problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
A regular octagon $ABCDEFGH$ has its sides' midpoints connected to form a smaller octagon. What fraction of the area of the larger octagon $ABCDEFGH$ is enclosed by the smaller octagon?
|
\frac{2 + \sqrt{2}}{4}
| 0.166667 |
Determine how many distinct five-digit positive integers make the product of their digits equal 18.
|
70
| 0.5 |
Let $a \star b = 2ab - 3b - a$. If $4 \star y = 80$, find the value of $y$.
|
16.8
| 0.083333 |
What is the smallest odd number with five different prime factors?
|
15015
| 0.833333 |
The average of four different positive integers is $5$. If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers?
|
2.5
| 0.75 |
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle XYZ$ are perpendicular. If $AD = 18$ and $BE = 24$, determine the area of $\triangle XYZ$.
|
288
| 0.5 |
Suppose all four numbers \[-1+\sqrt{5}, \; -1-\sqrt{5}, \; 2-\sqrt{3}, \; 2+\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?
|
4
| 0.916667 |
A sports conference comprises 16 teams divided equally into two divisions of 8 teams each. How many games are in a complete season for the conference if each team must play every other team in its own division three times and every team in the other division twice?
|
296
| 0.916667 |
The function $g(x)$ satisfies
\[xg(y) = yg(x)\]
for all real numbers $x$ and $y.$ If $g(20) = 30,$ find $g(5).$
|
7.5
| 0.416667 |
Let $A = (0,9),$ $B = (6,9),$ and $C = (p,q)$ be three points on the parabola $y = x^2 - 6x + 9,$ where $1 \le p \le 6.$ Find the largest possible area of triangle $ABC.$
|
27
| 0.583333 |
Let vectors $\mathbf{u}$ and $\mathbf{v}$ be such that $\|\mathbf{u}\| = 9$ and $\|\mathbf{v}\| = 13$. Find all possible values of $\mathbf{u} \cdot \mathbf{v}$ if the angle between them, $\theta$, is between $\frac{\pi}{6}$ and $\frac{\pi}{3}$.
|
[58.5, 58.5\sqrt{3}]
| 0.166667 |
Two standard dice are rolled. What is the expected number of 6's obtained? Express your answer as a common fraction.
|
\frac{1}{3}
| 0.916667 |
Consider two lines: line $l$ parametrized as
\begin{align*}
x &= 2 + 5t,\\
y &= 3 + 4t
\end{align*}
and the line $m$ parametrized as
\begin{align*}
x &= -3 + 5s,\\
y &= 5 + 4s.
\end{align*}
Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.
Assume $\overrightarrow{PA}$ is the projection of $\overrightarrow{BA}$ onto some vector $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ such that $3v_1 + v_2 = 4$. Find $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$.
|
\begin{pmatrix} \frac{16}{7} \\ -\frac{20}{7} \end{pmatrix}
| 0.416667 |
A $168 \times 350 \times 390$ rectangular solid is constructed by gluing together $1 \times 1 \times 1$ cubes. An internal diagonal of this solid passes through how many of the $1 \times 1 \times 1$ cubes?
|
880
| 0.916667 |
The deli has five kinds of bread, seven kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, turkey, cheddar cheese, and rye bread are each offered at the deli. If Al never orders a sandwich with a ham/cheddar cheese combination nor a sandwich with a rye bread/turkey combination, how many different sandwiches could Al order?
|
165
| 0.666667 |
Find the remainder when $123456789012$ is divided by $180$.
|
12
| 0.166667 |
Cylinder $C$'s height is equal to the radius of cylinder $D$, and cylinder $C$'s radius is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, express the volume of cylinder $D$ in the form $M \pi h^3$ cubic units. What is the value of $M$?
|
9
| 0.25 |
In a different sequence, the first term is $a_1 = 2000$ and the second term remains $a_2 = 2008$. The values of the remaining terms are set such that:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\]
for all $n \geq 1$. Determine $a_{1000}$.
|
2666
| 0.166667 |
In the expansion of $(x+1)^{50}$, what is the coefficient of the $x^3$ term?
|
19600
| 0.916667 |
In writing the integers from 100 through 999 inclusive, how many times is the digit 6 written?
