problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Four marbles are randomly selected, without replacement, from a bag containing three red, three blue, and two green marbles. What is the probability that exactly one marble of each color is selected, with an additional red marble?
|
\frac{9}{35}
| 0.833333 |
Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \[\frac{(x-a)(x-b)}{x-c} \le 0\] is true if and only if either $x < -1$ or $|x-10| \le 2.$ Given that $a < b$, find the value of $a + 2b + 3c.$
|
29
| 0.333333 |
The first few rows of a new sequence are given as follows:
- Row 1: $3$
- Row 2: $6, 6, 6, 6$
- Row 3: $9, 9, 9, 9, 9, 9$
- Row 4: $12, 12, 12, 12, 12, 12, 12, 12$
What is the value of the $40^{\mathrm{th}}$ number if this arrangement were continued?
|
18
| 0.333333 |
Find the largest integral value of \( x \) which solves: \( \frac{2}{5} < \frac{x}{7} < \frac{8}{11} \)
|
x = 5
| 0.833333 |
A certain ellipse is tangent to both the $x$-axis and the $y$-axis, and its foci are at $(-3 + \sqrt{5}, 2)$ and $(-3 - \sqrt{5}, 2).$ Find the length of the major axis.
|
6
| 0.75 |
If the system of equations
\[
4x - 2y = c, \quad
6y - 12x = d
\]
has a solution, find $\frac{c}{d},$ assuming $d \neq 0.$
|
\frac{c}{d} = -\frac{1}{3}
| 0.75 |
What is the largest possible median for the five-number set \(\{x, 2x, 6, 4, 7\}\) if \(x\) can be any integer?
|
7
| 0.833333 |
In $\triangle ABC$, point D is on side BC such that BD = 4 and DC = 14. Calculate the ratio of the area of $\triangle ABD$ to the area of $\triangle ADC$. Additionally, find what fraction of side BC does point D divide the side into.
|
\frac{2}{9}
| 0.583333 |
What is the value of $b$ for which $\frac{1}{\log_2 b} + \frac{1}{\log_3 b} + \frac{1}{\log_5 b} + \frac{1}{\log_6 b} = 1$?
|
180
| 0.75 |
The area of the parallelogram generated by the vectors $\mathbf{a}$ and $\mathbf{b}$ is 12. Find the area of the parallelogram generated by the vectors $3\mathbf{a} + 4\mathbf{b}$ and $2\mathbf{a} - 6\mathbf{b}$.
|
312
| 0.916667 |
A deck of 104 cards, consisting of two standard 52-card decks, is shuffled and dealt out in a circle. What is the expected number of pairs of adjacent cards which are both red?
|
\frac{2652}{103}
| 0.083333 |
A ball is dropped from a height of 150 feet. Each time it hits the ground, it rebounds to 40% of the height it fell. How many feet will the ball have traveled when it hits the ground the fifth time?
|
344.88
| 0.416667 |
Determine the leading coefficient in the polynomial $5(x^5 - 2x^3 + x^2) - 8(x^5 + x^4 + 3) + 6(3x^5 - x^3 - 2)$ after it is simplified.
|
15
| 0.916667 |
A rectangular dining table arrangement requires placing $48$ identical plates such that each row and each column contains the same number of plates, and each row and column has at least two plates. Determine how many different arrangements (configurations of rows and columns) are possible if we assume all plates are used and the arrangement respects the condition mentioned. Note that configurations are considered distinct based on the orientation (i.e., swapping rows and columns counts as different).
|
8
| 0.916667 |
Let $(x, y)$ be a solution to the system of equations
\[
\begin{aligned}
\lfloor x \rfloor - \{y\} &= 3.7, \\
\{x\} + \lfloor y \rfloor &= 6.2.
