruences \[\begin{cases} (x-16)(x+16)\equiv 0\pmod{125} \\ (x-16)(x+16)\equiv 0\pmod{8} \end{cases}\] Observe that $x\equiv {109} \pmod {125}$ or $x\equiv{16}\p... | 0.0625 | 8,141.125 | 7,974 | 8,152.266667 |
An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches.
If the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance
between the centers of the pulleys in inches is | 26 | 1. **Identify the setup**: We have two pulleys with radii $14$ inches and $4$ inches respectively. The distance between the points of contact of the belt with the pulleys is $24$ inches. We denote the centers of the smaller and larger pulleys as $A$ and $B$ respectively.
2. **Draw perpendicular radii**: Draw radii $AC... | 0.9375 | 1,504.0625 | 1,559.333333 | 675 |
$\frac{16+8}{4-2}=$ | 12 | 1. **Evaluate the Numerator and Denominator Separately:**
\[
\text{Numerator: } 16 + 8 = 24
\]
\[
\text{Denominator: } 4 - 2 = 2
\]
2. **Divide the Numerator by the Denominator:**
\[
\frac{24}{2} = 12
\]
3. **Identify the Correct Answer:**
The result of the division is 12, which correspo... | 1 | 1,740.9375 | 1,740.9375 | -1 |
Given a circle of radius 3, there are many line segments of length 4 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
A) $9\pi$
B) $\pi$
C) $4\pi$
D) $13\pi$
E) $16\pi$ | 4\pi | 0 | 7,556.375 | -1 | 7,556.375 | |
Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square! | 8/5 | 0.0625 | 7,970.125 | 4,642 | 8,192 | |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | 5416 | 0.625 | 5,178.25 | 3,835.3 | 7,416.5 | |
A point $Q$ is chosen within $\triangle DEF$ such that lines drawn through $Q$, parallel to the sides of $\triangle DEF$, divide it into three smaller triangles with areas $9$, $16$, and $25$ respectively. Determine the area of $\triangle DEF$. | 144 | 0.625 | 6,755.5 | 5,893.6 | 8,192 | |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | \dfrac{289}{15068} | 0 | 7,073.6875 | -1 | 7,073.6875 | |
In a store, there are 50 light bulbs in stock, 60% of which are produced by Factory A and 40% by Factory B. The first-class rate of the light bulbs produced by Factory A is 90%, and the first-class rate of the light bulbs produced by Factory B is 80%.
(1) If one light bulb is randomly selected from these 50 light bulb... | 1.08 | 0.5 | 4,878.625 | 2,890.125 | 6,867.125 | |
Let \( ABC \) be a right triangle with the hypotenuse \( BC \) measuring \( 4 \) cm. The tangent at \( A \) to the circumcircle of \( ABC \) meets the line \( BC \) at point \( D \). Suppose \( BA = BD \). Let \( S \) be the area of triangle \( ACD \), expressed in square centimeters. Calculate \( S^2 \). | 27 | 0.625 | 6,627.3125 | 5,688.5 | 8,192 | |
In the arithmetic sequence $\{a_n\}$, $a_2=4$, $a_4+a_7=15$.
(Ⅰ) Find the general term formula of sequence $\{a_n\}$.
(Ⅱ) Let $b_{n}= \frac {1}{a_{n}a_{n+1}}$, find the value of $b_1+b_2+b_3+\ldots+b_{10}$. | \frac{10}{39} | 1 | 2,577.5625 | 2,577.5625 | -1 | |
It is known that the ellipse $C_1$ and the parabola $C_2$ have a common focus $F(1,0)$. The center of $C_1$ and the vertex of $C_2$ are both at the origin. A line $l$ passes through point $M(4,0)$ and intersects the parabola $C_2$ at points $A$ and $B$ (with point $A$ in the fourth quadrant).
