problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
In this modified Number Wall, you still add the numbers next to each other to write the sum in the block directly above the two numbers. Which number will be in the block labeled '$m$'? [asy]
draw((0,0)--(8,0)--(8,2)--(0,2)--cycle);
draw((2,0)--(2,2));
draw((4,0)--(4,2));
draw((6,0)--(6,2));
draw((1,2)--(7,2)--(7,4)--(1,4)--cycle);
draw((3,2)--(3,4));
draw((5,2)--(5,4));
draw((2,4)--(2,6)--(6,6)--(6,4)--cycle);
draw((4,4)--(4,6));
draw((3,6)--(3,8)--(5,8)--(5,6));
label("$m$",(1,1));
label("6",(3,1));
label("12",(5,1));
label("10",(7,1));
label("36",(4,7));
[/asy]
|
-28
| 0.083333 |
Henry's little brother now has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to put all of the stickers on the sheets of paper, if only the number of stickers on each sheet matters and each sheet must have at least one sticker?
|
7
| 0.416667 |
Find the second smallest positive integer $x$ which is greater than $1$ and relatively prime to $210$.
|
13
| 0.916667 |
The interior of a right, circular cone is 12 inches tall with a 3-inch radius at the opening. The interior of the cone is filled with ice cream, and the cone has a right circular cylinder of ice cream, 2 inches tall with the same radius as the opening, exactly covering the opening of the cone. What is the volume of ice cream? Express your answer in terms of $\pi$.
|
54\pi \text{ cubic inches}
| 0.75 |
Find the number of lattice points that satisfy both $x^2 - y^2 = 75$ and $x - y = 5$ on the $xy$-plane.
|
1
| 0.916667 |
Find the minimum value of
\[ x^2 + 4xy + 5y^2 - 8x - 6y, \]
over all real numbers $x$ and $y$.
|
-41
| 0.833333 |
What is the remainder when 3,452,179 is divided by 7, after adding 50 to it?
|
4
| 0.75 |
What is the sum of the tens digit and the ones digit of the integer form of $(3+4)^{15}$?
|
7
| 0.916667 |
120 people were surveyed and asked the question: "Is teal kinda blue, or greenish?" Of them, 70 believe that teal is "kinda blue" and 35 believe it is both "kinda blue," and also "greenish." Another 20 think that teal is neither "kinda blue" nor "greenish."
How many of those 120 people believe that teal is "greenish"?
|
65
| 0.916667 |
Jacob has run fifteen half-marathons in his life. Each half-marathon is $13$ miles and $193$ yards. One mile equals $1760$ yards. If the total distance Jacob covered in these half-marathons is $m$ miles and $y$ yards, where $0 \le y < 1760$, what is the value of $y$?
|
1135
| 0.25 |
Twelve points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the twelve points as vertices?
|
4017
| 0.916667 |
In rhombus $ABCD$, angle $A$ measures $120^\circ$. What is the number of degrees in the measure of angle $C$?
|
120^\circ
| 0.916667 |
How many positive three-digit integers less than 600 have at least two digits that are the same?
|
140
| 0.833333 |
Suppose $a$, $b$, and $c$ are three positive numbers that satisfy the equations $abc = 1$, $a + \frac{1}{c} = 7$, and $b + \frac{1}{a} = 16$. Find $c + \frac{1}{b}$.
|
\frac{25}{111}
| 0.5 |
A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder?
|
10\sqrt{2} \text{ cm}
| 0.75 |
In rectangle $ABCD$, $AB = 10$ cm, $BC = 12$ cm, and $DE = DF$. The area of triangle $DEF$ is one-third the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.
|
4\sqrt{10}
| 0.416667 |
What is the smallest positive integer \( n \) such that all the roots of \( z^6 - z^3 + 1 = 0 \) are \( n^{\text{th}} \) roots of unity?
|
18
| 0.666667 |
What is the base ten equivalent of $23456_{8}$?
|
10030
| 0.083333 |
The sides of a triangle have lengths $5, 12,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse?