\end{aligned}
\]
Compute $|x - y|$.
|
2.1
| 0.833333 |
Each of $a_1,$ $a_2,$ $\dots,$ $a_{50}$ is equal to $1$ or $-1.$ Find the minimum positive value of
\[\sum_{1 \le i < j \le 50} a_i a_j.\]
|
7
| 0.916667 |
Jamie made a complete list of the prime numbers between 1 and 50. What is the sum of the second smallest prime number and the second largest prime number on her list?
|
46
| 0.916667 |
Let $m$ be the integer such that $0 \le m < 29$ and $4m \equiv 1 \pmod{29}$. What is $\left(5^m\right)^4 - 3 \pmod{29}$?
|
13
| 0.916667 |
The terms $250, b, \frac{81}{50}$ are the first, second, and third terms, respectively, of a geometric sequence. If $b$ is positive, what is the value of $b$?
|
9\sqrt{5}
| 0.416667 |
Determine the largest interval including the point $x=2$ where the function $g(x) = 3x^2 - 9x + 4$ is invertible.
|
[\frac{3}{2}, \infty)
| 0.083333 |
The product of three times the circumference of a circle and two is equal to twice the circle's area. What is the length of the radius of the circle, in inches?
|
6
| 0.916667 |
A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$.
|
2.7
| 0.166667 |
For $\mathbf{v} = \begin{pmatrix} 1 \\ y \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 8 \\ 4 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 7 \\ 3.5 \end{pmatrix}.\]Find $y$.
|
15.5
| 0.583333 |
Find the least common multiple (LCM) and the greatest common divisor (GCD) of 12 and 18.
|
6
| 0.916667 |
A block of measurements 9-inches by 7-inches by 12-inches is filled with as many solid 4-inch cubes as possible. What percentage of the volume of the block is occupied by these cubes?
|
50.79\%
| 0.333333 |
In a different hyperbola setting, the center is at $(1, 3),$ one focus is at $(1, 9),$ and one vertex is at $(1, 0).$ The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\]
Find $h + k + a + b.$
|
7 + 3\sqrt{3}
| 0.916667 |
Let $x$ and $y$ be positive real numbers, and $a$ be a positive constant. Find the maximum value of
\[
\frac{(x+y+a)^2}{x^2+y^2+a^2}.
\]
|
3
| 0.666667 |
If $\log_2 x^2 + \log_4 x = 6,$ compute $x.$
|
2^{\frac{12}{5}}
| 0.833333 |
The Boston weatherman says there is a 75 percent chance of rain for each day of a five-day holiday weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny exactly two of those days for a historical reenactment. What is the probability they get the weather they want? Give your answer as a fraction.
|
\frac{135}{512}
| 0.25 |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. What is the probability that $x^2 + y^2 < y$?
|
\frac{\pi}{32}
| 0.083333 |
Let $g(x) = 2x^6 + 3x^4 - x^2 + 7$. If $g(5) = 29$, find $g(5) + g(-5)$.
|
58
| 0.666667 |
The Estimable Mathematics Institute is organizing its freshman reception. The new freshman class has fewer than **700** people. When these students are arranged in columns of **20**, the last column contains **19** students. When they line up in columns of **25**, the last column has **24** students. When they line up in columns of **9**, there are **3** students in the last column. How many students are in the freshman class?
|
399
| 0.833333 |
The lengths of the sides of a triangle with positive area are $\log_{10}20$, $\log_{10}45$, and $\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$.
|
897
| 0.916667 |
What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect fifth power?
|
2592
| 0.833333 |
Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 12$ and $SR = 8$, what is the volume of pyramid $SPQR$?
|
192
| 0.833333 |
Triangle $DEF$ has sides of $7$ units, $10$ units, and $13$ units. A rectangle, whose area is equal to that of triangle $DEF$, has a length of $7$ units. What is the perimeter of this rectangle?
|
14 + \frac{40\sqrt{3}}{7}
| 0.916667 |
Determine the smallest possible median for the five number set $\{x, 3x, 4, 1, 6\}$ if $x$ can be any integer.
|
1
| 0.333333 |
It is currently $8\!:\!45\!:\!00 \text{ a.m.}$. What time will it be in $9876$ seconds?
|
11\!:\!29\!:\!36 \text{ a.m.}
| 0.416667 |
The expression $512x^3 + 27$ can be decomposed into the form $(ax+b)(cx^2+dx+e)$. Find the value of $a + b + c + d + e$.
|
60
| 0.083333 |
A number between $0000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
|
670
| 0.083333 |
The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 45 minutes? Express your answer in terms of $\pi$.
|
900\pi
| 0.833333 |
The function $y=\frac{x^3+11x^2+38x+35}{x+3}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$?
