problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
If \( 5x - 10 = 15x + 5 \), what is \( 5(x+3) \)?
|
\frac{15}{2}
| 0.916667 |
Convert $\rm{BFACE}_{16}$ to a base 10 integer, where the 'digits' A through F represent the values 10, 11, 12, 13, 14, and 15 respectively.
|
785102
| 0.583333 |
Convert the point $(0, 3, -3 \sqrt{3})$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$
|
\left( 6, \frac{\pi}{2}, \frac{5\pi}{6} \right)
| 0.833333 |
**
Consider a set of numbers $\{1, 2, 3, 4, 5, 6, 7\}$. Two different natural numbers are selected at random from this set. What is the probability that the greatest common divisor (gcd) of these two numbers is one? Express your answer as a common fraction.
**
|
\frac{17}{21}
| 0.416667 |
The foci of the ellipse \[\frac{x^2}{25} + \frac{y^2}{b^2} = 1\] and the foci of the hyperbola
\[\frac{x^2}{64} - \frac{y^2}{36} = \frac{1}{16}\] coincide. Find $b^2$.
|
b^2 = \frac{75}{4}
| 0.833333 |
Suppose the point $(2,3)$ is on the graph of $y = 2f(x)$. Determine the point that must be on the graph of $y = \frac{f^{-1}(x)}{3}$ and find the sum of that point's coordinates.
|
\frac{13}{6}
| 0.75 |
Simplify $8 \cdot \frac{15}{4} \cdot \frac{-40}{45}$.
|
-\frac{80}{3}
| 0.916667 |
What is the coefficient of \( x^5 \) when $$x^5 - 4x^4 + 6x^3 - 5x^2 + 2x + 1$$ is multiplied by $$3x^4 - 2x^3 + x^2 + 4x - 8$$ and the like terms are combined?
|
-2
| 0.166667 |
Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction.
|
\frac{5}{32}
| 0.833333 |
What is the sum of the positive even divisors of 180?
|
468
| 0.833333 |
At the beginning of my bike ride I feel good, so I can travel 15 miles per hour. Later, I get tired and travel only 10 miles per hour. If I travel a total of 100 miles in a total time of 8 hours, for how many hours did I feel good? Express your answer as a common fraction.
|
4
| 0.5 |
A 12-foot by 15-foot floor is tiled with square tiles of size 2 feet by 2 feet. Each tile has a pattern consisting of four white quarter circles of radius 1 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
|
180 - 45\pi
| 0.5 |
The positive integers \( A, B \) and \( C \) form an arithmetic sequence, while the integers \( B, C \) and \( D \) form a geometric sequence. If \( \frac{C}{B} = \frac{7}{4}, \) what is the smallest possible value of \( A + B + C + D \)?
|
97
| 0.916667 |
How many positive integers smaller than $10{,}000$ are powers of $2$, but are not powers of $4$?
|
7
| 0.833333 |
Determine the number of pairs $(x, y)$ such that $12, x, y, xy$ form an arithmetic progression.
|
2
| 0.25 |
Let points $A = (4, \phi_1)$ and $B = (10, \phi_2)$ in polar coordinates, where $\phi_1 - \phi_2 = \frac{\pi}{4}$. Determine the distance $AB$.
|
\sqrt{116 - 40\sqrt{2}}
| 0.666667 |
Find the area of a triangle with side lengths 10, 11, and 11.
|
20\sqrt{6}
| 0.916667 |
What is the sum of every third odd integer between $200$ and $500$?
|
17400
| 0.25 |
Starting with the number 200, Declan repeatedly divides his number by 3 and then takes the greatest integer less than or equal to that result. How many times must he do this before he reaches a number less than 2?
|
5
| 0.75 |
For the quadrilateral given, how many different whole numbers could be the length of the diagonal represented by the dashed line?
