problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 26$, and $\tan Z = 3 \sin Z$. What is $XZ$?
|
\frac{26}{3}
| 0.583333 |
A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
|
12
| 0.083333 |
At the end of the year, the Science Club decided to hold an election for which 4 equal officer positions were available. However, 20 candidates were nominated, of whom 8 were past officers. Of all possible elections of the officers, how many will have at least 2 of the past officers?
|
2590
| 0.833333 |
Suppose the numbers \[3 - \sqrt{8}, \;5+\sqrt{12}, \;16 - 2\sqrt{9}, \;-\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?
|
7
| 0.416667 |
In rectangle $ABCD$, $AB=8$ and $BC=5$. Points $F$ and $G$ are on $\overline{CD}$ such that $DF=3$ and $GC=1$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.
|
40
| 0.583333 |
In a given diagram, the area of triangle $ABC$ is 45 square units. The point $D$ still lies on the line $AC$, extending beyond $C$ such that $CD = 30$ units. If the length of $AC$ is 10 units, what is the area of triangle $BCD$?
|
135
| 0.916667 |
How many positive integers smaller than $500,000$ are powers of $2$ but are not powers of $4$?
|
9
| 0.833333 |
Simplify
\[
\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.
\]
|
2 \csc x
| 0.916667 |
Calculate $\dbinom{7}{5}$ and also find the number of permutations of 7 items taken 5 at a time.
|
2520
| 0.333333 |
Triangle \(PQR\) is similar to triangle \(XYZ\). If \(PQ = 8\), \(QR = 16\), and \(YZ = 24\) units, what is the length of segment \(XY\) if the perimeter of triangle \(XYZ\) is 60 units?
|
12
| 0.916667 |
If $5x - 7 = 15x + 21$, what is $3(x + 10)$?
|
21.6
| 0.333333 |
What is the base $2$ representation of $123_{10}$?
|
1111011_2
| 0.833333 |
What is the sum $(-1)^1+(-1)^2+\cdots+(-1)^{2007}$?
|
-1
| 0.916667 |
Evaluate the determinant of the following matrix:
\[
\begin{vmatrix}
0 & 2\sin \theta & -\cos \theta \\
-2\sin \theta & 0 & \sin \phi \\
\cos \theta & -\sin \phi & 0
\end{vmatrix}.
\]
|
0
| 0.833333 |
A cone is perfectly fitted inside a cube such that the cone's base is one face of the cube and its vertex touches the opposite face. A sphere is inscribed in the same cube. Given that one edge of the cube is 8 inches, calculate:
1. The volume of the inscribed sphere.
2. The volume of the inscribed cone.
Express your answer in terms of $\pi$.
|
\frac{128}{3}\pi
| 0.583333 |
If $c$ and $d$ are integers with $c > d$, what is the smallest possible positive value of $\frac{c+2d}{c-d} + \frac{c-d}{c+2d}$?
|
2
| 0.75 |
Let $p$ and $q$ be real numbers so that the roots of
\[3z^2 + (6 + pi) z + (50 + qi) = 0\]are complex conjugates. Enter the ordered pair $(p,q)$.
|
(0, 0)
| 0.833333 |
How many integers $-15 \leq n \leq 10$ satisfy $(n-1)(n+3)(n + 7) < 0$?
|
11
| 0.5 |
What integer \( n \) satisfies \( 0 \leq n < 137 \) and
$$ 12345 \equiv n \pmod{137}~? $$
|
15
| 0.916667 |
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\tan \angle APD = 2,$ then find $AB.$
|
27
| 0.583333 |
In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written?
|
20
| 0.833333 |
There are four containers:
- Container A holds 5 red balls and 7 green balls.
- Container B holds 7 red balls and 3 green balls.
- Container C holds 8 red balls and 2 green balls.
- Container D holds 4 red balls and 6 green balls.
The probability of choosing container A or D is $\frac{1}{4}$ each, and the probability of choosing containers B or C is $\frac{1}{4}$ each. If a container is selected at random and then a ball is randomly selected from that container, what is the probability that the ball selected is green? Express your answer as a common fraction.
|
\frac{101}{240}
| 0.333333 |
In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the trisection point of side $BC$ (closer to $B$ than to $C$). What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction.
|
\frac{1}{12}
| 0.75 |
Coach Grunt is preparing a 5-person starting lineup for his basketball team, the Grunters, which includes 15 players. Among them, Ace (a forward) and Zeppo (a guard), who are league All-Stars, will definitely be in the starting lineup. The remaining positions to be filled are one guard, one forward, and one center. If there are 4 additional guards, 5 additional forwards, and 3 centers available, how many different starting lineups are possible?
|
60
| 0.916667 |
Find the units digit of the sum, $$ 1! + 2! + 3! + \cdots + 10! + 2^1 + 2^2 + 2^3 + \ldots + 2^{10}. $$
|
9
| 0.333333 |
Find the first term in the geometric sequence $x, y, z, 81, 162$.
|
10.125
| 0.25 |
Determine all values of \( c \) for which the following system has a solution \( (x, y) \) in real numbers:
\begin{align*}
\sqrt{x^2y^2} &= c^{2c}, \\
\log_c (x^{\log_c y}) + \log_c (y^{\log_c x}) &= 8c^4.