|
20
| 0.666667 |
Let **a** and **b** be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{5} \\ -\frac{8}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$
|
\begin{pmatrix} \frac{24}{5} \\ -\frac{2}{5} \end{pmatrix}
| 0.833333 |
In a hexagon $ABCDEF$, where angle $A$ is unknown, the following angles are given:
$\angle B = 134^\circ$, $\angle C = 98^\circ$, $\angle D = 120^\circ$, $\angle E = 139^\circ$, $\angle F = 109^\circ$. What is the measure of angle $A$?
|
120^\circ
| 0.166667 |
Solve for \( x \): \( 10(5x + 4) - 4 = -4(2 - 15x) \).
|
\frac{22}{5}
| 0.916667 |
Find the smallest composite number that has no prime factors less than 20.
|
529
| 0.833333 |
In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. Determine the sum of all possible values for $k$.
|
51
| 0.916667 |
Find the remainder when $8x^5 - 10x^4 + 3x^3 + 5x^2 - 7x - 35$ is divided by $x - 5$.
|
19180
| 0.916667 |
Determine the values of $x$ for which $\frac{\log{(5-2x)}}{\sqrt{2x-3}}$ is defined.
|
\left(\frac{3}{2}, \frac{5}{2}\right)
| 0.916667 |
A teacher offers candy to her class of 50 students, with the mean number of pieces taken by each student being 7. If every student takes at least one candy but no more than 20 candies, what is the greatest number of pieces one student could have taken?
|
20
| 0.833333 |
What is the smallest positive integer $n$ such that $n^2$ is divisible by 24, $n^3$ is divisible by 900, and $n^4$ is divisible by 1024?
|
120
| 0.75 |
Find the value of $x$ if \[|x-30| + |x-20| = |3x-90|.\]
|
x = 40
| 0.5 |
If $q(x) = x^4 - 4x + 2$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^3$.
|
-64
| 0.833333 |
Given a quadrilateral $PQRS,$ side $\overline{PQ}$ is extended past $Q$ to $P'$ so that $P'Q = PQ.$ Points $Q'$, $R'$, and $S'$ are similarly constructed.
After this construction, points $P,$ $Q,$ $R,$ and $S$ are erased. You only know the locations of points $P',$ $Q',$ $R'$ and $S',$ and want to reconstruct quadrilateral $PQRS.$
There exist real numbers $u,$ $v,$ $w,$ and $x$ such that
\[\overrightarrow{P} = u \overrightarrow{P'} + v \overrightarrow{Q'} + w \overrightarrow{R'} + x \overrightarrow{S'}.\]
Find the ordered quadruple $(u,v,w,x).$
|
\left( \frac{1}{15}, \frac{2}{15}, \frac{4}{15}, \frac{8}{15} \right)
| 0.083333 |
Emily is thinking of a number. She gives the following 3 clues. "My number has 216 as a factor. My number is a multiple of 45. My number is between 1000 and 3000." What is Emily's number?
|
2160
| 0.833333 |
Let $\mathcal{S}$ be the set $\{1,2,3,\ldots,12\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by $1000$.
|
625
| 0.25 |
Line \( m \) has the equation \( y = 2x + 8 \). Line \( n \) has the equation \( y = kx - 9 \). Lines \( m \) and \( n \) intersect at the point \((-4, 0)\). What is the value of \( k \)?
|
k = -\frac{9}{4}
| 0.75 |
Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=2x-4$. Find $g^{-1}(f(-3))$.
|
\frac{1}{2}
| 0.833333 |
Find the smallest constant $n$, so that for any positive real numbers $a, b, c, d, e,$ we have
\[
\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > n.