|
6
| 0.75 |
Let $x = (3 + \sqrt{8})^{1000}$, let $n = \lfloor x \rfloor$, and let $f = x - n$. Find
\[x(1 - f).\]
|
1
| 0.916667 |
Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors (i.e., the smallest prime), and $n$ is the second largest integer less than 150 with exactly three positive divisors. Calculate $m+n$.
|
51
| 0.916667 |
Our school's girls volleyball team has 16 players, including a set of triplets: Alicia, Amanda, and Anna, and a set of twins: Beth and Brenda. In how many ways can we choose 6 starters if at most one of the triplets and at most one of the twins can be in the starting lineup?
|
4752
| 0.75 |
Find the sum of the values of $x$ which satisfy $x^2 + 2010x = 2011 + 18x$.
|
-1992
| 0.833333 |
Factor the expression $x(x+2)+(x+2)^2$.
|
2(x+2)(x+1)
| 0.583333 |
The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 110^{\circ}$ and $\angle AOB = 150^{\circ}$. Determine the degree measure of $\angle ABC$.
|
50^\circ
| 0.5 |
A point is randomly selected from the portion of the number line from 0 to 10. What is the probability that the point is closer to 7 than to 0? Express your answer as a decimal to the nearest tenth.
|
0.7
| 0.916667 |
Let \( g(x) = x^2 - 2x + 2022 \). What is the greatest common divisor of \( g(50) \) and \( g(52) \)?
|
2
| 0.916667 |
If the value of $x$ is doubled and then this increased value is divided by 4, the result is 12. What is the value of $x$?
|
24
| 0.916667 |
There are seven unmarked envelopes on a table, each with a letter for a different person. If the mail is randomly distributed to these seven people, with each person getting one letter, what is the probability that exactly two people get the correct letter?
|
\frac{11}{60}
| 0.916667 |
What is the smallest prime divisor of $3^{25} + 11^{19}$?
|
2
| 0.916667 |
A bag contains 4 red, 7 green, 9 yellow, and 10 blue jelly beans. A jelly bean is selected at random. What is the probability that it is not yellow?
|
\frac{7}{10}
| 0.916667 |
Two distinct numbers are selected simultaneously and at random from the set $\{2, 3, 4, 5, 6\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction.
|
\frac{3}{10}
| 0.916667 |
Sue now owns 16 pairs of shoes: seven identical black pairs, four identical brown pairs, three identical gray pairs, and two identical red pairs. If she picks two shoes at random, what is the probability that they are a matching color and one is a left shoe and the other is a right shoe? Express your answer as a common fraction.
|
\frac{39}{248}
| 0.583333 |
Simplify $\displaystyle \frac{2-2i}{3+4i}$, where $i^2 = -1$.
|
-\frac{2}{25} - \frac{14}{25}i
| 0.833333 |
The pages of a book are numbered from 1 to n. Due to an error, one of the page numbers was added twice, resulting in a total sum of 2076. Determine which page number was incorrectly added twice.
|
60
| 0.416667 |
A triangle is formed from wood sticks of lengths 10, 24, and 26 inches joined end-to-end. Pieces, with length $x$ from each of the sticks, are to be cut so that the remaining pieces can no longer form a triangle. Determine the smallest value of $x$ that meets this condition.
|
8\text{ inches}
| 0.833333 |
Let \( A \) and \( B \) be two points on the parabola \( y = x^2 + 1 \), such that the tangents at \( A \) and \( B \) are perpendicular. Find the \( y \)-coordinate of their point of intersection \( P \).
|
\frac{3}{4}
| 0.916667 |
Find the smallest possible perimeter of a scalene triangle where each side length is a different prime number, and the triangle's perimeter is also a prime number, given that the smallest side length is at least 5.
|
23
| 0.833333 |
Let $p$ and $q$ be the solutions of the equation $3x^2 - 9x - 15 = 0$. Calculate the value of $(3p-5)(6q-10)$.
|
-130
| 0.916667 |
What is the greatest four-digit number that is one more than a multiple of 7 and five more than a multiple of 8?
|
9997
| 0.083333 |
Given $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = 5$, find the value of:
\[\begin{vmatrix} x & 2x + 4y \\ z & 2z + 4w \end{vmatrix}\]
|
20
| 0.916667 |
If the product $(4x^2 - 6x + 5)(8 - 3x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are real numbers, then find $8a + 4b + 2c + d$.