|
280
| 0.583333 |
A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid).
|
34
| 0.166667 |
Note that $8^2 = 64$, which contains no zeros; $88^2 = 7744$, which contains no zeros; and $888^2 = 788,\!944$, which contains 1 zero. Assuming this pattern continues, how many zeros are in the expansion of $88,\!888,\!888^2$?
|
6
| 0.333333 |
How many non-similar convex quadrilaterals have angles whose degree measures are distinct positive integers in arithmetic progression?
|
29
| 0.75 |
Alan is 4 years younger than Bob. Bob is 5 years older than Carl. Donna is two years older than Carl. Emily is the sum of Alan's and Donna's age minus Bob's age. If Bob is 20 years old, how old is Emily?
|
13
| 0.916667 |
In the diagram, $O$ is the center of a circle with radii $OP=OQ=r$. What is the perimeter of the shaded region assuming the arc $PQ$ corresponds to exactly half of the circle?
[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(0,-1),S);
fill(Arc((0,0),1,-90,0)--cycle,mediumgray);
draw(Arc((0,0),1,-90,0));
fill((0,0)--(-1,0)--(0,-1)--cycle,white);
draw((-1,0)--(0,0)--(0,-1));
draw((-.1,0)--(-.1,-.1)--(0,-.1));
[/asy]
|
2r + \pi r
| 0.083333 |
What is the number of square meters in the area of a circle with a diameter of \(7.5\) meters? Express your answer in terms of \(\pi\).
|
14.0625\pi
| 0.916667 |
Calculate the number of digits in the value of $2^{15} \times 5^{10} \times 12$.
|
13
| 0.583333 |
A certain integer has $5$ digits when written in base $7$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?
|
17
| 0.833333 |
Find all values of $x$ such that $\arctan x > \arcsin x$.
|
[-1, 0)
| 0.083333 |
What is the simplified value of the sum: $-1^{2010} + (-1)^{2011} + 1^{2012} -1^{2013}$?
|
-2
| 0.916667 |
Compute $[(\dbinom{10}{3})!]$.
|
120!
| 0.916667 |
The graph of \(y = h(x)\) is defined in parts as follows:
1. For \(x \geq -4\) and \(x \leq 1\), \(y = 1 - x\).
2. For \(x \geq 1\) and \(x \leq 3\), \(y = \sqrt{4 - (x - 3)^2} + 1\).
3. For \(x \geq 3\) and \(x \leq 4\), \(y = 2(x - 3) + 1\).
The graph of \(y = j(x)\) is derived from the graph of \(y = h(x)\). First, the graph of \(y = h(x)\) is reflected over the y-axis and then shifted 6 units to the right. What is \(j(x)\) in terms of \(h(x)\)?
|
j(x) = h(6 - x)
| 0.166667 |
If $p$, $q$, and $r$ are positive integers such that $\gcd(p,q) = 210$ and $\gcd(p,r) = 770$, what is the smallest possible value of $\gcd(q,r)$?
|
70
| 0.916667 |
Determine how many times the graphs \( r = 3 \cos \theta \) and \( r = 6 \sin \theta \) intersect.
|
2
| 0.916667 |
Five years ago, there were 30 trailer homes on Maple Street with an average age of 12 years. At that time, a group of brand new trailer homes was then added to Maple Street. Today, the average age of all the trailer homes on Maple Street is 10 years. How many new trailer homes were added five years ago?
|
42
| 0.833333 |
Find the base \( b \) such that when the number \( 64_{10} \) is expressed in base \( b \), it has exactly 4 digits.
|
4
| 0.666667 |
In triangle $XYZ$, medians $\overline{XM}$ and $\overline{YN}$ intersect at right angles at point $O$. If $XM = 18$ and $YN = 24$, find the area of $\triangle XYZ$.
|
288
| 0.583333 |
Below is a portion of the graph of a quadratic function, $y=p(x)=ax^2+bx+c$:
\[ \text{The graph has symmetry such that } p(10) = p(20) \text{ and } p(9) = p(21) \text{, etc., with an axis of symmetry at } x = 15. \]
Given that $p(25) = 9$ and the graph goes through the point $(0, 1)$, calculate $p(5)$.