[asy]
draw((0,0)--(5,5)--(12,1)--(7,-8)--cycle,linewidth(0.7));
draw((0,0)--(12,1),dashed);
label("9",(2.5,2.5),NW);
label("11",(8.5,3),NE);
label("17",(9.5, -3.5),SE);
label("13",(3.5,-4),SW);
[/asy]
|
15
| 0.916667 |
How many positive integers less than $1000$ are either multiples of $5$ or multiples of $10$?
|
199
| 0.916667 |
April has five different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?
|
17280
| 0.916667 |
What is the area enclosed by the graph of $|x| + |3y| = 12$?
|
96
| 0.75 |
How many integers $n$ satisfy the double inequality $-12\sqrt{\pi} \le n^2 \le 15\pi$?
|
13
| 0.833333 |
What is the sum of the $x$-values that satisfy the equation $7=\frac{x^3 - 3x^2 - 10x}{x+2}$?
|
5
| 0.916667 |
Given that $\sec x + \tan x = \frac{5}{4},$ find all possible values of $\sin x.$
|
\frac{9}{41}
| 0.916667 |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is an ace?
|
\frac{1}{52}
| 0.25 |
If $\log_7 (x+4) = 3$, find $\log_{13} x$.
|
\log_{13} 339
| 0.916667 |
For some real numbers $c$ and $d$, the equation $9x^3 + 5cx^2 + 6dx + c = 0$ has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 6, what is the value of $c$?
|
-576
| 0.833333 |
In isosceles triangle $ABC$, the base $BC$ is $10$ units long and the height from $A$ perpendicular to $BC$ is $6$ units. What is the perimeter of triangle $ABC$?
|
2\sqrt{61} + 10
| 0.833333 |
Using the same rules for assigning values to letters as described (with a repeating pattern of $1, 2, 1, 0, -1, -2, -1, 0$):
Calculate the sum of the numeric values of the letters in the word "algebra".
|
4
| 0.333333 |
Determine the smallest possible perimeter of a scalene triangle where:
1. All sides are distinct prime numbers greater than 3.
2. The perimeter is a prime number.
|
23
| 0.75 |
When a polynomial is divided by $3x^2 - 5x + 12,$ what are the possible degrees of the remainder?
|
0, 1
| 0.666667 |
Four dwarf planets have been added to the solar system tally, each with distinct moon counts. Now, what is the median number of moons per celestial body in the expanded list? The counts are as follows:
\begin{tabular}{c|c}
Celestial Body & $\#$ of Moons \\
\hline
Mercury & 0 \\
Venus & 0 \\
Earth & 1 \\
Mars & 2 \\
Jupiter & 20 \\
Saturn & 22 \\
Uranus & 14 \\
Neptune & 2 \\
Pluto & 5 \\
Ceres & 0 \\
Eris & 1 \\
Haumea & 2 \\
Makemake & 3 \\
\end{tabular}
|
2
| 0.916667 |
In a classroom arranged in a row of 10 chairs, Alex and Jamie choose their seats at random. What is the probability that they do not sit next to each other?
|
\frac{4}{5}
| 0.916667 |
A pyramid has a rectangular base measuring $10 \times 12$. Each of the four edges joining the apex to the corners of the rectangular base has length $15$. Calculate the volume of the pyramid.
|
80\sqrt{41}
| 0.833333 |
Let $\mathbf{B} = \begin{pmatrix} x & 2 \\ -3 & y \end{pmatrix}$ for some real numbers $x$ and $y.$ If
\[\mathbf{B} + 2\mathbf{B}^{-1} = \mathbf{0},\] then find $\det \mathbf{B}.$
|
2
| 0.833333 |
If \( h(x) = 3x^2 + x - 4 \), what is the value of \( h(h(3)) \)?
|
2050
| 0.916667 |
Find the integer \( n \), \( -90 \le n \le 90 \), such that \( \sin n^\circ = \sin 721^\circ \).
|
1
| 0.75 |
How many distinct, positive factors does $1320$ have?
|
32
| 0.916667 |
What is the sum of all integer solutions to the inequality \(1 < (x-3)^2 < 36\)?
|
24
| 0.583333 |
What is the least integer greater than $\sqrt{500}$?
|
23
| 0.916667 |
Given the following data, determine how much cheaper, in cents, is the cheapest store's price for Camera $Y$ compared to the most expensive one?
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Camera $Y$} \\ \hline
Mega Deals & $\$12$~off the list price~$\$52.50$ \\ \hline
Budget Buys & $30\%$~off the list price~$\$52.50$ \\ \hline
Frugal Finds & $20\%$~off the list price~$\$52.50$ plus an additional $\$5$~off \\ \hline
\end{tabular}
|
375
| 0.916667 |
What is the shortest distance between the circles defined by $x^2 - 8x + y^2 + 6y + 9 = 0$ and $x^2 + 10x + y^2 - 2y + 25 = 0$?
|
\sqrt{97} - 5
| 0.75 |
The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?