\]
|
2
| 0.25 |
Two cross sections of a right pentagonal pyramid are obtained by cutting the pyramid with planes parallel to the pentagonal base. The areas of the cross sections are \(125\sqrt{3}\) square feet and \(500\sqrt{3}\) square feet. The two planes are \(12\) feet apart. How far from the apex of the pyramid is the larger cross section, in feet?
|
24
| 0.833333 |
The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.
|
813
| 0.916667 |
Find the equation of the oblique asymptote for the graph of the function $\frac{3x^2 + 8x + 12}{3x + 4}$.
|
y = x + \frac{4}{3}
| 0.916667 |
What is the sum of the digits of the base $8$ representation of $888_{10}$?
|
13
| 0.833333 |
If $x$ is an even number, then find the largest integer that always divides the expression \[(15x+3)(15x+9)(5x+10).\]
|
90
| 0.333333 |
Suppose that we have a right triangle $DEF$ with the right angle at $E$ such that $DF = \sqrt{85}$ and $DE = 7$. A circle is drawn with its center on $DE$ such that the circle is tangent to $DF$ and $EF.$ If $Q$ is the point where the circle and side $DF$ meet, then what is $FQ$?
|
FQ = 6
| 0.25 |
How many distinct arrangements of the letters in the word "balloon" are there?
|
1260
| 0.666667 |
Each of the numbers $b_1, b_2, \dots, b_{97}$ is $\pm 1$. Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 97} b_i b_j.\]
|
12
| 0.75 |
What is the smallest positive integer that can be added to 725 to make it a multiple of 5?
|
5
| 0.5 |
Define \( g \) by \( g(x)=3x-2 \). If \( g(x)=f^{-1}(x)-2 \) and \( f^{-1}(x) \) is the inverse of the function \( f(x)=ax+b \), find \( 3a+4b \).
|
1
| 0.916667 |
Find all values of \( x > 6 \) which satisfy
\[
\sqrt{x - 6 \sqrt{x - 9}} + 3 = \sqrt{x + 6 \sqrt{x - 9}} - 3.
\]
|
[18, \infty)
| 0.166667 |
A square has a side length of 8 cm, and each vertex of the square is the center of a circle. Each circle has a radius of 3 cm. Calculate the area of the shaded region formed in the square, assuming that the circles intersect similar to the original problem.
|
64 - 9\pi \text{ cm}^2
| 0.666667 |
How many positive perfect square integers are factors of the product $\left(2^{14}\right)\left(3^{9}\right)\left(5^{20}\right)$?
|
440
| 0.833333 |
Each of the first ten prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number greater than 10?
|
\frac{1}{15}
| 0.75 |
A cylindrical container of honey, which is 4 inches in diameter and 5 inches high, sells for $\$$0.80. At the same rate, what would be the price for a container that is 8 inches in diameter and 10 inches high?
|
\$6.40
| 0.916667 |
What is the remainder when $1520 \cdot 1521 \cdot 1522$ is divided by 17?
|
11
| 0.833333 |
Point \(Q\) lies on the line \(x= 1\) and is 8 units from the point \((-4, -3)\). Find the product of all possible \(y\)-coordinates that satisfy the given conditions.
|
-30
| 0.916667 |
Simplify the product \[\frac{10}{5} \cdot \frac{15}{10} \cdot \frac{20}{15} \dotsm \frac{5n + 5}{5n} \dotsm \frac{2520}{2515}.\]
|
504
| 0.5 |
Yan is somewhere between his home and the stadium. He can either walk directly to the stadium or walk back home and then ride his bicycle to the stadium. He rides 9 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
|
\frac{4}{5}
| 0.833333 |
Let $T$ be the set of 12-tuples $(b_0, b_1, \dots, b_{11})$, where each entry is 0 or 1, so $T$ contains $2^{12}$ 12-tuples. For each 12-tuple $t = (b_0, b_1, \dots, b_{11})$ in $T$, let $q_t(x)$ be the polynomial of degree at most 11 such that
\[q_t(n) = b_n\]for $0 \le n \le 11.$
Find
\[\sum_{t \in T} q_t(12).\]
|
2048
| 0.833333 |
Simplify
\[
\frac{\sin{25^\circ}+\sin{35^\circ}}{\cos{25^\circ}+\cos{35^\circ}}.
\]
Enter your answer as a trigonometric function evaluated at an integer, such as "sin 7". (The angle should be positive and as small as possible.)
|
\tan 30^\circ
| 0.666667 |
Let $A$ be the greatest common factor and let $B$ be the least common multiple of 12, 18, and 30. Find twice the value of $A$ added to $B$. What is the value of $2A + B$?
|
192
| 0.833333 |
Let \(a\), \(b\), and \(c\) be angles such that
\[
\cos a = \tan b, \quad \cos b = \tan c, \quad \cos c = \tan a.