1. If $|MB| = 4|AM|$, find... | \sqrt{34} | 0 | 8,192 | -1 | 8,192 | |
Bobbo starts swimming at 2 feet/s across a 100 foot wide river with a current of 5 feet/s. Bobbo doesn't know that there is a waterfall 175 feet from where he entered the river. He realizes his predicament midway across the river. What is the minimum speed that Bobbo must increase to make it to the other side of the ri... | 3 \text{ feet/s} | When Bobbo is midway across the river, he has travelled 50 feet. Going at a speed of 2 feet/s, this means that Bobbo has already been in the river for $\frac{50 \text{ feet}}{2 \text{ feet/s}} = 25 \text{ s}$. Then he has traveled 5 feet/s $\cdot$ 25 s = 125 feet down the river. Then he has 175 feet - 125 feet = 50 fee... | 0 | 7,584.5625 | -1 | 7,584.5625 |
A line $l$ with an inclination angle of $45^\circ$ passes through the focus $F$ of the parabola $y^2=8x$ and intersects the parabola at points $A$ and $B$. Find the value of $|\overrightarrow{FA}| \cdot |\overrightarrow{FB}|$. | 32 | 1 | 5,163.75 | 5,163.75 | -1 | |
How many ten digit positive integers with distinct digits are multiples of $11111$ ? | 3456 | 0 | 8,192 | -1 | 8,192 | |
Find the constant term in the expansion of $\left( \sqrt {x}+ \dfrac {1}{2 \sqrt {x}}\right)^{8}$. | \dfrac {35}{8} | 1 | 2,686.8125 | 2,686.8125 | -1 | |
A triangle with side lengths $5,7,8$ is inscribed in a circle $C$. The diameters of $C$ parallel to the sides of lengths 5 and 8 divide $C$ into four sectors. What is the area of either of the two smaller ones? | \frac{49}{18} \pi | Let $\triangle P Q R$ have sides $p=7, q=5, r=8$. Of the four sectors determined by the diameters of $C$ that are parallel to $P Q$ and $P R$, two have angles equal to $P$ and the other two have angles equal to $\pi-P$. We first find $P$ using the law of cosines: $49=25+64-2(5)(8) \cos P$ implies $\cos P=\frac{1}{2}$ i... | 0 | 8,061.6875 | -1 | 8,061.6875 |
What is the largest $4$ digit integer congruent to $15 \pmod{22}?$ | 9981 | 1 | 4,381.375 | 4,381.375 | -1 | |
Let \(\left\{a_{n}\right\}\) be a sequence of positive integers such that \(a_{1}=1\), \(a_{2}=2009\) and for \(n \geq 1\), \(a_{n+2} a_{n} - a_{n+1}^{2} - a_{n+1} a_{n} = 0\). Determine the value of \(\frac{a_{993}}{100 a_{991}}\). | 89970 | 0.4375 | 6,790.8125 | 5,429.571429 | 7,849.555556 | |
What is the sum of the digits of \(10^{2008} - 2008\)? | 18063 | 0.25 | 7,406.8125 | 5,310.5 | 8,105.583333 | |
Let $f$ be a function defined on the positive integers, such that
\[f(xy) = f(x) + f(y)\]for all positive integers $x$ and $y.$ Given $f(10) = 14$ and $f(40) = 20,$ find $f(500).$ | 39 | 0.9375 | 2,439.25 | 2,349.066667 | 3,792 | |
Out of 1500 people surveyed, $25\%$ do not like television, and out of those who do not like television, $15\%$ also do not like books. How many people surveyed do not like both television and books? | 56 | 0 | 4,707.5 | -1 | 4,707.5 | |
Find all solutions to the equation\[ \sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}.\] | 81, 256 | 0 | 2,631.375 | -1 | 2,631.375 | |
Three balls are lying on a table, each pair touching each other. The radii of the balls form a geometric progression with a common ratio \( q \neq 1 \). The radius of the middle ball is 2012. Find the ratio of the sum of the squares of the sides of the triangle formed by the points of contact of the balls with the tabl... | 4024 | 0.25 | 7,759.375 | 6,461.5 | 8,192 | |
Given the function $f(x) = x^{3} + ax^{2} - 2x + 1$ has an extremum at $x=1$.