|
105
| 0.916667 |
A bizarre weighted coin comes up heads with probability $\frac{1}{4}$, tails with probability $\frac{1}{2}$, and rests on its edge with probability $\frac{1}{4}$. If it comes up heads, you win 1 dollar. If it comes up tails, you win 3 dollars. But if it lands on its edge, you lose 8 dollars. What are your expected winnings from flipping this coin? Express your answer as a dollar value.
|
-\$0.25
| 0.25 |
Find the value of $x$ if the fourth power of the square root of $x$ is 256.
|
16
| 0.916667 |
What is the smallest positive integer $n$ such that the product $n(n+1)$ is divisible by some but not all integer values of $k$ within the range $1\leq k \leq n$?
|
4
| 0.833333 |
A cylinder has a radius of 5 cm and a height of 12 cm. Calculate both the longest segment that would fit inside this cylinder and the volume of the cylinder.
|
300\pi \text{ cm}^3
| 0.25 |
The first $20$ numbers of an arrangement are shown below. What would be the value of the $50^{\mathrm{th}}$ number if the arrangement were continued?
$\bullet$ Row 1: $2,$ $2$
$\bullet$ Row 2: $4,$ $4,$ $4,$ $4$
$\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$
$\bullet$ Row 4: $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8$
|
14
| 0.75 |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 4x + 3|$ and $y = 7 - 2x$.
|
2
| 0.75 |
A sports conference is organized into three divisions. Division A contains 6 teams, Division B contains 7 teams, and Division C contains 5 teams. Each team must play every other team in its own division twice and every team in the other divisions exactly twice. How many games are in a complete season for the conference?
|
306
| 0.416667 |
How many positive integers less than $300$ are multiples of $5$, but not multiples of $15$?
|
40
| 0.916667 |
Similarly, by restricting the domain of the function $f(x) = 3x^2 + 6x - 8$ to an interval, we can make it invertible. What is the largest such interval that includes the point $x=2$?
|
[-1, \infty)
| 0.916667 |
In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 48. What is the perimeter of the shaded region formed similarly as before with the triangular region where each circle touches the other two?
|
24
| 0.25 |
The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 4 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -\frac{3}{2} \\ -2 \end{pmatrix}\] lie on a line. Find the equation of this line in the form "$y = mx + b$".
|
y = -\frac{3}{4}x - \frac{25}{8}
| 0.916667 |
The graph below indicates the number of home runs hit in May by the top hitters in a baseball league. Calculate the mean (average) number of home runs hit by these players for the month.
[asy]
draw((18,0)--(0,0)--(0,18));
label("5",(3,-1));
label("6",(6,-1));
label("8",(9,-1));
label("9",(12,-1));
label("11",(15,-1));
fill((3,.5)..(3.5,1)..(3,1.5)..(2.5,1)..cycle);
fill((3,2)..(3.5,2.5)..(3,3)..(2.5,2.5)..cycle);
fill((6,.5)..(6.5,1)..(6,1.5)..(5.5,1)..cycle);
fill((6,2)..(6.5,2.5)..(6,3)..(5.5,2.5)..cycle);
fill((6,3.5)..(6.5,4)..(6,4.5)..(5.5,4)..cycle);
fill((9,.5)..(9.5,1)..(9,1.5)..(8.5,1)..cycle);
fill((9,2)..(9.5,2.5)..(9,3)..(8.5,2.5)..cycle);
fill((12,.5)..(12.5,1)..(12,1.5)..(11.5,1)..cycle);
fill((15,.5)..(15.5,1)..(15,1.5)..(14.5,1)..cycle);
label("Number of Home Runs",(9,-3));
picture perpLabel;
label(perpLabel,"Number of Top Hitters");
add(rotate(90)*perpLabel,(-1,9));
[/asy]
|
\frac{64}{9}
| 0.5 |
What is the area, in square units, of a triangle with vertices at $A(1, 3), B(6, 3), C(4, 9)$?
|
15 \text{ square units}
| 0.916667 |
If $5x + 3 \equiv 1 \pmod{18}$, then $3x + 8$ is congruent $\pmod{18}$ to what integer between $0$ and $17$, inclusive?
|
14
| 0.666667 |
In a mixed-doubles tennis exhibition, there were four teams, each consisting of one man and one woman. After the exhibition, each player shook hands exactly once with every other player except with their team partner. How many handshakes occurred?
|
24
| 0.916667 |
Find the sum of the coefficients in the polynomial $-3(x^8 - 2x^5 + 4x^3 - 6) + 5(x^4 + 3x^2) - 4(x^6 - 5)$.
