problem
stringlengths 18
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stringlengths 1
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float64 0.08
0.92
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|---|---|---|
Given that $\frac{a}{30-a} + \frac{b}{70-b} + \frac{c}{80-c} = 9$, evaluate $\frac{6}{30-a} + \frac{14}{70-b} + \frac{16}{80-c}$.
|
2.4
| 0.083333 |
If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 10? Express your answer as a common fraction.
|
\frac{27}{216}
| 0.083333 |
Determine how many positive odd integers greater than 1 and less than $200$ are square-free.
|
80
| 0.25 |
Suppose there are 6 red plates, 5 blue plates, and 3 green plates. If I randomly select two plates to serve dinner on, what is the probability that they are both the same color?
|
\frac{28}{91}
| 0.083333 |
A three-row triangle is constructed using unit rods and connectors. The first row uses 3 rods and 4 connectors. Each subsequent row has 3 more rods and one additional connector than the previous row. Determine how many pieces would be needed to create a ten-row triangle.
|
250
| 0.75 |
The sequence $8820, 2940, 980, \ldots$ is made by repeatedly dividing by 3. How many integers are in this sequence?
|
3
| 0.916667 |
Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{AB}$ at $P$ and line segment $\overline{AC}$ at $Q$, forming $\triangle APQ$ with area 18. Compute $s$.
|
10 - 2\sqrt{15}
| 0.833333 |
Find the least common multiple and the greatest common divisor of 1365 and 910.
|
455
| 0.916667 |
Factor the following expression: $75x^{19} + 165x^{38}$.
|
15x^{19}(5 + 11x^{19})
| 0.833333 |
All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the tenth number in that list?
|
147
| 0.5 |
A $270^\circ$ rotation around the origin in the counter-clockwise direction is applied to $-4 + 2i.$ Followed by a scaling of the result by a factor of 2. What is the resulting complex number?
|
4 + 8i
| 0.916667 |
$ABCD$ is a trapezoid where the measure of base $\overline{AB}$ is three times the measure of the base $\overline{CD}$. Additionally, the perimeter of trapezoid $ABCD$ is 36 units. Point $E$ is the intersection of the diagonals. If the length of diagonal $\overline{AC}$ is 18 units, determine the length of segment $\overline{EC}$.
|
\frac{9}{2}
| 0.083333 |
The area of the parallelogram generated by the vectors $\mathbf{u}$ and $\mathbf{v}$ is 12. Find the area of the parallelogram generated by the vectors $3\mathbf{u} - 2\mathbf{v}$ and $4\mathbf{u} + 5\mathbf{v}$.
|
276
| 0.833333 |
Determine the remainder when \(1 + 5 + 5^2 + \cdots + 5^{1002}\) is divided by \(500\).
|
31
| 0.5 |
Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.001, 0.01).$
|
287
| 0.916667 |
Simplify this expression to a common fraction: $\frac{1}{\frac{1}{(\frac{1}{3})^{1}}+\frac{1}{(\frac{1}{3})^{2}}+\frac{1}{(\frac{1}{3})^{3}}}$.
|
\frac{1}{39}
| 0.916667 |
In right triangle $ABC$, $\angle B = \angle C$ and $AB = 10$. What is the area of $\triangle ABC$?
|
50
| 0.75 |
Given the points $(7, -9)$ and $(1, 7)$ as the endpoints of a diameter of a circle, calculate the sum of the coordinates of the center of the circle, and also determine the radius of the circle.
|
\sqrt{73}
| 0.083333 |
Find the number of positive integers $n \le 2000$ such that $10n$ is a perfect square.
|
14
| 0.416667 |
Jane and her brother each spin this spinner once. The spinner has six congruent sectors, numbered 1 through 6. If the non-negative difference of their numbers is less than or equal to 3, Jane wins; otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.
|
\frac{5}{6}
| 0.333333 |
How many 3-digit numbers have the property that the units digit is at least three times the tens digit?
|
198
| 0.833333 |
Carol hypothesizes that the length of her exercise session the night before a test and her score on the test are inversely related. On a previous test, after exercising for 45 minutes, she scored 80 points. Carol aims for an average score of 85 over two tests. To the nearest tenth, how long should Carol exercise the night before her next test so that her hypothesis remains valid and she achieves her score goal?