|
9
| 0.583333 |
The equation \( y = -16t^2 + 64t \) describes the height (in feet) of a projectile launched from the ground at 64 feet per second. At what \( t \) will the projectile reach 25 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth.
|
0.4
| 0.916667 |
In the diagram, each circle is divided into two equal areas, and $O$ is the center of the larger circle. The area of the larger circle is $100\pi$. Two smaller circles of equal size lie within the larger circle such that each smaller circle touches $O$ and the perimeter of the larger circle. What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((1,0),1,180,360)--cycle,mediumgray); fill(Arc((-1,0),1,180,360)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((1,0),1));
draw(Circle((-1,0),1));
dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1,-1)--(1,-1));
[/asy]
|
75\pi
| 0.166667 |
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $12$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.
|
12
| 0.75 |
Ben throws five identical darts. Each hits one of five identical dartboards on the wall. After throwing the five darts, he lists the number of darts that hit each board, from greatest to least. How many different lists are possible?
|
7
| 0.916667 |
Define the function \( g \) on the positive integers as:
\[
g(n) = \left\{
\begin{array}{cl}
n^2 + 20 & \text{if } n \le 12, \\
g(n - 7) & \text{if } n > 12.
\end{array}
\right.
\]
Find the maximum value of function \( g \).
|
164
| 0.916667 |
Billy is hiking in Colorado. He walks eastward five miles, then turns $45$ degrees northward and walks eight miles. How far is he from his starting point? Express your answer in simplest radical form.
|
\sqrt{89 + 40\sqrt{2}}
| 0.916667 |
Find the remainder when $3x^5 - 2x^3 + 5x - 8$ is divided by $x^2 + 2x + 1$.
|
14x
| 0.5 |
Compute the distance between the parallel lines given by
\[\begin{pmatrix} 3 \\ -4 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -14 \\ 0 \end{pmatrix}\]and
\[\begin{pmatrix} 2 \\ -7 \\ 4 \end{pmatrix} + s \begin{pmatrix} 2 \\ -14 \\ 0 \end{pmatrix}.\]
|
\sqrt{11}
| 0.666667 |
Factor $64 - 16y^2$.
|
16(2-y)(2+y)
| 0.25 |
Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. If set $C$ intersects only with set $A$ adding 300 more elements to the union, how many elements are in set $A$?
[asy]
label("$A$", (2,67));
label("$B$", (80,67));
label("$C$", (41,10));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
draw(Circle((44, 27), 22));
label("1200", (44, 45));
label("300", (44, 27));
[/asy]
|
3825
| 0.083333 |
Eight painters working at the same rate take $2.5$ work-days to finish a job. If six painters are available and they take a rest day for every five consecutive workdays, how many total calendar days will it take them to finish the job, working at the same rate?
|
4
| 0.166667 |
Let \(h(x) = x^5 + x^4 + x^3 + x^2 + x + 1.\) What is the remainder when the polynomial \(h(x^{18})\) is divided by the polynomial \(h(x)\)?
|
6
| 0.916667 |
Calculate the least number of digits in the repeating block of the decimal expansion of $7/13$.
|
6
| 0.916667 |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. If $n$ is a multiple of 18, what is the smallest value of $n$ such that $f(n) > 18$?
|
n = 342
| 0.833333 |
How many degrees are in the sum of the measures of the nine numbered angles pictured in the diagram? In the diagram, there are three triangles, each sharing sides but not overlapping:
1. Triangle A with angles labeled 1, 2, 3.
2. Triangle B with angles labeled 4, 5, 6.
3. Triangle C, newly added, with angles labeled 7, 8, 9.
Each triangle has its vertices connected such that no vertex lies inside another triangle.
|
540^\circ
| 0.833333 |
Given a cylinder with a fixed volume \( V \), the total surface area (including the two circular ends) is minimized for a radius \( R \) and height \( H \). Find \( \frac{H}{R} \) when the volume is doubled.
|
2
| 0.583333 |
For positive integers $n$, let $g(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point in base $3$. How many positive integer divisors does $g(2023)$ have?