|
18
| 0.833333 |
I randomly pick an integer $p$ between $1$ and $15$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction.
|
\frac{4}{15}
| 0.833333 |
A rectangular tank holds 216 cubic feet of water. Determine the volume of this tank in cubic meters, knowing that 1 cubic meter is approximately 35.315 cubic feet.
|
6.12
| 0.083333 |
A circle with center $A$ and radius four feet is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
|
48\pi
| 0.916667 |
For how many values of the digit $A$ is it true that $75$ is divisible by $A$ and $536{,}1A4$ is divisible by $4$?
|
0
| 0.916667 |
A three-inch cube ($3\times3\times3$) of titanium weighs 5 pounds and is worth $\$300$. How much is a five-inch cube of titanium worth? Assume that, due to material characteristics, the price weight ratio increases by 20% with every inch increase in dimension.
|
\$2000
| 0.5 |
The values of \(x\) and \(y\) are always positive, and \(x^3\) and \(y\) vary inversely. If \(y = 5\) when \(x = 2\), find \(x\) when \(y = 2000\).
|
x = \frac{1}{\sqrt[3]{50}}
| 0.916667 |
Find the smallest composite number that has no prime factors less than 20.
|
529
| 0.833333 |
A regular decagon \(Q_1 Q_2 \dotsb Q_{10}\) is drawn in the coordinate plane with \(Q_1\) at \((2,0)\) and \(Q_6\) at \((-2,0)\). If \(Q_n\) is the point \((x_n,y_n)\), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).
\]
|
-1024
| 0.583333 |
Given the following stem and leaf plot, calculate the positive difference between the median and the mode. In this plot, $6|7$ represents $67$.
\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
2 & $1 \hspace{2mm} 3 \hspace{2mm} 3 \hspace{2mm} 5 \hspace{2mm} 5$ \\ \hline
3 & $2 \hspace{2mm} 2 \hspace{2mm} 2 \hspace{2mm} \hspace{2mm} \hspace{2mm} \hspace{1.5mm}$ \\ \hline
4 & $0 \hspace{2mm} 0 \hspace{2mm} 7 \hspace{2mm} 8 \hspace{2mm} \hspace{1.9mm}$ \\ \hline
5 & $1 \hspace{2mm} 2 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} \hspace{1.9mm}$ \\ \hline
6 & $3 \hspace{2mm} 7 \hspace{2mm} 8 \hspace{2mm} \hspace{2mm} \hspace{2mm} \hspace{1.5mm}$ \\\hline
\end{tabular}
|
8
| 0.916667 |
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane is $15$, and the sum of the $y$-coordinates of the vertices is also $15$. Find the sum of the $x$-coordinates and the sum of the $y$-coordinates of the midpoints of the sides of the triangle.
|
15
| 0.166667 |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that the penny and dime both come up the same, and the nickel and quarter also both come up the same?
|
\dfrac{1}{4}
| 0.833333 |
A line with slope $-3$ intersects the positive $x$-axis at a point $A$ and the positive $y$-axis at $B$. Another line passes through the points $C(6,0)$ on the $x$-axis and $D$ on the $y$-axis. The lines intersect at $E(3,3)$. What is the area of the shaded quadrilateral formed by $OBEC$?
|
27
| 0.916667 |
Let $D$ be the circle with equation $x^2 + 4y - 16 = -y^2 + 12x + 16$. Find the values of $(c,d)$, the center of $D$, and $s$, the radius of $D$, and calculate $c+d+s$.
|
4 + 6\sqrt{2}
| 0.75 |
I received \$40 in pocket money and spent it as indicated in the pie graph below. How many dollars did I spend on video games?