|
40
| 0.833333 |
A basketball player made the following number of successful free throws in 10 successive games: 8, 17, 15, 22, 14, 12, 24, 10, 20, and 16. He attempted 10, 20, 18, 25, 16, 15, 27, 12, 22, and 19 free throws in those respective games. Calculate both the median number of successful free throws and the player's best free-throw shooting percentage game.
|
90.91\%
| 0.666667 |
Let $A$ equal the number of four-digit odd numbers. Let $B$ equal the number of four-digit multiples of 3. Find $A+B$.
|
7500
| 0.916667 |
How many 4-letter words can we make from the letters A, B, C, D, and E, if we are allowed to repeat letters, and we must use each of the letters A and E at least once?
|
194
| 0.166667 |
In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written?
|
20
| 0.833333 |
Let $g(n)$ be a function that, given an integer $n$, returns an integer $m$, where $m$ is the smallest possible integer such that $m!$ is divisible by $n$. Given that $n$ is a multiple of 21, what is the smallest value of $n$ such that $g(n) > 21$?
|
n = 483
| 0.75 |
What is the least positive whole number divisible by four different prime numbers, none of which is smaller than 5?
|
5005
| 0.666667 |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 8?
|
8
| 0.916667 |
For all integers $n$ greater than 1, define $a_n = \dfrac{1}{\log_n 5000}$. Let $b = a_2 + a_3 + a_5$ and $c = a_{8} + a_{9} + a_{10} + a_{12}$. Find $b - c.$
|
-\log_{5000} 288
| 0.5 |
In right triangle $DEF$ where $\angle D = 90^\circ$, and side lengths are $DE = 9$ and $EF = 40$. Find $\cos D$.
|
0
| 0.916667 |
How many positive integers smaller than $500{,}000$ are powers of $3$, but are not powers of $27$? It may help to know that $3^6 = 729$.
|
8
| 0.666667 |
If $p$, $q$, and $r$ are positive integers satisfying $pq+r = qr+p = rp+q = 47$, what is the value of $p+q+r$?
|
48
| 0.666667 |
A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$?
|
225
| 0.583333 |
What is the remainder when $2011 \cdot 1537$ is divided by $450$?
|
307
| 0.333333 |
The diameter, in inches, of a sphere with three times the volume of a sphere of radius 6 inches can be expressed in the form $c\sqrt[3]{d}$ where $c$ and $d$ are positive integers and $d$ contains no perfect cube factors. Compute $c+d$.
|
15
| 0.916667 |
Let $\mathbf{c} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$ and $\mathbf{d} = \begin{pmatrix} 6 \\ 1 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{c},$ and $\mathbf{d}.$
|
19.5
| 0.166667 |
Let $\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at $W$. If $XW = 3$ and $YW = 9$, find the length of $WZ$.
|
27
| 0.25 |
The equation $x^2 - 2x = i$ has two complex solutions. Determine the product of their real parts.
|
\frac{1 - \sqrt{2}}{2}
| 0.166667 |
A rectangle is termed "great" if the number of square units in its area equals three times the number of units in its perimeter, and it has integer side lengths. Calculate the sum of all different possible areas of such great rectangles.
|
942
| 0.916667 |
Suppose you have a cube with a side length of 2x. A regular octahedron is formed by joining the centers of adjoining faces of this larger cube. Calculate the ratio of the volume of this octahedron to the volume of the cube.
|
\frac{1}{6}
| 0.333333 |
Suppose $\cos Q = 0.6$ in the right-angled triangle below. If the length of $QP$ is 18 units, find the length of $QR$.
|
30
| 0.916667 |
How many three-digit whole numbers have no 5's, 7's, or 9's as digits?
|
294
| 0.833333 |
How many zeros are in the expansion of $999,\!999,\!999,\!997^2$?
|
11
| 0.083333 |
Consider the following expansion using the binomial theorem: $(1+0.1)^{2000}$. This can be expanded as:
\[\sum_{k=0}^{2000} {2000 \choose k}(0.1)^k = B_0 + B_1 + B_2 + \cdots + B_{2000},\]
where $B_k = {2000 \choose k}(0.1)^k$. Identify the value of $k$ for which $B_k$ is the largest.
|
181
| 0.916667 |
In a right triangle with integer length sides, the hypotenuse has length 65 units. What is the length of the shorter leg?
|
16
| 0.75 |
Compute the integer \( k > 2 \) for which
\[
\log_{10} (k - 2)! + \log_{10} (k - 1)! + 3 = 2 \log_{10} k!.
