problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
What is the area enclosed by the graph of $|x| + |3y| = 9$?
|
54
| 0.833333 |
Medians $\overline{AP}$ and $\overline{BQ}$ of $\triangle ABC$ are perpendicular. If $AP = 15$ and $BQ = 20$, what is ${AC}$?
|
\frac{20\sqrt{13}}{3}
| 0.083333 |
I have 7 books, two of which are identical copies of a science book and another two identical copies of a math book, while the rest of the books are all different. In how many ways can I arrange them on a shelf, and additionally, how many of these arrangements can be made if I decide to highlight exactly two books (not necessarily different)?
|
26460
| 0.083333 |
A region \(S\) in the complex plane is defined by:
\[
S = \{x + iy: -1 \leq x \leq 1, -1 \leq y \leq 1\}.
\]
A complex number \(z = x + iy\) is chosen uniformly at random from \(S\). What is the probability that \((\frac{1}{2} + \frac{1}{2}i)z\) is also in \(S\)?
|
1
| 0.75 |
What is the sum of the squares of the coefficients of $3(x^4 + 2x^3 + 5x^2 + 2)$?
|
306
| 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 this cylinder?
|
10\sqrt{2} \text{ cm}
| 0.833333 |
Determine the last three digits of $7^{1987}$.
|
543
| 0.083333 |
The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?
|
105
| 0.916667 |
A right pyramid with a square base has a lateral face area of 120 square meters. If the slant height of this pyramid is 24 meters, find the length of a side of its base.
|
10
| 0.833333 |
Determine the 50th number in the row of Pascal's triangle that contains 52 numbers.
|
1275
| 0.916667 |
Compute $0.045 \div 0.0005$.
|
90
| 0.916667 |
Find the quotient when $x^6 - 4x^5 + 5x^4 - 27x^3 + 13x^2 - 16x + 12$ is divided by $x-3$.
|
x^5 - x^4 + 2x^3 - 21x^2 - 50x - 166
| 0.583333 |
The equation \( y = -8t^2 - 12t + 72 \) describes the height (in feet) of a ball thrown downward at 12 feet per second from a height of 72 feet from the surface of Mars. In how many seconds will the ball hit the ground? Express your answer as a decimal rounded to the nearest hundredth.
|
2.34
| 0.916667 |
Simplify $\sqrt[3]{40a^5b^8c^{14}}$ and find the sum of the exponents of the variables that are outside the radical.
|
7
| 0.833333 |
The number of liters of tea a writer drinks on any given day is inversely proportional to the number of hours she spends writing. On Sunday, she spent 8 hours writing and consumed 3 liters of tea. On Wednesday, she spent 4 hours writing. How many liters of tea did she drink on Wednesday? And if she drinks only 2 liters of tea on Thursday, how many hours does she spend writing?
|
12
| 0.166667 |
The digits $1, 2, 3, 4, 5, 6$ can be arranged to form different $6$-digit positive integers with six distinct digits. In how many such integers is the digit $1$ to the left of the digit $2$ and the digit $3$ to the left of the digit $4$?
|
180
| 0.666667 |
How many ways are there to distribute 6 balls into 4 boxes if the balls are distinguishable but the boxes are not?
|
187
| 0.166667 |
If $x$, $y$, and $z$ are positive integers satisfying $xy+z = yz+x = zx+y = 47$, what is the value of $x+y+z$?
|
48
| 0.833333 |
Calculate the angle $\theta$ in degrees for the expression in the form $r \text{cis} \theta$ where
\[\text{cis } 60^\circ + \text{cis } 70^\circ + \text{cis } 80^\circ + \dots + \text{cis } 160^\circ.\]
|
110^\circ
| 0.916667 |
Let $x, y, z$ be real numbers such that $x + y + z = 2$, and $x \ge -\frac{1}{2}$, $y \ge -2$, and $z \ge -3$. Find the maximum value of
\[
\sqrt{6x + 3} + \sqrt{6y + 12} + \sqrt{6z + 18}.
