problem
stringlengths 18
4.46k
| answer
stringlengths 1
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| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Find the number of zeros between the decimal point and the first non-zero digit when $\frac{7}{5000}$ is written as a decimal.
|
2
| 0.333333 |
Calculate the product:
\[
\begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 5 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 7 \\ 0 & 1 \end{pmatrix} \dotsm
\begin{pmatrix} 1 & 101 \\ 0 & 1 \end{pmatrix}.
\]
|
\begin{pmatrix} 1 & 2600 \\ 0 & 1 \end{pmatrix}
| 0.833333 |
Given that $f(x) = x^{k/n}$ where $k < 0$ and $n > 0$, determine the range of $f(x)$ on the interval $[2, \infty)$.
|
(0, 2^{k/n}]
| 0.833333 |
Find the ones digit of $13^{13(7^{7})}$.
|
7
| 0.916667 |
Alice rolls six fair 6-sided dice, each numbered with numbers from 1 to 6. What is the probability that exactly three of the dice show a prime number?
|
\frac{5}{16}
| 0.916667 |
According to Moore's law, the number of transistors on a chip doubles every 18 months. In 1995, a typical CPU contained about $2,\!500,\!000$ transistors. How many transistors did a typical CPU contain in the year 2010?
|
2,\!560,\!000,\!000 \text{ transistors}
| 0.916667 |
Multiply the base-10 numbers 354 and 78. Write the product in base-7. What is the units digit of the base-7 representation?
|
4
| 0.833333 |
Determine the number of ways to arrange the letters of the word "MATHEMATICS".
|
4989600
| 0.916667 |
What is the remainder when the sum $3 + 11 + 19 + \cdots + 291 + 299$ is divided by $8$?
|
2
| 0.833333 |
Solve
\[\arcsin x + \arcsin 3x = \frac{\pi}{2}.\]
|
\frac{\sqrt{10}}{10}
| 0.5 |
How many integers from 100 through 999, inclusive, do not contain any of the digits 0, 1, 8, or 9?
|
216
| 0.833333 |
A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and that each number rolled is a prime number? Express your answer as a common fraction.
|
\frac{1}{6561}
| 0.833333 |
A valid license plate in Yonderville consists of three letters followed by three digits. The first letter must be a vowel (A, E, I, O, U), which cannot repeat in the license plate, while the other two letters can be any letter except the first chosen vowel. How many valid license plates are possible?
|
3,\!125,\!000
| 0.083333 |
A very large number $y$ is equal to $2^33^44^55^66^77^88^99^{10}$. What is the smallest positive integer that, when multiplied with $y$, produces a product that is a perfect square?
|
6
| 0.666667 |
What is the greatest integer less than 200 for which the greatest common divisor of that integer and 18 is 9?
|
189
| 0.833333 |
Find \( d \) given that \( \lfloor d \rfloor \) satisfies
\[ 3x^2 + 11x - 20 = 0 \]
and \( \{ d \} = d - \lfloor d \rfloor \) satisfies
\[ 4x^2 - 12x + 5 = 0. \]
|
-\frac{9}{2}
| 0.833333 |
I have two 12-sided dice that each have 3 red sides, 4 green sides, 2 blue sides, and 3 yellow sides. If I roll both dice, what is the probability that they come up the same?
|
\frac{19}{72}
| 0.416667 |
What common fraction (i.e., a fraction reduced to its lowest terms) is equivalent to $.4\overline{37}$?
|
\frac{433}{990}
| 0.833333 |
How many positive integers smaller than $2,000,000$ are powers of $2$, but are not powers of $4$?
|
10
| 0.916667 |
What is the smallest four-digit integer $n$ that satisfies $$75n \equiv 225 \pmod{450}~?$$
|
1005
| 0.75 |
Two numbers \(180\) and \(n\) share exactly three positive divisors. What is the greatest of these three common divisors?
|
9
| 0.75 |
Consider the quadratic function \( g(x) = 3x^2 - 9x + 4 \). Determine the largest interval that includes the point \( x=-1 \) on which \( g \) can be made invertible.
|
(-\infty, \frac{3}{2}]
| 0.583333 |
In 1970, there were 700,000 cases of a hypothetical disease reported in a particular country. By 2010, the cases decreased to 1,000. Calculate how many cases would have been reported in 1995 if the number of cases had decreased linearly over this period.
|
263,125
| 0.75 |
The chart below provides the air distance in miles between selected world cities. If two different cities from the chart 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
& Bangkok & Cape Town & Honolulu & London \\ \hline
Bangkok & & 6300 & 6609 & 5944 \\ \hline
Cape Town & 6300 & & 11,535 & 5989 \\ \hline
Honolulu & 6609 & 11,535 & & 7240 \\ \hline
London & 5944 & 5989 & 7240 & \\ \hline
\end{tabular}
|
\frac{5}{6}
| 0.75 |
Given positive integers $x$ and $y$ such that $x \neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$, what is the smallest possible value for $x + y$?