\]
|
10
| 0.833333 |
Let \( r_1, r_2, \) and \( r_3 \) be the roots of the polynomial equation
\[ x^3 - 4x^2 + 5x + 12 = 0. \]
Find the monic polynomial, in \( x \), whose roots are \( 3r_1, 3r_2, \) and \( 3r_3 \).
|
x^3 - 12x^2 + 45x + 324
| 0.75 |
Compute \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$
|
8
| 0.833333 |
What is the measure of the smaller angle between the hands of a 12-hour clock at 3:40 pm, and also find the larger angle? Express your answer as a decimal to the nearest tenth of a degree.
|
230.0 \text{ degrees}
| 0.166667 |
A triangle has vertices at coordinates (3,3), (7,7), and (8,3). What is the number of units in the length of the longest side of this triangle?
|
4\sqrt{2}
| 0.083333 |
In the diagram, $AB$ is a line segment, and $CD$ is perpendicular to $AB$. A line $CE$ forms an angle of $65^\circ$ with $AB$. Find the value of $x$.
[asy]
draw((0,0)--(10,0),black+linewidth(1));
draw((4,0)--(4,8),black+linewidth(1));
draw((4,0)--(3.5,0)--(3.5,0.5)--(4,0.5)--cycle,black+linewidth(1));
draw((4,0)--(9,7),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(10,0),E);
label("$x^\circ$",(4.75,2.25));
label("$65^\circ$",(5.5,0.75));
label("$C$",(4,0),S);
label("$D$",(4,8),N);
label("$E$",(9,7),NE);
[/asy]
|
25^\circ
| 0.916667 |
In how many ways can I arrange 4 different math books and 6 different history books on my bookshelf, if I require a math book on both ends and two specific history books must not be adjacent?
|
362,\!880
| 0.416667 |
Two cylindrical cans have the same volume. The height of one can is double the height of the other. If the radius of the narrower can is 10 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form.
|
10\sqrt{2}
| 0.666667 |
Consider the system of equations
\[
4x - 3y = a,
\]
\[
6y - 8x = b.
\]
Assuming \( b \neq 0 \), find the value of \(\frac{a}{b}\).
|
-\frac{1}{2}
| 0.916667 |
If we let $g(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $j$ exist such that $1 \le j \le 5041$ and $g(j) = 1 + \sqrt{j} + j$?
|
20
| 0.916667 |
Points $C$ and $D$ are on a graph where both have the same $x$-coordinate of 7, but different $y$-coordinates. Point $C$ has a $y$-coordinate of 5 and point $D$ has a $y$-coordinate of -3. What is the value of the $x$-coordinate where this line intersects the $x$-axis?
|
7
| 0.333333 |
Find \( h(x) \), with terms in order of decreasing degree, if
\[ 12x^4 + 5x^3 + h(x) = -3x^3 + 4x^2 - 7x + 2. \]
|
-12x^4 - 8x^3 + 4x^2 - 7x + 2
| 0.916667 |
Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to her friend Rachel. Tracy and her mom then each ate 20 candies from what Tracy had left. Finally, Tracy’s brother took anywhere from two to six candies, leaving Tracy with ten candies. How many candies did Tracy have at the start?
|
112
| 0.166667 |
If I roll a fair, regular ten-sided die six times, what is the probability that I will roll the number $1$ exactly four times?
|
\frac{243}{200000}
| 0.583333 |
Let $\omega$ be a nonreal root of $z^4 = 1.$ Find the number of ordered pairs $(a,b)$ of integers such that $|a \omega + b| = 1.$
|
4
| 0.916667 |
A circular piece of paper with a radius of 5 inches has a section cut out, forming a \(225^\circ\) sector. This sector is then used to create a right circular cone by gluing points A and B together. Determine the circumference of the cone's base, expressed in terms of \(\pi\).