[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.707,.707);
B=(-.707,.707);
C=(-.707,-.707);
D=(.707,-.707);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(0,.6);
label("Books", W, N);
label("$\frac{2}{5}$", W, S);
X=(-.6, 0);
label("Video Games", X, S);
Y=(0,-.6);
label("Snacks", Y, N);
label("$\frac{1}{4}$", Y, S);
Z=(.6, 0);
label("Toys", Z, N);
label("$\frac{1}{5}$", Z, S);
[/asy]
|
6
| 0.083333 |
A line is described by the equation $y - 3 = 6(x - 5)$. What is the sum of its $x$-intercept and $y$-intercept?
|
-22.5
| 0.083333 |
Compute \[
\left\lfloor \frac{2017! + 2014!}{2016! + 2015!}\right\rfloor.
\]
|
2016
| 0.583333 |
Find the total number of odd integers \( n \) that satisfy
\[ 25 < n^2 < 144. \]
|
6
| 0.5 |
Convert the point $(2, -2\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2\pi.$
|
\left( 4, \frac{5\pi}{3} \right)
| 0.916667 |
Evaluate $103^4 - 4 \cdot 103^3 + 6 \cdot 103^2 - 4 \cdot 103 + 1$.
|
108243216
| 0.5 |
Given that $b$ is a multiple of $2700$, find the greatest common divisor of $b^2 + 27b + 75$ and $b + 25$.
|
25
| 0.833333 |
If $m$, $n$, and $p$ are positive integers such that $\gcd(m,n) = 180$ and $\gcd(m,p) = 240$, then what is the smallest possible value of $\gcd(n,p)$?
|
60
| 0.833333 |
If $|x| + x + y = 12$ and $x + |y| - y = 10$, find $x + y + z$ given that $x - y + z = 5$.
|
\frac{9}{5}
| 0.583333 |
How many distinct sequences of five letters can be made from the letters in PROBLEM if each letter can be used only once, each sequence must begin with L, and does not end with P?
|
300
| 0.666667 |
Calculate $\frac{4 \cdot 6! + 24\cdot 5!}{7!}$
|
\frac{8}{7}
| 0.833333 |
If $q(x) = x^4 - 4x + 5$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^3$.
|
-64
| 0.5 |
Each day, Jenny eats $25\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the third day, 27 remained. How many jellybeans were in the jar originally?
|
64
| 0.833333 |
The function $g(x)$ satisfies $g(1) = 2$ and
\[g(x + y) = 5^y g(x) + 3^x g(y)\]
for all real numbers $x$ and $y.$ Find the function $g(x).$
|
5^x - 3^x
| 0.666667 |
What is the arithmetic mean of the integers from -6 through 7, inclusive? Express your answer as a decimal.
|
0.5
| 0.916667 |
Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations:
\[
17x + by + cz = 0, \\
ax + 31y + cz = 0, \\
ax + by + 53z = 0.
\]
Assuming that $ a \ne 17$ and $ x \ne 0$, what is the value of
\[
\frac{a}{a - 17} + \frac{b}{b - 31} + \frac{c}{c - 53} \,?
\]
|
1
| 0.416667 |
Consider the function $f(x) = x^2 + 4\sqrt{x}$. Evaluate $2f(3) - f(9)$.
|
-75 + 8\sqrt{3}
| 0.666667 |
Suppose that \(x,\) \(y,\) and \(z\) are three positive numbers such that \(xyz = 1,\) \(x + \frac{1}{z} = 8,\) and \(y + \frac{1}{x} = 20.\) Find \(z + \frac{1}{y}.\)
|
\frac{10}{53}
| 0.583333 |
What is the smallest four-digit palindrome that is divisible by 3 and has an odd first digit?
|
1221
| 0.666667 |
The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $60^{\text{th}}$ integer in the list?
|
32541
| 0.166667 |
A rectangle with even integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area?
|
116
| 0.166667 |
Suppose \(x^2\) varies inversely with \(y^3\). If \(x = 10\) when \(y = 2\), find the value of \(x^2\) when \(y = 4\).
|
12.5
| 0.333333 |
Consider the sequence where $x_1+1=x_2+2=x_3+3=\cdots=x_{100}+100=x_1+x_2+x_3+\cdots+x_{100}+101$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{100}x_n$.
|
51
| 0.333333 |
Evaluate the ratio $\frac{10^{2010}+10^{2013}}{10^{2011}+10^{2014}}$ and determine which whole number it is closest to.
|
0
| 0.916667 |
What is the distance between points (-3, 7) and (4, -9), and what is the midpoint of the segment connecting these points?
