problem
stringlengths 18
4.46k
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stringlengths 1
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float64 0.08
0.92
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|---|---|---|
A certain club consists of seven leaders and some number of regular members. Each year, the current leaders leave the club. Every regular member then recruits three new people to join as regular members. Following this, seven new leaders are elected from outside the club to serve for the next year. Initially, there are twenty-one people in total in the club. How many people in total will be in the club five years from now?
|
14343
| 0.583333 |
The Jackson High School Football Club has 24 players, including 4 goalkeepers. The club organizes a competition to discover which goalkeeper can save the most goals from penalty kicks. For each penalty kick, one goalkeeper will defend the goal while the remaining players (including the other goalkeepers) attempt to score.
How many penalty kicks are required so that every player has the opportunity to shoot against each goalkeeper?
|
92
| 0.166667 |
A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row, and each column contains the same number of chairs as every other column. With the condition that there must be at least two chairs in every row and column, and all the chairs in the room must be included, how many arrays are possible in a classroom containing $48$ chairs?
|
4 \times 2 = 8
| 0.833333 |
What is the positive difference between the $1002^{\mathrm{th}}$ term and the $1008^{\mathrm{th}}$ term of the arithmetic sequence $-11,$ $-4,$ $3,$ $10,$ $\ldots$?
|
42
| 0.916667 |
If $5 \cot \theta = 4 \sin \theta$ and $0 < \theta < \pi$, find the value of $\cos \theta$.
|
\cos \theta = \frac{-5 + \sqrt{89}}{8}
| 0.916667 |
Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line.
|
-10
| 0.25 |
How many three-digit numbers remain if we exclude all three-digit numbers where exactly two digits are the same and these two digits are adjacent?
|
738
| 0.25 |
If \( f(x) = 3x^2 + x - 4 \), what is the value of \( f(f(3)) \)?
|
2050
| 0.916667 |
An 8-sided die is rolled. If \( n \) is the number rolled, I win \( n^3 \) dollars. What is the expected value of my winnings? Express your answer as a dollar value rounded to the nearest cent.
|
\$162.00
| 0.833333 |
Calculate $8! - 7\cdot7! - 2\cdot7!$.
|
-5040
| 0.916667 |
Let \( a \), \( b \), and \( c \) be the roots of the polynomial equation \( x^3 - 2x^2 + x - 1 = 0 \). Calculate \( \frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2} \).
|
-5
| 0.5 |
If $2x - 3y = 18$ and $x + 2y = 8$, what is the value of $x$?
|
\frac{60}{7}
| 0.75 |
A circle with a radius of 5 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Find the length of the longer side of the rectangle, in centimeters. Express your answer in terms of $\pi$.
|
7.5\pi
| 0.75 |
When the binary number $110110111010_2$ is divided by 8, what is the remainder (give your answer in base 10)?
|
2
| 0.416667 |
A mini-domino set is made in a similar way to a standard domino set, but only uses integers from 0 to 6. Each integer within this range is paired with every other and itself exactly once to form a complete mini-set. A $\textit{double}$ in this set is a domino where the two numbers are identical. What is the probability that a domino randomly selected from this mini-set will be a $\textit{double}$? Express your answer as a common fraction.
|
\frac{1}{4}
| 0.833333 |
The expression \(512y^3 - 27\) is a difference of cubes. Express \(512y^3 - 27\) in the form \((ay + b)(cy^2 + dy + e)\) and find \(a+b+c+d+e\).
|
102
| 0.25 |
Factor the following expression: $58a^2 + 174a$.
|
58a(a+3)
| 0.833333 |
Starting with the number 250, Tim repeatedly divides his number by 3 and then takes the greatest integer less than or equal to that number. How many times must he perform this operation before the result is less than or equal to 2?
|
5
| 0.833333 |
A rectangle has a perimeter of 40 inches and each side must have an integer length. Determine how many non-congruent rectangles can be formed under these conditions.
|
10
| 0.916667 |
In right triangle $PQR$ with $\angle Q = 90^\circ$, we have $$5\sin R = 4\cos R.$$ What is $\sin R$?