$(1)$ Find the value of $a$;
$(2)$ Determine the monotonic intervals and extremum of $f(x)$. | -\frac{1}{2} | 0.875 | 3,058.9375 | 3,058.357143 | 3,063 | |
Given $f(x)=\cos x\cdot\ln x$, $f(x_{0})=f(x_{1})=0(x_{0}\neq x_{1})$, find the minimum value of $|x_{0}-x_{1}|$ ___. | \dfrac {\pi}{2}-1 | 0.125 | 7,977.25 | 6,736 | 8,154.571429 | |
If an integer of two digits is $k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by | 11-k | 1. **Define the number and its properties**: Let the two-digit integer be represented as $n = 10a + b$, where $a$ and $b$ are the tens and units digits respectively. According to the problem, this number $n$ is $k$ times the sum of its digits. Therefore, we have the equation:
\[
10a + b = k(a + b)
\]
2. **Exp... | 0.8125 | 5,193.625 | 4,501.692308 | 8,192 |
The diagonals of a rhombus are 6 inches and 8 inches. What is the perimeter, in inches, of the rhombus below? [asy]
import geometry; size(150); defaultpen(linewidth(0.8));
pair A = (0,0),B = (8,6),C = (16,0),D = (8,-6), E = (8,0);
draw(A--B--C--D--cycle);
draw(A--C,StickIntervalMarker(2,2,10)); draw(B--D,StickIntervalM... | 20 | 1 | 1,154.125 | 1,154.125 | -1 | |
Ten points are spaced evenly along the perimeter of a rectangle measuring $3 \times 2$ units. What is the probability that the two points are one unit apart? | \frac{2}{9} | 0.3125 | 7,704.75 | 6,632.8 | 8,192 | |
In how many different places in the $xy$-plane can a third point, $R$, be placed so that $PQ = QR = PR$ if points $P$ and $Q$ are two distinct points in the $xy$-plane? | 2 | If point $R$ is placed so that $PQ = QR = PR$, then the resulting $\triangle PQR$ is equilateral. Since points $P$ and $Q$ are fixed, then there are two possible equilateral triangles with $PQ$ as a side - one on each side of $PQ$. One way to see this is to recognize that there are two possible lines through $P$ that m... | 0.875 | 3,534.25 | 2,909 | 7,911 |
How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$? | 31 |
To solve the problem, we are tasked with finding the number of integers \( n > 1 \) such that \( n \) divides \( x^{13} - x \) for every positive integer \( x \).
First, observe that if \( n \mid x^{13} - x \) for every integer \( x \), then \( n \mid x^{13} - x \) for each \( x \) in particular values, such as \( x ... | 0.4375 | 6,275.9375 | 3,812.428571 | 8,192 |
Find the smallest sum of distances from an arbitrary point on the plane to the vertices of a unit square.
A problem of the shortest connection for four points. Four points: \(A, B, C,\) and \(D\) are the vertices of a square with side length 1. How should these points be connected by roads to ensure that it is possibl... | 2\sqrt{2} | 0.125 | 8,018.3125 | 6,802.5 | 8,192 | |
A class is scheduled to have 6 classes in one day: Chinese, Mathematics, Politics, English, PE, and Art. It is required that the Mathematics class is scheduled within the first 3 periods, and the PE class cannot be scheduled in the first period. The number of different scheduling methods is ______. (Answer in digits). | 312 | 0.125 | 7,931.4375 | 6,107.5 | 8,192 | |
Determine the number of six-letter words where the first and last two letters are the same (e.g., "aabbaa"). | 456976 | 0.4375 | 6,520.3125 | 4,998.857143 | 7,703.666667 | |
In square \(ABCD\) with a side length of 10, points \(P\) and \(Q\) lie on the segment joining the midpoints of sides \(AD\) and \(BC\). Connecting \(PA\), \(PC\), \(QA\), and \(QC\) divides the square into three regions of equal area. Find the length of segment \(PQ\). | 20/3 | 0 | 8,192 | -1 | 8,192 | |
A sports equipment store purchaser needs to purchase a total of 100 basketballs and volleyballs from the manufacturer, with a total payment not exceeding $11,815. The wholesale prices from the two types of balls and the retail prices at the store are shown in the table below. Solve the following questions:
| Product |... | 2,600 | 0 | 3,925.4375 | -1 | 3,925.4375 |
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