|
\left(\frac{1}{2}, -1\right)
| 0.5 |
How many divisors of $9!$ are multiples of $10$?
|
70
| 0.75 |
Find the smallest solution to \[\lfloor x \rfloor = 3 + 50 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$
|
3.00
| 0.833333 |
The graph of the rational function $\frac{p(x)}{q(x)}$ has a horizontal asymptote of $y = 0$ and a vertical asymptote at $x = -2$. If $q(x)$ is quadratic, $p(2)=4$, and $q(2) = 8$, and there is a hole at $x = 3$, find $p(x) + q(x)$.
|
-2x^2 - 2x + 24
| 0.666667 |
Define a new operation $ \diamond $ for non-zero integers where $a \diamond b = \frac{1}{a} + \frac{1}{b}$. Given that $a * b = 10$ and $ab = 24$, what is $a \diamond b$? Express your answer as a common fraction.
|
\frac{5}{12}
| 0.75 |
For what value of $k$ does $6 \times 9 \times 2 \times k = 8!$?
|
\frac{1120}{3}
| 0.25 |
What is the greatest integer less than 200 for which the greatest common factor of that integer and 24 is 4?
|
196
| 0.916667 |
Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]Let $\phi$ be the angle between $\mathbf{v}$ and $\mathbf{w}.$ Find $\sin \phi.$
|
\frac{\sqrt{3}}{2}
| 0.666667 |
In a regular nonagon (a 9-sided polygon), two diagonals are chosen randomly. What is the probability that these diagonals intersect inside the nonagon?
|
\frac{14}{39}
| 0.666667 |
Find the area of the parallelogram generated by the vectors $3\mathbf{a} + 4\mathbf{b}$ and $2\mathbf{a} - 6\mathbf{b}$ if the area of the parallelogram generated by $\mathbf{a}$ and $\mathbf{b}$ is 15 units\(^2\).
|
390
| 0.833333 |
Let $x, y,$ and $z$ be three positive real numbers such that $x + y + z = 60.$ Find the ordered triple $(x, y, z)$ for which $x^3 y^2 z^4$ is maximized.
|
(20, \frac{40}{3}, \frac{80}{3})
| 0.5 |
Find the function $g(x)$ if $g(2) = 2$ and for all real numbers $x$ and $y$,
\[ g(x + y) = 5^y g(x) + 3^x g(y) \]
|
g(x) = \frac{5^x - 3^x}{8}
| 0.416667 |
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k$ such that $Q$ is divisible by $3^k.$
|
76
| 0.333333 |
Let the line \( p \) be the perpendicular bisector of points \( A = (20, 12) \) and \( B = (-4, 3) \). Determine the point \( C = (x, y) \) where line \( p \) meets segment \( AB \), and calculate \( 3x - 5y \).
|
-13.5
| 0.333333 |
What is the greatest positive integer that must divide the sum of the first twelve terms of any arithmetic sequence whose terms are positive integers?
|
6
| 0.916667 |
A sequence consists of $2000$ terms. Each term after the first is $1$ larger than the previous term. The sum of the $2000$ terms is $5010$. When every second term is added up, starting with the second term and ending with the last term, what is the sum?
|
3005
| 0.333333 |
Let \( A \), \( B \), and \( C \) be nonnegative integers such that \( A + B + C = 15 \). What is the maximum value of \[A\cdot B\cdot C + A\cdot B + B\cdot C + C\cdot A?\]
|
200
| 0.75 |
Consider a sequence $(v_n)$ where each term follows the relation:
\[ v_{n+2} = 3v_{n+1} - 2v_n \]
Given that $v_3 = 5$ and $v_6 = -76$, find the value of $v_5$.
|
-\frac{208}{7}
| 0.833333 |
The diagonal lengths of a rhombus are 18 units and 26 units. Calculate both the area and the perimeter of the rhombus.
|
20\sqrt{10}
| 0.25 |
A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side?
|
14
| 0.916667 |
When the decimal point of a certain positive decimal number is moved two places to the right, the new number is nine times the reciprocal of the original number. What is the original number?
|
0.3
| 0.166667 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.