|
\frac{25\pi}{4}
| 0.75 |
Jason borrowed money from his parents to buy a new surfboard. His parents have agreed to let him work off his debt by babysitting under the following conditions: his first hour of babysitting is worth $\$1$, the second hour worth $\$2$, the third hour $\$3$, the fourth hour $\$4$, the fifth hour $\$5$, the sixth hour $\$6$, the seventh hour $\$7$, the eighth hour $\$1$, etc. If he repays his debt by babysitting for 39 hours, how many dollars did he borrow?
|
\$150
| 0.25 |
Carlos and Nina play a game where Carlos picks an integer between 1 and 4500 inclusive. Nina divides 4500 by that integer and checks if the result is an integer. How many integers can Carlos pick so that the quotient Nina receives is an integer? Additionally, Carlos should only pick numbers that are divisible by 3.
|
24
| 0.833333 |
There are 60 chips in a box. Each chip is either small or large. If the number of small chips is greater than the number of large chips by twice a prime number, what is the greatest possible number of large chips?
|
28
| 0.916667 |
In the increasing sequence of positive integers $a_1,$ $a_2,$ $a_3,$ $\dots,$ it's given that
\[a_{n + 2} = a_{n + 1} + a_n\] for all $n \ge 1$. If $a_7 = 210,$ find $a_9$.
|
550
| 0.25 |
Consider the cubic polynomial $p(x) = x^3 - (a^2 + b^2 + c^2)x^2 + (a^4 + b^4 + c^4)x - abc$ whose roots are real numbers, and the matrix
\[ \begin{pmatrix} a^2 & b & c \\ b^2 & c & a \\ c^2 & a & b \end{pmatrix} \]
is not invertible. List all possible values of
\[ \frac{a^2}{b^2 + c} + \frac{b^2}{a^2 + c} + \frac{c^2}{a^2 + b^2} \]
|
\frac{3}{2}
| 0.25 |
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 |
Factor $x^2 + 6x + 9 - 64x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2 + bx + c)(dx^2 + ex + f)$, with $a < d$.
|
(-8x^2 + x + 3)(8x^2 + x + 3)
| 0.416667 |
Let $P_1$ be a regular $r~\mbox{gon}$ and $P_2$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1$ is $\frac{61}{60}$ as large as each interior angle of $P_2$. What is the largest possible value of $s$?
|
121
| 0.083333 |
Each of $a_1,$ $a_2,$ $\dots,$ $a_{50}$ is equal to $1$ or $-1.$ Find the minimum positive value of
\[\sum_{1 \le i < j \le 50} a_i a_j.\]
|
7
| 0.916667 |
Let $r$ and $s$ be the two distinct solutions to the equation $$\frac{5x-15}{x^2+3x-18}=x+3.$$ If $r > s$, what is the value of $r - s$?
|
\sqrt{29}
| 0.666667 |
If 520 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?
|
12
| 0.166667 |
Let \( p(x) \) be a monic polynomial of degree 7 such that \( p(1) = 1, \) \( p(2) = 2, \) \( p(3) = 3, \) \( p(4) = 4, \) \( p(5) = 5, \) \( p(6) = 6, \) and \( p(7) = 7. \) Find \( p(8). \)
|
5048
| 0.916667 |
Suppose $a, b, c$ are positive numbers such that $abc = 1$, $a + \frac{1}{c} = 7$, and $b + \frac{1}{a} = 31$. Find $c + \frac{1}{b}$.
|
\frac{5}{27}
| 0.583333 |
Find the maximum integer value of
\[
\frac{3x^2 + 9x + 21}{3x^2 + 9x + 7}
\]
where $x$ is a real number.
|
57
| 0.416667 |
Let \(a_1, a_2, \dots\) be a sequence of positive real numbers such that
\[a_n = 15a_{n-1} - 2n\] for all \(n > 1\). Find the smallest possible value of \(a_1\).