|
45
| 0.833333 |
Suppose $179\cdot 933 / 7 \equiv n \pmod{50}$, where $0 \le n < 50$.
|
1
| 0.75 |
$ABCD$ is a trapezoid where $\overline{AB}$ is three times the length of $\overline{CD}$, and the height from $D$ to line $AB$ is 5 units. Point $E$ is the intersection of the diagonals. If the length of diagonal $\overline{AC}$ is 15, find the length of segment $\overline{EC}$. Express your answer as a common fraction.
|
\frac{15}{4}
| 0.916667 |
**Sarah has misplaced her friend Layla's phone number. Sarah recalls that the first four digits are either 3086, 3089, or 3098. The remaining digits are 0, 1, 2, and 5, but she is not sure about their order. If Sarah randomly calls a seven-digit number based on these criteria, what is the probability that she correctly dials Layla's phone number? Express the answer as a common fraction.**
|
\frac{1}{72}
| 0.583333 |
Let $f(x) = 5x^2 - 4$ and $g(f(x)) = x^2 + x + x/3 + 1$. Find the sum of all possible values of $g(49)$.
|
\frac{116}{5}
| 0.75 |
How many ways can 2023 be factored as a product of two two-digit numbers? (Two factorizations of the form $a\cdot b$ and $b\cdot a$ are considered the same).
|
0
| 0.666667 |
In a diagram, there are two concentric circles where the visible gray area between the larger and smaller circles is equal to four times the area of the smaller white circular region enclosed by the smaller circle. Determine the ratio of the radius of the small circle to the radius of the large circle. Express your answer as a common fraction.
|
\frac{1}{\sqrt{5}}
| 0.083333 |
What is the remainder when \( 4x^8 - 3x^7 + 2x^6 - 8x^4 + 5x^3 - 9 \) is divided by \( 3x - 6 \)?
|
671
| 0.833333 |
Rationalize the denominator for the expression $\sqrt[3]{\frac{4}{9}}.$
|
\frac{\sqrt[3]{324}}{9}
| 0.5 |
Simplify the product \[\frac{10}{5}\cdot\frac{15}{10}\cdot\frac{20}{15} \dotsm \frac{5n+5}{5n} \dotsm \frac{1030}{1025}.\]
|
206
| 0.666667 |
Let $a$ and $b$ be integers such that $ab = 144$. Find the minimum value of $a + b$.
|
-145
| 0.916667 |
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 $pq - 5p - 3q = 3$? Express your answer as a common fraction.
|
\frac{7}{15}
| 0.416667 |
Each letter represents a non-zero digit. What is the value of $s?$ \begin{align*}
a + b &= x \\
x + c &= s \\
s + a &= z \\
b + c + z &= 16
\end{align*}
|
8
| 0.916667 |
Find the distance between the centers of the inscribed and the circumscribed circles of a right triangle with sides 6, 8, and 10 units.
|
\sqrt{5}
| 0.666667 |
A number $x$ is equal to $6 \cdot 18 \cdot 42$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?
|
441
| 0.833333 |
Find the value of $k$ so that:
\[ 5 + \frac{5 + 2k}{4} + \frac{5 + 4k}{4^2} + \frac{5 + 6k}{4^3} + \dotsb = 10. \]
|
\frac{15}{4}
| 0.416667 |
Warren wishes to solve the equation \(25x^2+30x-45=0\) by completing the square. He reformulates the equation as \((ax+b)^2=c\), where \(a\), \(b\), and \(c\) are integers and \(a>0\). Determine the value of \(a + b + c\).
|
62
| 0.916667 |
A circle centered at $O$ is circumscribed about $\triangle ABC$ as shown in the diagram. Assume angle measurements $\angle AOC = 140^\circ$ and $\angle BOC = 90^\circ$. What is the measure of $\angle BAC$, in degrees?
|
45^\circ
| 0.916667 |
Find the remainder when \( x^5 + 2x^3 + x + 3 \) is divided by \( x-4 \).
|
1159
| 0.5 |
Compute $\binom{19}{10}$. You are told that $\binom{17}{7} = 19448$ and $\binom{17}{9} = 24310$.
|
92378
| 0.916667 |
Add $91.234$ to $42.7689$ and round your answer to the nearest hundredth.
|
134.00
| 0.833333 |
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? Additionally, calculate the total surface area of this cylinder.
|
150\pi
| 0.5 |
Let $B,$ $S,$ $N,$ and $K$ be positive real numbers such that
\begin{align*}
\log_{10} (BK) + \log_{10} (BN) &= 3, \\
\log_{10} (NK) + \log_{10} (NS) &= 4, \\
\log_{10} (SB) + \log_{10} (SK) &= 5.