\]
|
3\sqrt{15}
| 0.166667 |
What is the 42nd number in the row of Pascal's triangle that has 46 numbers?
|
148995
| 0.25 |
In triangle $XYZ$, $YZ = 10$. The length of median $XM$ is 7. Let $N$ be the largest possible value of $XZ^2 + XY^2$, and let $n$ be the smallest possible value. Find $N - n$.
|
0
| 0.916667 |
Sides $\overline{BC}$ and $\overline{DE}$ of regular octagon $ABCDEFGH$ are extended to meet at point $Q$. What is the degree measure of angle $Q$?
|
90^\circ
| 0.166667 |
In a certain hyperbola, the center is at $(1, 0),$ one focus is at $(1 + \sqrt{41}, 0),$ and one vertex is at $(4, 0).$ The equation of this hyperbola can be written as
\[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1.\]
Find $h + k + a + b.$
|
4 + 4\sqrt{2}
| 0.916667 |
Determine the next year after 2021 where the sum of its digits equals '5'.
|
2030
| 0.833333 |
Determine the number of terms in the simplified expression of \[(x+y+z)^{2010} + (x-y-z)^{2010}.\]
|
1,012,036
| 0.333333 |
How many values of \(x\), \(0 < x < 3\pi\), satisfy \(3\cos^2 x + 2\sin^2 x = 2?\) (Note: \(x\) is measured in radians.)
|
3
| 0.75 |
Find the smallest value of $x$ such that \[\frac{x^2 - x - 72}{x-9} = \frac{3}{x+6}.\]
|
-9
| 0.666667 |
What is the smallest prime divisor of $3^{19} + 6^{21}$?
|
3
| 0.75 |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 10?
|
\frac{5}{6}
| 0.916667 |
Given that \( x \) is a multiple of \( 7263 \), determine the greatest common divisor (GCD) of \( g(x) = (3x+4)(9x+5)(17x+11)(x+17) \) and \( x \).
|
1
| 0.75 |
In a cube $ABCDEFGH$ where each side has length $2$ units. Find $\sin \angle GAC$. (Consider this by extending the calculations needed for finding $\sin \angle HAC$)
|
\frac{\sqrt{3}}{3}
| 0.583333 |
Let $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$
|
\begin{pmatrix} 6 \\ 2 \\ 3 \end{pmatrix}
| 0.333333 |
Find the quadratic polynomial $q(x)$ such that $q(-6) = 0,$ $q(3) = 0,$ and $q(4) = -45.$
|
-4.5x^2 - 13.5x + 81
| 0.083333 |
In a right-angled triangle $PQR$ with angle $PQR = 90^{\circ}$, suppose $\cos Q = \frac{5}{13}$. Given $PR = 13$, calculate the length of $PQ$.
|
12
| 0.333333 |
A convex pentagon has interior angles with measures \(x+2\), \(2x+3\), \(3x-4\), \(4x+5\), and \(5x-6\) degrees. What is the measure of the largest angle?
|
174
| 0.916667 |
John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa's score? Remember, the AHSME consists of 30 multiple choice questions, and the score, $s$, is given by $s = 30 + 4c - w$, where $c$ is the number of correct answers, and $w$ is the number of wrong answers (no penalty for unanswered questions).
|
85
| 0.916667 |
A company incurs a daily maintenance cost of \$600 and pays each worker \$20 per hour. Each worker manufactures 6 widgets per hour, which are then sold at \$3.50 each. Determine the minimum number of workers the company must employ to turn a profit during an 8-hour workday.
|
76
| 0.916667 |
Find the remainder when the sum \(8930 + 8931 + 8932 + 8933 + 8934\) is divided by 13.
|
5
| 0.75 |
Consider $\mathcal{T}$ as the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are digits but can be the same (i.e., not necessarily distinct). Calculate the sum of the elements of $\mathcal{T}$.
|
500
| 0.5 |
What is the remainder when $3211$ is divided by $103$?
|
18
| 0.666667 |
Find the greatest common divisor of 100 and 250.
|
50
| 0.916667 |
The Chess Club at a high school has 25 members and needs to select 3 officers: president, treasurer, and secretary. Each member may hold at most one position. Two members, Alice and Bob, will only accept the positions if both are chosen as officers. Another member, Charlie, will only agree to be an officer if Alice is not chosen. In how many ways can the club select its officers?