|
\frac{29}{98}
| 0.833333 |
Tom has a red marble, a green marble, a blue marble, a purple marble, and four identical yellow marbles. How many different groups of two marbles can Tom choose?
|
11
| 0.416667 |
Find the greatest common divisor of $7429$ and $13356$.
|
1
| 0.666667 |
If $n = 2^{12} \times 3^{15} \times 5^9$, how many of the natural-number factors of $n$ are multiples of 300?
|
1320
| 0.75 |
Given \( w \) and \( z \) are complex numbers such that \( |w+z|=2 \) and \( |w^2+z^2|=8 \), find the smallest possible value of \( |w^3+z^3| \).
|
20
| 0.75 |
Calculate the sum of the $2023$ roots of $(x-1)^{2023} + 2(x-2)^{2022} + 3(x-3)^{2021} + \cdots + 2022(x-2022)^2 + 2023(x-2023)$.
|
2021
| 0.333333 |
Find the number of complex numbers $z$ satisfying $|z| = 1$ and
\[\left| \frac{z^3}{\overline{z}^3} + \frac{\overline{z}^3}{z^3} \right| = 3.\]
|
0
| 0.833333 |
A hexagon is formed by joining, in order, the points $(0,0)$, $(1,1)$, $(3,1)$, $(4,0)$, $(3,-1)$, $(1,-1)$, and back to $(0,0)$. The perimeter of the hexagon can be written in the form $a + b\sqrt{2} + c\sqrt{5}$, where $a$, $b$, and $c$ are whole numbers. Find $a+b+c$.
|
8
| 0.833333 |
Compute the sum of the series:
\[ 2(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4))))))) \]
|
43690
| 0.5 |
Select two distinct numbers simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smaller one divides the larger one and both numbers are either both even or both odd?
|
\frac{4}{15}
| 0.416667 |
At a gathering, there are 8 married couples. Each person is seated in a circular arrangement and shakes hands with every other person except for his or her spouse and the person directly next to them (neighbors). How many handshakes occur?
|
96
| 0.666667 |
When the graph of $y = 2x^2 - x + 5$ is shifted seven units to the right and then shifted three units up, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$.
|
86
| 0.333333 |
What is the modulo $7$ remainder of the sum $1+2+3+4+\ldots+140?$
|
0
| 0.916667 |
In triangle $ABC$, $AB = 6$, $BC = 8$, and $CA = 10$. Triangle ABC is a right triangle with the right angle at $C$.
Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to $C$ than it is to either $A$ or $B$?
|
\frac{1}{2}
| 0.083333 |
For each positive integer $n$, let $n!$ denote the product $1\cdot 2\cdot 3\cdot\,\cdots\,\cdot (n-1)\cdot n$.
What is the remainder when $10!$ is divided by $13$?
|
6
| 0.583333 |
If $x$, $y$, and $z$ are positive with $xy=32$, $xz=64$, and $yz=96$, what is the value of $x+y+z$?
|
\frac{44\sqrt{3}}{3}
| 0.416667 |
A triangle has three different integer side lengths and a perimeter of 24 units. What is the maximum length of any one side?
|
11
| 0.916667 |
What is the sum of all integer solutions to $|n| < |n-4| < 10$?
|
-14
| 0.583333 |
When a polynomial is divided by $3x^3 - 4x^2 + x - 5,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas.
|
0, 1, 2
| 0.916667 |
Find the largest prime divisor of $36^2 + 49^2$.
|
3697
| 0.083333 |
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $15$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.
|
15
| 0.916667 |
The Lucas sequence is defined by \( L_1 = 1 \), \( L_2 = 3 \), and each subsequent term is the sum of the previous two terms. Calculate the remainder when the $150^\text{th}$ term of the Lucas sequence is divided by 5.
|
3
| 0.916667 |
The diameter, in inches, of a sphere with three times the volume of a sphere of radius 9 inches can be expressed in the form \(a\sqrt[3]{b}\) where \(a\) and \(b\) are positive integers, and \(b\) contains no perfect cube factors. Compute \(a+b\).
|
21
| 0.583333 |
How many lattice points lie on the graph of the equation \(x^2 - y^2 = 65\)?
|
8
| 0.833333 |
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