\]
Find the largest possible value of \(\sin a\).
|
\frac{\sqrt{5} - 1}{2}
| 0.333333 |
Place each of the digits 3, 4, 7, and 8 in exactly one square to make the smallest possible product. [asy]draw((0,.5)--(10,.5),linewidth(1));
draw((4,1)--(6,1)--(6,3)--(4,3)--(4,1),linewidth(1));
draw((7,1)--(9,1)--(9,3)--(7,3)--(7,1),linewidth(1));
draw((7,4)--(9,4)--(9,6)--(7,6)--(7,4),linewidth(1));
draw((4,4)--(6,4)--(6,6)--(4,6)--(4,4),linewidth(1));
draw((1,3)--(2,4),linewidth(1));
draw((1,4)--(2,3),linewidth(1)); [/asy]
|
1776
| 0.916667 |
Consider the arithmetic sequence $2$, $5$, $8$, $11$, $\ldots$. Find the $20^\text{th}$ term and the sum of the first $20$ terms.
|
610
| 0.25 |
What is the smallest five-digit positive integer that is divisible by 53?
|
10017
| 0.75 |
In Mr. Kennedy's class, 8 of the 20 students received an A on the latest exam. If the same ratio of students received an A in Mr. Holden's exam where he teaches 30 students, how many students in Mr. Holden's class received an A? If Mr. Holden had an additional quiz where the ratio of students who did not receive an A was half as much as those in the exam, how many students did not receive an A in the quiz?
|
9
| 0.833333 |
How many ways are there to choose 4 cards from a standard deck of 52 cards, if all four cards must be of different suits?
|
28561
| 0.916667 |
What is the smallest positive perfect square that is divisible by both 5 and 7?
|
1225
| 0.75 |
At the end of a game, each of the six members of a basketball team shakes hands with each of the six members of the other team, each player shakes hands with each of his own team members, and all of the players shake hands with three referees. How many handshakes occur?
|
102
| 0.666667 |
James now has 10 apples, 6 of which are red and 4 of which are green. If he chooses 3 apples at random, what is the probability that at least one of the apples he chooses is green?
|
\frac{5}{6}
| 0.916667 |
Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}.$
|
\begin{pmatrix} \frac{9}{14} & -\frac{3}{14} & \frac{6}{14} \\ -\frac{3}{14} & \frac{1}{14} & -\frac{2}{14} \\ \frac{6}{14} & -\frac{2}{14} & \frac{4}{14} \end{pmatrix}
| 0.25 |
A line segment begins at $(2, 2)$ and ends at the point $(x, 5)$, with $x > 0$. If the length of the line segment is 6 units, what is the value of $x$?
|
2 + 3\sqrt{3}
| 0.833333 |
Starting with the number 250, Shaffiq repeatedly divides his number by three and then takes the greatest integer less than or equal to that number. How many times must he do this before he reaches the number 1?
|
5
| 0.916667 |
Given $ab + ac + bd + cd = 40$ and $a+d = 6$, find $b+c$.
|
\frac{20}{3}
| 0.416667 |
Determine the smallest integer $y$ such that $\frac{y}{4} + \frac{3}{7} > \frac{4}{7}$.
|
1
| 0.916667 |
Compute $\frac{x^8 + 16x^4 + 64 + 4x^2}{x^4 + 8}$ when $x = 3$.
|
89 + \frac{36}{89}
| 0.083333 |
Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is the length of $QS$?
|
5
| 0.666667 |
A point is selected at random from the portion of the number line from 0 to 8. What is the probability that the point is closer to 6 than to 0? Express your answer as a decimal to the nearest tenth.
|
0.6
| 0.916667 |
Ryosuke drives his friend from a cafe to a library. The odometer reads 85,340 when he picks his friend up at the cafe, and it reads 85,368 when he drops his friend off at the library. Ryosuke's car gets 32 miles per gallon and the price of one gallon of gas is $\$3.95$. Calculate the cost of the gas used for this journey.
|
\$3.46
| 0.75 |
Real numbers $a$ and $b$ are chosen with $2<a<b$ such that no triangle with positive area has side lengths $2, a,$ and $b$. Additionally, it is given that $a + b = 7$. What is the smallest possible value of $b$?
|
\frac{9}{2}
| 0.333333 |
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