|
49
| 0.833333 |
What is the remainder when $2002 \cdot 1493$ is divided by $300$?
|
86
| 0.75 |
Factor the expression $5x(x-2) + 9(x-2) - 4(x-2)$.
|
5(x-2)(x+1)
| 0.833333 |
Simplify $\frac{210}{18} \cdot \frac{6}{150} \cdot \frac{9}{4}$.
|
\frac{21}{20}
| 0.75 |
Find the greatest common divisor of $7384$ and $12873$.
|
1
| 0.916667 |
The graph of $y = ax^2 + bx + c$ has a maximum value of 36, and passes through the points $(-3,0)$ and $(3,0).$ Find $a + b + c.$
|
32
| 0.916667 |
If an integer is divisible by both $4$ and $3$, and the sum of its last two digits is $12$, then what is the product of its last two digits?
|
32
| 0.916667 |
If the six digits 1, 2, 3, 5, 7, and 8 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by both 15 and 2? Express your answer as a common fraction.
|
0
| 0.916667 |
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?
|
10\sqrt{2}
| 0.833333 |
A rectangle has a perimeter of 40 units and one of its sides must be an even number. What is the maximum possible area of the rectangle if its dimensions are whole numbers?
|
100
| 0.75 |
Compute the distance between the vertices of the parabolas formed by the graph of the equation \[\sqrt{x^2+y^2} + |y-1| = 5.\]
|
5
| 0.5 |
The average of four different positive whole numbers 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
| 0.916667 |
If $a, b, c$ are integers from the set of positive integers less than $5$ such that
\begin{align*}
abc &\equiv 1\pmod{5}, \\
3c &\equiv 2\pmod{5}, \\
4b &\equiv 3 + b\pmod{5},
\end{align*}
then what is the remainder when $a+b+c$ is divided by $5$?
|
4
| 0.666667 |
Find the sum of the coefficients in the polynomial \[ -3(x^8 - 2x^5 + 4x^3 - 6) + 5(x^4 + 3x^2) - 2(x^6 - 5) \].
|
37
| 0.916667 |
When the base-12 integer $2615_{12}$ is divided by $9$, what is the remainder?
|
8
| 0.5 |
In Pascal's Triangle, each number is the result of the sum of the number above it to the left and the number above it to the right. Consider Row 10, calculate the sum of the numbers in this row, and also determine the sum for Row 11. Verify the pattern mentioned in the previous problem.
|
2048
| 0.583333 |
What is the period and phase shift for the function \( y = \sin(3x + \frac{\pi}{4}) \)?
|
-\frac{\pi}{12}
| 0.25 |
Express $8.\overline{9}$ as a common fraction.
|
9
| 0.916667 |
Calculate the expected value of rolling an 8-sided die where the numbers 1 through 4 have a probability of $\frac{1}{10}$ each, and the numbers 5 through 8 have a probability of $\frac{3}{20}$ each.
|
4.9
| 0.75 |
A student must choose a program of five courses from a list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Biology. This program must contain English and at least two mathematics courses. In how many ways can this program be chosen?
|
6
| 0.333333 |
Add $518_{12} + 276_{12}$. Express your answer in base 12, using $A$ for $10$ and $B$ for $11$ if necessary.
|
792_{12}
| 0.333333 |
Diane has borrowed $200$ clams from Harold at a $4\%$ simple daily interest. Ian has borrowed $100$ clams from Jackie at a $8\%$ simple daily interest. How many days will it take until the total amount they both owe is $400$ clams, assuming that they will not make any repayments during this time?
|
6.25 \text{ days}
| 0.833333 |
Calculate the domain of the function \( f(x) = \log_5(\log_3(\log_2(x^2))) \).
|
(-\infty, -\sqrt{2}) \cup (\sqrt{2}, \infty)
| 0.583333 |
In a new laboratory experiment, a colony of bacteria triples in number every day. The colony begins with 4 bacteria. Determine the first day when the number of bacteria exceeds 200.
|
4
| 0.833333 |
Suppose that $a,$ $b,$ and $c$ are three positive numbers that satisfy the equations $abc = 1,$ $a + \frac {1}{c} = 8,$ and $b + \frac {1}{a} = 20.$ Find $c + \frac {1}{b}.$
|
\frac{10}{53}
| 0.75 |
Given that the point \((6,10)\) is on the graph of \(y=f(x)\), there is one point that must be on the graph of \(2y=5f(3x)+7\). What is the sum of the coordinates of that point?
|
30.5
| 0.333333 |
Given the function \( h(x) = 3x^3 + 3x^2 - x - 1 \), find the value of \( h(h(3)) \).