|
\frac{4\sqrt{41}}{41}
| 0.75 |
In triangle $ABC,$ $AB = 4,$ $AC = 7,$ $BC = 9,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Find $\sin \angle BAD.$
|
\frac{3\sqrt{14}}{14}
| 0.833333 |
Find the positive solution to
\[\sqrt{x + \sqrt{x + \sqrt{x + \dotsb}}} = \sqrt{x \sqrt{x \sqrt{x \dotsm}}}.\]
|
2
| 0.833333 |
Suppose $a$ and $b$ are positive integers such that $(a+bi)^3 = 2 + 11i$. What is $a+bi$?
|
2 + i
| 0.666667 |
Find the sum of all solutions to the equation $2^{|x|^2} + 2|x|^2 = 34.$
|
0
| 0.916667 |
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 60, how many such house numbers are possible?
|
156
| 0.583333 |
What is the greatest divisor of 540 that is smaller than 100 and also a factor of 180?
|
90
| 0.75 |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers all of which are nonprime?
|
97
| 0.75 |
Let $S$ be the set of 12-tuples $(a_0, a_1, \dots, a_{11})$, where each entry is 0 or 1, so $S$ contains $2^{12}$ 12-tuples. For each 12-tuple $s = (a_0, a_1, \dots, a_{11})$ in $S$, let $p_s(x)$ be the polynomial of degree at most 11 such that
\[p_s(n) = a_n\] for $0 \le n \le 11.$ Find
\[\sum_{s \in S} p_s(12).\]
|
2048
| 0.916667 |
Find the sum of all integral values of \( c \) with \( c \leq 30 \) for which the equation \( y = x^2 - 9x - c \) has two rational roots.
|
-28
| 0.25 |
Let $a$ and $b$ be integers such that $ab = -72.$ Find the maximum value of $a + b.$
|
71
| 0.916667 |
What is the smallest positive number that is a multiple of $45$ and also exceeds $100$ by a multiple of $7$?
|
135
| 0.916667 |
A clock has a second hand that is 10 cm long. Calculate the distance in centimeters that the tip of the second hand travels in 15 minutes. Express your answer in terms of $\pi$.
|
300\pi
| 0.916667 |
A gardener plants four maple trees, five oak trees, and three birch trees in a row. He plants them randomly, each arrangement being equally likely. Calculate the probability, in lowest terms, that no two birch trees are adjacent. Let this probability be $\frac{m}{n}$ and find $m+n$.
|
17
| 0.166667 |
What is the value of $a + b$ if the sequence $3, ~9, ~15, \ldots, ~a, ~b, ~33$ is an arithmetic sequence?
|
48
| 0.916667 |
Twenty teams play a tournament where each team plays against every other team exactly once. No ties occur, and each team has a \(50\%\) chance of winning any game it plays. Calculate the probability that no two teams win the same number of games, expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Determine \(\log_2 n\).
|
172
| 0.166667 |
When two fair 8-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two-digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 4?
|
\frac{1}{16}
| 0.75 |
Determine the values of $p$ and $q$ if $18^3 = \frac{27^2}{3} \cdot 2^{9p} \cdot 3^{3q}$.
|
\frac{1}{3}
| 0.083333 |
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
| 0.166667 |
If \( A = -3 + 2i \), \( O = 3i \), \( P = 1 + 3i \), and \( S = -2 - i \), then compute \( 2A - O + 3P + S \).
|
-5 + 9i
| 0.833333 |
Triangles $PQR$ and $STU$ are similar. The lengths of $\overline{PQ}$ and $\overline{PR}$ are 7 cm and 9 cm, respectively, while $\overline{ST}$ is 4.2 cm. Additionally, the area of triangle $PQR$ is 18 cm². Calculate the length of $\overline{SU}$. Express your answer in cm to the nearest tenth.
|
5.4 \text{ cm}
| 0.833333 |
In triangle $XYZ$, $\angle X = 90^\circ$, $XZ = 4$, and $YZ = \sqrt{17}$. What is $\tan Y$?
|
4
| 0.666667 |
How many positive integers smaller than $1{,}000{,}000{,}000$ are powers of $2$, but are not powers of $8$?