|
10758
| 0.083333 |
The equation $y = -15t^2 + 75t$ represents the height (in feet) of a projectile launched from the ground with an initial velocity of 75 feet per second. Determine at what time $t$ will the projectile reach a height of 30 feet for the first time. Express your answer as a decimal rounded to the nearest tenth.
|
0.4
| 0.75 |
Compute the sum of the series:
\[ 2(1+4(1+4(1+4(1+4(1+4))))) \]
|
2730
| 0.833333 |
It is now 3:00:00 PM, as read on a 12-hour digital clock. In 287 hours, 18 minutes and 53 seconds the time will be $X:Y:Z$. What is the value of $X + Y + Z$?
|
73
| 0.75 |
The function $g(x)$ satisfies $g(1) = 2$ and
\[g(x + y) = 4^y g(x) + 3^x g(y)\] for all real numbers $x$ and $y.$ Find the function $g(x).$
|
2(4^x - 3^x)
| 0.416667 |
A box contains seven cards. Four of the cards are black on both sides, two cards are black on one side and red on the other, and one card is red on both sides. You pick a card uniformly at random from the box and see one side of the card. Given that the side you see is red, what is the probability that the other side is also red? Express your answer as a common fraction.
|
\frac{2}{4} = \frac{1}{2}
| 0.75 |
Determine the point on the plane $2x + 3y - z = 15$ that is closest to the point $(4,1,-2).$
|
\left(\frac{30}{7}, \frac{10}{7}, -\frac{15}{7}\right)
| 0.916667 |
How many non-empty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ consist entirely of numbers that are odd or divisible by 3?
|
63
| 0.833333 |
Triangle $ABC$ has side-lengths $AB = 12, BC = 26,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$
|
30
| 0.333333 |
Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x = 12$.
Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.
|
-\frac{23}{12}
| 0.666667 |
In the diagram, each circle is divided into unequal areas by a horizontal line, and $O$ is the center of the larger circle. The area of the larger circle is $100\pi.$ The line divides the larger circle such that the shaded lower semi-circle is $\frac{2}{3}$ of its total area. The smaller circle’s center is located 2 units above $O$ and just touches the boundary of the larger circle. What is the total area of the shaded regions? [asy]
size(120);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((0,2),1,0,180)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((0,2),1));
dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1,2)--(1,2));
[/asy]
|
\frac{296\pi}{3}
| 0.666667 |
Chandra now has six bowls of different colors (red, blue, yellow, green, orange, purple) and the same six colored glasses. She decides to choose a bowl and a glass wherein a pairing is valid if they are the same color or different colors. How many valid pairings are possible?
|
36
| 0.666667 |
In the circle with center $O$, the measure of $\angle QIS$ is $45^\circ$ and $OQ=15$ cm. Find the number of centimeters in the length of arc $QS$. Express your answer in terms of $\pi$.
|
7.5\pi \text{ cm}
| 0.416667 |
Given that $b$ is an odd multiple of $997$, find the greatest common divisor of $3b^2 + 17b + 31$ and $b + 7$.
|
1
| 0.666667 |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 3?
|
32
| 0.583333 |
What is the remainder when $6^{700}$ is divided by $72$?
|
0
| 0.916667 |
What is the tenth number in the row of Pascal's triangle that has 100 numbers?
|
\binom{99}{9}
| 0.166667 |
A solid box measures 20 cm by 15 cm by 10 cm. From each corner of this box, a cube measuring 4 cm on each side is removed. What percentage of the original volume is removed?
|
17.07\%
| 0.75 |
The following table presents the air distances in miles between selected world cities. If two different cities from the updated table are chosen at random, what is the probability that the distance between them is less than $8000$ miles? Express your answer as a common fraction.
\begin{tabular}{|c|c|c|c|c|}
\hline
& Tokyo & Cairo & Sydney & Paris \\ \hline
Tokyo & & 5900 & 4800 & 6200 \\ \hline
Cairo & 5900 & & 8700 & 2133 \\ \hline
Sydney & 4800 & 8700 & & 10400 \\ \hline
Paris & 6200 & 2133 & 10400 & \\ \hline
\end{tabular}
|
\frac{2}{3}
| 0.916667 |
For what base is the representation of $346_{10}$ a four-digit number whose final digit is odd?