|
3406935
| 0.583333 |
Compute the value of the expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \]
|
1022
| 0.333333 |
In a panel discussion at a sports conference, five athletes from different teams are participating: two are from the Lakers, two are from the Celtics, and one is from the Warriors. If athletes from the same team insist on sitting together, how many ways can the five athletes be seated in a row?
|
24
| 0.833333 |
For how many integers $n$ is it true that $\sqrt{3n} \le \sqrt{5n - 8} < \sqrt{3n + 7}$?
|
4
| 0.833333 |
The average age of 40 fifth-graders is 10 years. The average age of 60 of their parents is 35 years. Additionally, there are 10 teachers with an average age of 45 years. What is the average age of all these fifth-graders, parents, and teachers combined?
|
26.82
| 0.166667 |
How many even perfect square factors does $2^6 \cdot 5^4 \cdot 7^3$ have?
|
18
| 0.833333 |
Simplify $\frac{6}{4y^{-4}} \cdot \frac{5y^{3}}{3}$ and add $\frac{7y^5}{8}$.
|
\frac{20y^7 + 7y^5}{8}
| 0.916667 |
What is the probability that the square root of a randomly selected two-digit whole number is less than nine? Express your answer as a common fraction.
|
\frac{71}{90}
| 0.833333 |
Math City is now designed with ten streets, all of which are straight. No two streets are parallel to one another. However, two of the ten streets have been built with tunnels that cross other streets without creating intersections. Assuming these tunnels are uniquely positioned such that each only bypasses one potential intersection with any other street, find the greatest number of police officers needed for the intersections.
|
43
| 0.916667 |
Fifteen prime numbers are randomly selected without replacement, from the first fifteen prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction.
|
\frac{4}{15}
| 0.416667 |
A sphere is perfectly inscribed in a cube. If the edge of the cube measures 10 inches, determine the volume of the sphere in cubic inches. Express your answer in terms of \(\pi\).
|
\frac{500}{3}\pi
| 0.166667 |
In a triangle $PQR$, where $P=(0,8)$, $Q=(0,0)$, and $R=(10,0)$, point $S$ is the midpoint of $\overline{PQ}$, and point $T$ is the midpoint of $\overline{QR}$. Determine the sum of the $x$ and $y$ coordinates of $U$, the point of intersection of $\overline{PT}$ and $\overline{RS}$.
|
\frac{18}{3} = 6
| 0.916667 |
Factor completely: $x^6 + 2x^4 - x^2 - 2$.
|
(x-1)(x+1)(x^2+1)(x^2+2)
| 0.583333 |
A mineralogist is hosting a competition to guess the age of an ancient mineral sample. The age is provided by the digits 2, 2, 3, 3, 5, and 9, with the condition that the age must start with an odd number.
|
120
| 0.5 |
The operation $\star$ is redefined as $a \star b = a^2 + \frac{a}{b}$. What is the value of $5 \star 2$?
|
27.5
| 0.916667 |
Cecilia is solving two quadratic equations simultaneously. The first equation is $9x^2 - 36x - 81 = 0$. She wants to solve it by completing the square. In addition, she is also considering the equation $y^2 + 6y + 9 = 0$. Find the values of $x + y$.
|
-1 - \sqrt{13}
| 0.25 |
What is the radius of the circle inscribed in triangle $ABC$ if $AB = 6, AC = 8, BC = 10$? Express your answer in simplest radical form.
|
2
| 0.916667 |
What is the coefficient of $a^3b^2$ in $(a+b)^5\left(c+\dfrac{1}{c}\right)^6$?
|
200
| 0.75 |
Find \(n\) such that \(2^6 \cdot 3^3 \cdot n = 10!\).
|
2100
| 0.833333 |
What is the slope of the line defined by any two solutions to the equation $\frac{3}{x} + \frac{4}{y} = 0$? Express your answer as a common fraction.
|
-\frac{4}{3}
| 0.916667 |
How many three-digit numbers are there in which the second digit is greater than the third digit?
|
405
| 0.583333 |
Find the sum of the distinct prime factors of $7^7 - 7^4$.
|
31
| 0.583333 |
James rode 40 miles at 8 miles per hour and 20 miles at 40 miles per hour. Additionally, James took a 30-minute break during his ride. What was his average speed, in miles per hour, for the entire ride excluding the break?
|
\frac{120}{11}
| 0.916667 |
What is the fifteenth term in the geometric sequence $12, 4, \frac{4}{3}, \ldots$?
|
\frac{4}{1594323}
| 0.333333 |
Simplify $\dfrac{222}{8888} \cdot 44.$
|
\dfrac{111}{101}
| 0.75 |
When the base-15 integer $2643_{15}$ is divided by 9, what is the remainder?