|
20
| 0.833333 |
Larry now starts with the number 4 on his pinky finger. He applies function $f$ to 4 and writes the output on his ring finger. Then, instead of applying $f$ again, he uses a new function $g$ defined on the same set of points. If Larry alternates between applying $f$ and $g$, what number will Larry write on his tenth finger? The functions $f$ and $g$ at different points are defined as follows based on the previous graph data:
- $f(4) = 3$
- $g(3) = 1$
- $f(1) = 8$
- $g(8) = 7$
- $f(7) = 2$
- $g(2) = 1$
- $f(1) = 8$ (repetition starts here)
|
2
| 0.75 |
The surface area of a sphere is \(256\pi\text{ cm}^2\). What is the volume, in cubic centimeters, of the sphere? Express your answer in terms of \(\pi\).
|
\frac{2048}{3}\pi
| 0.083333 |
The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?
|
105
| 0.916667 |
What is the total number of digits used when the first 1500 positive even integers are written?
|
5448
| 0.416667 |
A new survey was conducted where 150 men and 750 women were questioned about their support for increased healthcare funding. It was found that 55% of the men and 85% of the women surveyed are in favor of it. What percentage of the total surveyed population supports the proposed funding increase?
|
80\%
| 0.583333 |
Determine the radius of the circle inscribed in triangle $XYZ$ where $XY = 8, XZ = 13, YZ = 15$.
|
\frac{5\sqrt{3}}{3}
| 0.916667 |
As $x$ ranges over all real numbers, find the range of the function
\[ f(x) = \sin^4 x + \cos^4 x. \]
Enter your answer using interval notation.
|
\left[ \frac{1}{2}, 1 \right]
| 0.916667 |
Compute
\[
S = \cos^4 0^\circ + \cos^4 30^\circ + \cos^4 60^\circ + \cos^4 90^\circ + \cos^4 120^\circ + \cos^4 150^\circ + \cos^4 180^\circ.
\]
|
\frac{13}{4}
| 0.75 |
Evaluate $\log_{\sqrt{8}} (512\sqrt{8})$.
|
7
| 0.916667 |
Let $a$ and $b$ be the roots of $x^2 - 5x + 6 = 0.$ Compute
\[
a^3 + a^4b^2 + a^2b^4 + b^3 + ab(a+b).
\]
|
533
| 0.416667 |
The weather forecast predicts a 75 percent chance of rain each day for the upcoming five-day holiday weekend. If it doesn't rain, the weather will be sunny. Sam and Alex want exactly two of those days to be sunny for their outdoor activities. What is the probability they get the weather they want?
|
\frac{135}{512}
| 0.083333 |
Find the real root of the equation \[\sqrt{x+4} + \sqrt{x+6} = 12.\]
|
\frac{4465}{144}
| 0.833333 |
Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $RP=9$. 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 $QS$?
|
QS = 5
| 0.75 |
Evaluate $(3-w)(3-w^2)\cdots(3-w^{11})$ where $w=e^{2\pi i/12}.$
|
265720
| 0.916667 |
What is the units digit of the product of all the odd positive integers between 20 and 50 that are not multiples of 5?
|
9
| 0.416667 |
A standard deck of 52 cards consists of 13 ranks and 4 suits (Spades, Hearts, Diamonds, Clubs), each suit represented equally. If two cards are drawn simultaneously from the deck, what is the probability that both cards are Diamonds?
|
\frac{1}{17}
| 0.916667 |
How many different 8-digit positive integers are there such that the last digit is not zero?
|
81,\!000,\!000
| 0.833333 |
If $\lceil{\sqrt{x}}\rceil=17$, how many possible integer values of $x$ are there?
|
33
| 0.916667 |
Suppose that a real number \( y \) satisfies \[\sqrt{64-y^2}-\sqrt{36-y^2}=4.\] What is the value of \(\sqrt{64-y^2}+\sqrt{36-y^2}\)?
|
7
| 0.75 |
Two concentric circles have the radius of the inner circle as \( r \) feet and the outer circle as \( 3r \) feet. If the width of the gray region between these circles is \( 4 \) feet, what is the area of the gray region, expressed in terms of \( \pi \) and \( r \)?