|
7
| 0.416667 |
In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 23, respectively. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a common fraction.
|
\frac{100}{429}
| 0.666667 |
The local theater has one ticket window. Seven people, including a family of three who must stay together, line up to buy a ticket. In how many ways can they line up given this condition?
|
720
| 0.916667 |
How many integer values of \(n\) satisfy the inequality \(-100 < n^3 + n^2 < 100\)?
|
9
| 0.833333 |
Let \( x \) be a real number such that \( x + \frac{1}{x} = 4 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_5 \).
|
724
| 0.916667 |
The median of a set of consecutive odd integers is 153. If the greatest integer in the set is 167, what is the smallest integer in the set?
|
139
| 0.833333 |
What is the base 4 representation of the base 2 number $101001110010_2$?
|
221302_4
| 0.916667 |
A street has parallel curbs 60 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 20 feet and each stripe is 50 feet long. Find the distance, in feet, between the stripes.
|
24
| 0.166667 |
Given that \( P = (\sqrt{2010}+\sqrt{2011}), Q = (-\sqrt{2010}-\sqrt{2011}), R = (\sqrt{2010}-\sqrt{2011}), \) and \( S = (\sqrt{2011}-\sqrt{2010}), \) find \( PQRS. \)
|
1
| 0.833333 |
What is the largest 5-digit integer congruent to 31 modulo 26?
|
99975
| 0.916667 |
The third term of a geometric sequence of positive numbers is 27, and the ninth term is 3. What is the sixth term of the sequence?
|
9
| 0.916667 |
Suppose $\mathbf{a}$ and $\mathbf{b}$ are unit vectors with an angle of $\frac{\pi}{4}$ between them. Calculate the volume of the parallelepiped formed by the vectors $\mathbf{a}, \mathbf{b},$ and $\mathbf{b} + \mathbf{b} \times \mathbf{a}$.
|
\frac{1}{2}
| 0.833333 |
A street has parallel curbs 60 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 20 feet and each stripe is 50 feet long. Find the distance, in feet, between the stripes.
|
24 \text{ feet}
| 0.166667 |
Express $\frac{63}{2^3 \cdot 5^4}$ as a terminating decimal.
|
0.0126
| 0.916667 |
What is the slope of the line determined by any two solutions to the equation $\frac{4}{x} + \frac{6}{y} = 0$? Express your answer as a common fraction.
|
-\frac{3}{2}
| 0.916667 |
In Pascal's Triangle, each number is the sum of the number just above it to the left and the number just above it to the right. Calculate the sum of the numbers in Row 10 of Pascal's Triangle.
|
1024
| 0.833333 |
A parabola has a vertex at $V = (0, 0)$ and a focus at $F = (0, 2)$. Let $P$ be a point on this parabola in the first quadrant such that the distance from $P$ to the focus $F$ is 50. Find $P$.
|
(8\sqrt{6}, 48)
| 0.833333 |
$\triangle ABC$ is inscribed in a circle of radius $r$, with $A$ and $B$ lying diametrically opposite each other. Point $C$ is placed on the circle but not on diameter $AB$. Determine the maximum possible value of $s^2$, where $s = AC + BC$.
|
8r^2
| 0.75 |
In Gridtown, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Tom runs on the inner edge of a street surrounding one of these blocks, while Jerry runs on the outer edge of the same street. Calculate how many more feet than Tom does Jerry run for each lap around the block.
|
240
| 0.583333 |
Evaluate the series
\[
\sum_{n=1}^\infty \frac{n^3 + 2n^2 - n + 1}{(n+3)!}.