|
0
| 0.666667 |
Consider a new sequence of numbers arranged in rows where the number in each row starts with $3n$, where $n$ is the row number, and each row $n$ contains $n^3$ times this number. Given this pattern, what is the $40^{\mathrm{th}}$ number in the sequence?
|
12
| 0.333333 |
Given $ab + ac + bd + cd = 40$ and $a + d = 6$, find the value of $b + c$ assuming $a \neq d$.
|
\frac{20}{3}
| 0.75 |
Solve for $x$:
\[
\frac{x^2 - x - 2}{x + 2} = x - 1
\]
|
0
| 0.416667 |
For how many values of $c$ in the interval $[0, 2000]$ does the equation \[5 \lfloor x \rfloor + 4 \lceil x \rceil = c\] have a solution for $x$?
|
445
| 0.416667 |
Express \( 5.\overline{317} \) as a common fraction in lowest terms.
|
\frac{5312}{999}
| 0.916667 |
What is the base ten equivalent of $54123_{6}$?
|
7395
| 0.75 |
The polynomial $g(x)=x^4+px^3+qx^2+rx+s$ has real coefficients, and $g(3i)=g(3+2i)=0$. What is $p+q+r+s$?
|
79
| 0.666667 |
What is the largest divisor of 540 that is less than 80 and also a factor of 180?
|
60
| 0.583333 |
Mandy Monkey starts hopping on a number line at 0. She aims to reach 1, but can only hop 1/4 of the distance towards the goal at each jump. Each subsequent hop proceeds as 1/4 of the remaining distance to 1. Calculate the total distance Mandy has traveled after six hops. Express your answer as a common fraction.
|
\frac{3367}{4096}
| 0.583333 |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}.$ Given $\mathbf{m} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10,$ find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$
|
144
| 0.916667 |
Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $5$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $20$ distinct primes. Additionally, let the sum of the distinct prime factors of $a$ be greater than $50$.
If $a$ has fewer distinct prime factors than $b$, then what is the maximum number of distinct prime factors that $a$ can have?
|
12
| 0.75 |
What is the sum of the digits of the base $8$ representation of $888_{10}$?
|
13
| 0.833333 |
Convert $2 e^{15 \pi i/4}$ to rectangular form.
|
\sqrt{2} - i\sqrt{2}
| 0.25 |
Every week, Judy goes to the supermarket and buys the following: $5$ carrots at $\$1$ each, $3$ bottles of milk at $\$3$ each, $2$ pineapples at $\$4$ each, $2$ bags of flour at $\$5$ each, and a giant $\$7$ container of ice cream. This week the store has a sale, offering 25% off on flour. Judy also has a coupon for $\$10$ off any order of $\$30$ or over. How much money does Judy spend on this shopping trip?
|
\$26.5
| 0.75 |
The line \( x = k \) intersects the graph of the parabola \( x = -3y^2 + 2y + 7 \) at exactly one point. What is \( k \)?
|
\frac{22}{3}
| 0.916667 |
Find all $c$ which satisfy $$\frac{c}{4} \le 3+c < -3(1+c).$$ Express your answer in interval notation, simplifying any fractions which occur in your answer.
|
[-4, -\frac{3}{2})
| 0.666667 |
Consider the sum $$2+33+444+5555+66666+777777+8888888+99999999.$$ Calculate its congruence modulo 9, denoted as $m$, where $0 \le m < 9$.
|
6
| 0.75 |
If 18 bahs are equal to 30 rahs, and 6 rahs are equivalent to 10 yahs, how many bahs are equal to 1200 yahs?
|
432
| 0.916667 |
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 8 and an altitude of 10, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.
|
\frac{20}{9}
| 0.333333 |
Triangle $ABC$ has vertices $A(1, 10)$, $B(3, 0)$, and $C(9, 0)$. A horizontal line with equation $y=t$ intersects line segment $\overline{AB}$ at $J$ and line segment $\overline{AC}$ at $K$, forming $\triangle AJK$ with area 7.5. Compute $t$.
|
t=5
| 0.833333 |
Fido's leash is tied to a stake at the center of his yard, which is now in the shape of a square. His leash is long enough to reach exactly the midpoint of each side of the yard. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest form as \(\frac{a}{b}\pi\), what is the value of \(a+b\)?
|
5
| 0.833333 |
A certain integer has $5$ digits when written in base $3$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$?
|
15
| 0.833333 |
Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smallest number divides the other two numbers? Express your answer as a common fraction.
|
\frac{11}{20}
| 0.333333 |
Let the operation $\&$ be defined as $\&(a, b, c) = b^3 - 3abc - 4ac^2$, for all real numbers $a, b$ and $c$. What is the value of $\&(2, -1, 4)$?
|
-105
| 0.916667 |
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