|
8\pi r^2
| 0.25 |
What is the greatest common divisor of 256, 162, and 450?
|
2
| 0.916667 |
When $\frac{7}{8000}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?
|
3
| 0.833333 |
Simplify $$(2x^5 - 3x^4 + 5x^3 - 9x^2 + 8x - 15) + (5x^4 - 2x^3 + 3x^2 - 4x + 9).$$ After simplification, evaluate the polynomial at \(x = 2\).
|
98
| 0.25 |
For a nonnegative integer $n$, let $r_8(n)$ stand for the remainder left when $n$ is divided by $8$. Define a sequence where each term $n$ satisfies $$r_8(7n) \leq 3.$$ What is the $15^{\text{th}}$ entry in this list, starting with the first entry as $0$?
|
30
| 0.416667 |
Evaluate the complex number \[\frac{\tan \frac{\pi}{6} + i}{\tan \frac{\pi}{6} - i}\] and determine if it is a twelfth root of unity. If so, find the specific root it represents (i.e., find $m$ such that the number equals $\cos \frac{2m \pi}{12} + i \sin \frac{2m \pi}{12}$).
|
4
| 0.25 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$.
|
\frac{1}{24}
| 0.416667 |
Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2+y^2=18x+40y$. Find the minimum value of $y$ and the maximum value of $x$.
|
9 + \sqrt{481}
| 0.416667 |
Find the remainder when $7 \times 17 \times 27 \times \ldots \times 87 \times 97$ is divided by $3$.
|
0
| 0.916667 |
Circles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in a 1.5-foot length of this pattern?
|
13.5\pi
| 0.583333 |
Find the matrix $\mathbf{M}$ that triples every element of a given matrix $\mathbf{A}$. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 3a & 3b \\ 3c & 3d \end{pmatrix}.\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.
|
\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}
| 0.583333 |
Let \(x,\) \(y,\) and \(z\) be real numbers such that
\[ x^3 + y^3 + z^3 - 3xyz = 8. \]
Find the minimum value of \( x^2 + y^2 + z^2 \).
|
4
| 0.75 |
What integer $n$ satisfies $0 \leq n < 251$ and $$250n \equiv 123 \pmod{251}~?$$
|
128
| 0.5 |
The bottoms of two vertical poles are 18 feet apart and are placed on flat ground. The height of the first pole is 9 feet and the second pole is 24 feet tall. Find the length of the wire stretched from the top of the first pole to the top of the second pole.
|
\sqrt{549}
| 0.083333 |
What is the remainder when $3x^2 - 22x + 63$ is divided by $x - 4 + 1$?
|
24
| 0.083333 |
In trapezoid \(ABCD\), the lengths of the bases \(AB\) and \(CD\) are 10 and 25 respectively. The legs of the trapezoid are extended beyond \(A\) and \(B\) to meet at point \(E\). What is the ratio of the area of triangle \(EAB\) to the area of trapezoid \(ABCD\)? Express your answer as a common fraction.
|
\frac{4}{21}
| 0.583333 |
Let $X$, $Y$, and $Z$ be points on a circle of radius $12$. If $\angle XZY = 90^\circ$, what is the circumference of the minor arc $XY$? Express your answer in terms of $\pi$.
|
12\pi
| 0.833333 |
Let $a,$ $b,$ $c$ be real numbers such that $a + 3b + c = 5.$ Find the maximum value of
\[ab + ac + bc.\]
|
\frac{25}{3}
| 0.916667 |
Determine the value of $r$ if $8 = 3^{4r-2}$.
|
\frac{3\log_3(2) + 2}{4}
| 0.083333 |
Find the value of $x$ that satisfies $\frac{\sqrt{2x+7}}{\sqrt{8x+10}}=\frac{2}{\sqrt{5}}$. Express your answer as a common fraction.
|
\frac{-5}{22}
| 0.083333 |
Point $Q$ lies on the line $y = 4$ and is 15 units from the point $(-2,-3)$. Find the product of all possible $x$-coordinates that satisfy the given conditions.
|
-172
| 0.75 |
In $\triangle ABC$, we have $AC=BC=10$ and $AB=4$. Suppose $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$, and $CD=12$. What is $BD$?