\]
|
\frac{1}{3}
| 0.083333 |
Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit primes ``$WX$'' and ``$YZ$'' where the digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible?
|
110
| 0.666667 |
There are four points that are $4$ units from the line $y=10$ and $10$ units from the point $(5,10)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?
|
60
| 0.75 |
A function $g$ is defined on the complex numbers by $g(z)=(c+di)z,$ where $c$ and $d$ are real numbers. This function has the property that for each complex number $z$, $g(z)$ is equidistant from both $z$ and the origin. Given that $|c+di|=7$, find $d^2.$
|
\frac{195}{4}
| 0.916667 |
Xanthia buys hot dogs that come in packages of 5, and she buys hot dog buns that come in packages of 7. What is the smallest number of hot dog packages she can buy in order to be able to buy an equal number of hot dogs and hot dog buns?
|
7
| 0.75 |
**A square is divided into six congruent rectangles. If the perimeter of each of these six rectangles is 42 inches, what is the perimeter of the square, in inches?**
|
72 \text{ inches}
| 0.333333 |
How many four-digit positive integers $x$ satisfy $3874x + 481 \equiv 1205 \pmod{31}$?
|
290
| 0.5 |
A spherical bubble collapses into a right circular cone while maintaining its volume. The height of the cone is 6 times the radius of its base. If the radius of the base of the cone is $3\sqrt[3]{2}$ cm, find the radius of the original sphere before it changed shape.
|
3\sqrt[3]{3}
| 0.833333 |
Evaluate \begin{align*} (5a^2 - 13a + 4)(2a - 3) \end{align*} for $a = 2$.
|
-2
| 0.666667 |
Evaluate the product $\frac{1}{2}\cdot\frac{4}{1}\cdot\frac{1}{8}\cdot\frac{16}{1} \dotsm \frac{1}{16384}\cdot\frac{32768}{1}$.
|
256
| 0.166667 |
Find the matrix $\mathbf{M}$ that scales the first row of any matrix by 5 and triples the second row. Specifically, determine $\mathbf{M}$ such that
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 5a & 5b \\ 3c & 3d \end{pmatrix}.\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.
|
\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix}
| 0.916667 |
How many distinct, positive factors does $1320$ have?
|
32
| 0.916667 |
Sequence \(A\) is a geometric sequence and sequence \(B\) is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than \(300\). What is the least positive difference between a term from sequence \(A\) and a term from sequence \(B\)?
\(\bullet\) Sequence \(A\): \(3\), \(9\), \(27\), \(81\), \(243\), \(\ldots\)
\(\bullet\) Sequence \(B\): \(100\), \(110\), \(120\), \(130\), \(\ldots\)
|
3
| 0.166667 |
In the following diagram, what is the value of $x$?
[asy]
draw((0,0)--(20,0),black+linewidth(1));
draw((20,0)--(20,-8),black+linewidth(1));
draw((0,0)--(5,8)--(20,-8),black+linewidth(1));
draw((20,0)--(20,-0.5)--(19.5,-0.5)--(19.5,0)--cycle,black+linewidth(1));
label("$70^{\circ}$",(5.5,7),S);
label("$50^{\circ}$",(1,0),NE);
label("$x^{\circ}$",(20.25,-7),NW);
[/asy]
|
30^\circ
| 0.166667 |
The value $2^{10} - 1$ is divisible by how many prime numbers, and what is their sum?
|
45
| 0.916667 |
The values of $f$, $g$, $h$, and $j$ are 6, 7, 8, and 9, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$, and $fj$?
|
225
| 0.666667 |
Evaluate $\lfloor\sqrt{120}\rfloor$.
|
10
| 0.916667 |
Find the area of triangle $ABC$ below which is a right triangle with a $45^\circ$ angle at $A$.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (5,0);
R = (0,5);
draw (P--Q--R--P,linewidth(0.9));
label("$A$",P,S);
label("$B$",Q,S);
label("$C$",R,W);
label("$5$",Q/2,S);
label("$5$",R/2,W);
label("$45^\circ$",(-0.2,0.2),S);
[/asy]
|
12.5
| 0.25 |
The 5-digit number $63\,47\square$ is a multiple of 9. Which digit is represented by $\square$?
|
7
| 0.833333 |
Let \( T = 1 - 2 + 3 - 4 + \cdots + 2017 - 2018 \). What is the residue of \( T \), modulo 2018?
|
1009
| 0.833333 |
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