\end{align*}
Compute the value of the product $BSNK.$
|
10000
| 0.75 |
In a row of 10 chairs, two chairs (numbered 5 and 6) are broken and cannot be used. Mary and James each choose their remaining seats at random. What is the probability that they do not sit next to each other?
|
\frac{11}{14}
| 0.166667 |
The line $y = c-x$ where $0 < c < 6$ intersects the $y$-axis at $P$ and the line $x=6$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 4:16, what is the value of $c$?
|
4
| 0.25 |
Find $n$ such that $2^6 \cdot 3^3 \cdot n = 9!$.
|
210
| 0.75 |
Let $Q$ be a cubic polynomial such that $Q(0) = m$, $Q(1) = 3m$, and $Q(-1) = 4m$. Find the value of $Q(2) + Q(-2)$.
|
22m
| 0.833333 |
Consider license plates consisting of a sequence of four digits followed by two letters. Assume each arrangement is equally likely for these plates. What is the probability that such a license plate contains at least one palindrome sequence (either the four-digit sequence or the two-letter sequence)? Express your result as a simplified fraction.
|
\frac{5}{104}
| 0.083333 |
Chris and Dana each arrive at a cafe at a random time between 3:00 PM and 4:00 PM. Each stays for 20 minutes. What is the probability that Chris and Dana meet at the cafe?
|
\frac{5}{9}
| 0.916667 |
Two spinners are divided into fifths and sixths, respectively. Spinner A (in fifths) has numbers 2, 4, 5, 7, and 9, while Spinner B (in sixths) displays 3, 6, 7, 8, 10, and 12. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number?
|
\frac{4}{5}
| 0.75 |
A geologist has discovered a peculiar mineral that has an age represented by the digits 1, 1, 2, 3, 7, and 9. He claims that the age of the mineral begins with a prime number. How many different ages can be formed under these conditions?
|
180
| 0.833333 |
Evaluate $\log_3 81\sqrt{9}$. Express your answer as an improper fraction.
|
5
| 0.916667 |
Line segment $\overline{AB}$ is extended past $B$ to point $Q$ such that $AQ:QB = 5:2.$ Then
\[\overrightarrow{Q} = s \overrightarrow{A} + v \overrightarrow{B}\] for some constants $s$ and $v.$ Enter the ordered pair $(s,v).$
|
\left(-\frac{2}{3}, \frac{5}{3}\right)
| 0.666667 |
Compute \[
\left\lfloor \frac{2008! + 2005!}{2007! + 2006!}\right\rfloor.
\]
|
2007
| 0.833333 |
Rationalize the denominator of $\frac{7}{3+\sqrt{15}}$ and then subtract $\frac{1}{2}$ from the result.
|
-4 + \frac{7\sqrt{15}}{6}
| 0.25 |
Suppose we have two numbers, $29_{10}$ and $45_{10}$. If $29_{10}$ is first converted to base 4 and $45_{10}$ to base 5, what is the sum of these two numbers in base 5?
|
244_5
| 0.916667 |
For how many integer values of \(b\) does the equation \(x^2 + bx + 9b = 0\) have integer solutions for \(x\)?
|
6
| 0.916667 |
For how many positive integers $n$ less than or equal to 500 is $$(\sin (t+\frac{\pi}{4})+i\cos (t+\frac{\pi}{4}))^n=\sin (nt+\frac{n\pi}{4})+i\cos (nt+\frac{n\pi}{4})$$ true for all real $t$?
|
125
| 0.416667 |
Consider rows 1003, 1004, and 1005 of Pascal's triangle.
Let \((a_i)\), \((b_i)\), \((c_i)\) be the sequences, from left to right, of elements in the 1003rd, 1004th, and 1005th rows, respectively, with the leftmost element occurring at \(i = 0\). Compute
\[
\sum_{i = 0}^{1004} \frac{b_i}{c_i} - \sum_{i = 0}^{1003} \frac{a_i}{b_i}.
\]
|
\frac{1}{2}
| 0.75 |
Let \(x\) and \(y\) be positive real numbers such that \(2x + 3y = 4.\) Find the minimum value of
\[
\frac{2}{x} + \frac{3}{y}.
\]
|
\frac{25}{4}
| 0.916667 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.