|
2024
| 0.666667 |
Evaluate the following expression:
\[\binom{50}{0} - 2\binom{50}{1} + 3\binom{50}{2} - \dots + (-1)^{50}51\binom{50}{50}\]
|
0
| 0.583333 |
The National Courier Company charges an extra $\$0.15$ in postage if the length of an envelope in inches divided by its height in inches is less than $1.2$ or greater than $2.8$. Additionally, an extra $\$0.10$ is charged if the thickness of the envelope in inches exceeds $0.25$. For how many of these envelopes must the extra postage be paid?
\begin{tabular}[t]{cccc}
Envelope & Length in inches & Height in inches & Thickness in inches\\\hline
A &7 &5 &0.2\\
B &10 &2 &0.3\\
C &7 &7 &0.1\\
D &12 &4 &0.26
\end{tabular}
|
3
| 0.583333 |
A cylinder-shaped container of cookies is 4 inches in diameter and 5 inches high and sells for $\$$2.00. If the price of the container increases quadratically with volume, what would be the price for a container that is 8 inches in diameter and 10 inches high?
|
\$128.00
| 0.75 |
A circular pizza with a diameter of $16\text{ cm}$ is divided into four equal sectors. Determine the square of the length of the longest line segment that can be drawn within one of these sectors.
|
128
| 0.5 |
Compute
$$\sum_{k=1}^{500} k(\lceil \log_{\sqrt{3}}{k}\rceil- \lfloor\log_{\sqrt{3}}{k} \rfloor).$$
|
124886
| 0.833333 |
In triangle $DEF$, the measure of $\angle D$ is $88^\circ$. The measure of $\angle E$ is $20^\circ$ more than four times the measure of $\angle F$. What is the measure, in degrees, of $\angle F$?
|
14.4^\circ
| 0.916667 |
Let $g$ be a function defined by $g(x) = \frac{px + q}{rx + s}$, where $p$, $q$, $r$ and $s$ are nonzero real numbers, and the function has the properties $g(31)=31$, $g(41)=41$, and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Determine the unique number that is not in the range of $g$.
|
36
| 0.166667 |
Find the sum of the distinct prime factors of $7^7 - 7^4$.
|
31
| 0.583333 |
When $\sqrt[3]{8000}$ is simplified, the result is $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ is as small as possible. What is $c+d$?
|
21
| 0.833333 |
A fair 8-sided die is rolled. If I roll $n$, then I win $n^3$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.
|
\$162.00
| 0.916667 |
Given that \(a - b = 7\) and \(a^2 + b^2 = 53\), find \(a^3 - b^3\).
|
385
| 0.916667 |
We are allowed to remove exactly one integer from the list $$-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,$$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?
|
6
| 0.5 |
What is the largest integer $n$ for which $\binom{10}{3} + \binom{10}{4} = \binom{11}{n}$?
|
7
| 0.75 |
Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \).
|
81
| 0.75 |
There are three complex numbers $a+bi$, $c+di$, and $e+fi$. Given $b=5$, $e = -2(a+c)$, and the sum of the complex numbers is $4i$. Find $d+f$.
|
-1
| 0.916667 |
For how many positive integers $n$ does the sum $1+2+\cdots+n$ evenly divide $10n$?
|
5
| 0.916667 |
Find $100^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)
|
100
| 0.75 |
If 72 cards are dealt to 10 people as evenly as possible, how many people will end up with fewer than 8 cards?
|
8
| 0.666667 |
Determine the sum of the squares of the coefficients when the expression $3(x^2-3x+3)-8(x^3-2x^2+x-1)$ is fully simplified.
|
1003
| 0.083333 |
How many ways are there to put 5 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?
|
6
| 0.916667 |
What is the remainder when $1001+1003+1005+1007+1009+1011+1013+1015$ is divided by $16$?
|
0
| 0.833333 |
What is the 150th digit to the right of the decimal point in the decimal representation of $\frac{17}{150}$?
|
3
| 0.916667 |
In a right triangle $\triangle PQR$, we know that $\tan Q = 0.5$ and the length of $QP = 16$. What is the length of $QR$?
|
8 \sqrt{5}
| 0.666667 |
Find the remainder when \(5x^4 - 9x^3 + 3x^2 - 7x - 30\) is divided by \(3x - 9\).
|
138
| 0.75 |
How many license plates consist of 3 letters followed by 3 digits, where the digits must include exactly one odd number and two even numbers?