|
4\sqrt{3} - 2
| 0.666667 |
If $4x + 5 \equiv 3 \pmod{17}$, what is $2x + 8$ congruent to $\pmod{17}$?
|
7
| 0.916667 |
How many positive integers less than $1000$ are either a perfect cube or a perfect square?
|
37
| 0.75 |
What is the modulo $17$ residue of $255 + 7 \cdot 51 + 9 \cdot 187 + 5 \cdot 34$?
|
0
| 0.416667 |
Given the graph of the rational function $\frac{1}{q(x)}$, where $q(x)$ is a quadratic polynomial, and it's known that $q(2) = 2$. Further, there are vertical asymptotes at $x = -1$ and $x = 1$. Additionally, $q(1) = 0$. Determine the quadratic polynomial $q(x)$.
|
\frac{2}{3}x^2 - \frac{2}{3}
| 0.916667 |
A bookstore has a sale on days of the month that are multiples of 3 (such as June 3, June 6...). A shoe store has a sale every 7 days, starting from June 1. How many times in the months of June, July, and August do the two stores have sales on the same date?
|
5
| 0.25 |
In how many ways can 10 people sit around a round table if two specific individuals must sit next to each other? (Two seatings are considered the same if one is a rotation of the other.)
|
80,\!640
| 0.916667 |
Simplify
\[\frac{4 + 7i}{4 - 7i} + \frac{4 - 7i}{4 + 7i}.\]
|
-\frac{66}{65}
| 0.916667 |
Simplify $\sqrt{18} \times \sqrt{72}$.
|
36
| 0.916667 |
Four congruent equilateral triangles, each of a different color, are used to construct a regular tetrahedron. How many distinguishable ways are there to construct the tetrahedron? (Two colored tetrahedrons are distinguishable if neither can be rotated to look just like the other.)
[asy] import three; import math; unitsize(1.5cm); currentprojection=orthographic(2,0.2,1); triple A=(1,1,1); triple B=(-1,-1,1); triple C=(-1,1,-1); triple D=(1,-1,-1); draw(A--B--C--cycle); draw(A--C--D--cycle); draw(A--D--B--cycle); draw(B--C--D--cycle); [/asy]
|
2
| 0.833333 |
Compute $3(i^{603} + i^{602} + \cdots + i + 1)$, where $i^2 = -1$.
|
0
| 0.5 |
A ball is dropped from a height of 20 meters above the ground. On each bounce, it rises to $\frac{2}{3}$ of the height it fell from previously. The ball is caught when it reaches the high point after hitting the ground for the fourth time. To the nearest meter, how far has it travelled?
|
80
| 0.583333 |
In $\triangle ABC$, lines $CE$ and $AD$ are drawn such that $\dfrac{CD}{DB}=\dfrac{4}{1}$ and $\dfrac{AE}{EB}=\dfrac{4}{3}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Compute $r$.
|
7
| 0.333333 |
A "super ball" is dropped from a window 20 meters above the ground. On each bounce, it rises to $\frac{2}{3}$ of the height it fell from. The ball is caught when it reaches the high point after hitting the ground for the fourth time. To the nearest meter, how far has it travelled?
|
80
| 0.666667 |
Let $x$, $y$, and $z$ be positive numbers such that:
\begin{align*}
x^2/y &= 3, \\
y^2/z &= 4, \\
z^2/x &= 5.
\end{align*}
Find $x$.
|
\sqrt[7]{6480}
| 0.583333 |
Determine the value of $m$ such that $100^m = 100^{-3} \times \sqrt{\frac{100^{55}}{0.0001}}$.
|
25.5
| 0.416667 |
When the base-10 integers 500 and 1800 are expressed in base 2, how many more digits does 1800 have than 500 (after being converted)?
|
2
| 0.416667 |
The deli has now increased its variety and offers five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich still consists of one type of bread, one type of meat, and one type of cheese. Turkey, roast beef, Swiss cheese, and rye bread are each available at the deli. If Al never orders a sandwich with a turkey/Swiss cheese combination nor a sandwich with a rye bread/roast beef combination, how many different sandwiches could Al order?
|
199
| 0.583333 |
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