|
6,\!591,\!000
| 0.25 |
The function $g(x)$ satisfies
\[g(x) + 3g(1 - x) = 4x^2\] for all real numbers $x.$ Find $g(4)$.
|
\frac{11}{2}
| 0.916667 |
What is the least positive multiple of 17 that is greater than 450?
|
459
| 0.916667 |
In two concentric circles, the radius of the outer circle is twice the radius of the inner circle. What is the area of the gray region, in square feet, if the width of the gray region is now 3 feet instead of 2 feet? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),4),gray);
filldraw(circle((0,0),1),white);
draw((1,0)--(4,0),linewidth(1));
label("$3^{\prime}$",(2.5,0),N);
[/asy]
|
27\pi
| 0.583333 |
What is the area enclosed by the graph of $|5x| + |3y| = 30$?
|
120
| 0.75 |
Let \( f(x) = x^2 + px + q \) and \( g(x) = x^2 + rx + s \) be two distinct quadratic polynomials where the \( x \)-coordinate of the vertex of \( f \) is a root of \( g \), and the \( x \)-coordinate of the vertex of \( g \) is a root of \( f \), also both \( f \) and \( g \) have the same minimum value. If the graphs of the two quadratic polynomials intersect at the point \( (50,-200), \) what is the value of \( p + r \)?
|
-200
| 0.583333 |
I have 8 shirts, 4 pairs of pants, and 8 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, plus white, yellow, and red. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?
|
252
| 0.916667 |
Triangle $ABC$ has side-lengths $AB = 15, BC = 25,$ and $AC = 20.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects the extensions of $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. Determine the perimeter of $\triangle AMN$.
|
35
| 0.416667 |
Let \[f(x) = \left\{
\begin{array}{cl}
\sqrt{x} &\text{ if }x\geq 5, \\
x^2 &\text{ if }x < 5.
\end{array}
\right.\]Find $f(f(f(3)))$.
|
9
| 0.916667 |
What is the remainder when the sum of the first 150 counting numbers is divided by 11200?
|
125
| 0.583333 |
In how many ways can five people sit in a row of seven chairs if one specific person must always sit in the middle chair?
|
360
| 0.5 |
In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 10 and 20, respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. The height of the trapezoid from $AB$ to $CD$ is 12 units. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.
|
\frac{1}{3}
| 0.166667 |
If $81^4 = 27^y$, what is the value of $3^{-y}$? Express your answer as a common fraction.
|
\frac{1}{3^{\frac{16}{3}}}
| 0.083333 |
In a certain ellipse, the endpoints of the major axis are $(3, -5)$ and $(23, -5)$. Also, the ellipse passes through the point $(19, -2)$. Find the area of the ellipse.
|
37.5 \pi
| 0.083333 |
Find the integer $n$, such that $0 \le n \le 8$, and it satisfies:
\[n \equiv -4567 + x \pmod{9}\]
where $x$ is the smallest positive integer that makes $n$ non-negative.
|
0
| 0.083333 |
What is the product of all real numbers that are tripled when added to their reciprocals?
|
-\frac{1}{2}
| 0.916667 |
Triangle $PQR$ is an obtuse, isosceles triangle. Angle $P$ measures $30^\circ$. What is the number of degrees in the measure of the largest interior angle of triangle $PQR$?
|
120^\circ
| 0.916667 |
What is the remainder when \( x^4 - 8x^3 + 12x^2 + 5x - 20 \) is divided by \( x + 2 \)?
|
98
| 0.333333 |
Define \( A \diamond B \) as \( A \diamond B = \frac{(A-B)}{5} \). What is the value of \( (7 \diamond 15) \diamond 2 \)?
|
-\frac{18}{25}
| 0.75 |
A line segment has one endpoint at $(10,-5)$. The midpoint of the segment is scaled by a factor of 2 along each axis, resulting in a point at $(12, -18)$. Determine the sum of the coordinates of the other endpoint.
|
-11
| 0.916667 |
At the beginning of every session of Geometry class, Mr. Slate picks a random student to help him with a demonstration. There are 15 students in his class, and the class meets three times a week. How many different sequences of student helpers are possible in a week?
|
3375
| 0.833333 |
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