\end{align*}
|
\left(0, \frac{1}{2}\right]
| 0.833333 |
Consider two lines: line $r$ parametrized as
\begin{align*}
x &= 2 + 5t,\\
y &= 3 - 2t,
\end{align*}
and the line $s$ parametrized as
\begin{align*}
x &= 1 + 5u,\\
y &= -2 - 2u.
\end{align*}
Let $C$ be a point on line $r$, $D$ be a point on line $s$, and let $Q$ be the foot of the perpendicular from $C$ to line $s$.
Then $\overrightarrow{QC}$ is the projection of $\overrightarrow{DC}$ onto some vector $\begin{pmatrix} w_1\\w_2\end{pmatrix}$ such that $w_1 - w_2 = 3$. Find $\begin{pmatrix}w_1 \\ w_2 \end{pmatrix}$.
|
\begin{pmatrix} -2 \\ -5 \end{pmatrix}
| 0.333333 |
What is the units digit of the sum of the squares of the first 2500 odd, positive integers?
|
0
| 0.916667 |
Adam and Simon start their bicycle trips from the same point at the same time. Adam travels east at 12 mph and Simon travels south at 16 mph. After how many hours are they 100 miles apart?
|
5
| 0.833333 |
I won a trip for six to an exclusive beach resort. I can bring five of my friends. I have 12 friends to choose from. At least one of the friends I choose must be among the 6 who have previously traveled with me. In how many ways can I form my travel group?
|
786
| 0.583333 |
Cameron, Dean, and Olivia each have a 6-sided cube. All of the faces on Cameron’s cube have a 6. The faces on Dean's cube are numbered 1, 1, 2, 2, 3, 3. Four of the faces on Olivia’s cube have a 3, and two of the faces have a 6. All three cubes are rolled. What is the probability that Cameron's roll is greater than each of Dean’s and Olivia's?
|
\frac{2}{3}
| 0.333333 |
How many four-digit numbers \( N \) have the property that the three-digit number obtained by removing the leftmost digit is one sixth of \( N \)?
|
4
| 0.583333 |
What is $3w + 4 - 6w - 5 + 7w + 8 - 9w - 10 + 2w^2$?
|
2w^2 - 5w - 3
| 0.583333 |
Calculate $7 \cdot 9\frac{2}{5}$.
|
65\frac{4}{5}
| 0.416667 |
Find the minimum value of
\[
\sin^4 x + 2 \cos^4 x,
\]
as \( x \) varies over all real numbers.
|
\frac{2}{3}
| 0.916667 |
Let $O$ be the origin, and let $(ka, kb, kc)$ be a fixed point for some non-zero constant $k$. A plane passes through $(ka, kb, kc)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ and $O.$ Find
\[
\frac{ka}{p} + \frac{kb}{q} + \frac{kc}{r}.
\]
|
2
| 0.583333 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 9.$ Find the minimum value of
\[\frac{x^2 + y^2}{x + y} + \frac{x^2 + z^2}{x + z} + \frac{y^2 + z^2}{y + z}.\]
|
9
| 0.916667 |
Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2 + y^2 = 20x + 36y$. What is the minimum value of $y$?
|
18 - 2\sqrt{106}
| 0.916667 |
Arthur walks 8 blocks east and then 10 blocks north. If each block is one-fourth of a mile, how many miles did Arthur walk in total?
|
4.5
| 0.5 |
What is the maximum number of consecutive positive integers starting from 3 that can be added together before the sum exceeds 500?
|
29
| 0.833333 |
Sandy plans to paint her daughter's playhouse including two trapezoidal sides of the roof and the front triangular face, all shaded in the diagram. The paint covers 100 square feet per gallon and costs $\$15$ per gallon. The roof sides are each 8 feet wide at the base and 5 feet tall, tapering linearly to a top width of 4 feet. The front face is an equilateral triangle with a 6 feet side. Calculate the total cost of the paint needed.
|
\$15
| 0.75 |
Suppose you have a drawer containing 8 forks, 7 spoons, and 5 knives. If you reach in and randomly remove four pieces of silverware, what is the probability that you get two forks, one spoon, and one knife?
|
\frac{196}{969}
| 0.833333 |
Find the number of real solutions to
\[(x^{2010} + 1)(x^{2008} + x^{2006} + x^{2004} + \dots + x^2 + 1) = 2010x^{2009}.\]
|
1
| 0.583333 |
Pascal's Triangle starting with row 1 has the sum of elements in row $n$ given by $2^{n-1}$. What is the sum of the interior numbers of the ninth row, considering interior numbers are all except the first and last numbers in the row?
|
254
| 0.916667 |
How many distinct, positive factors does $1320$ have?
|
32
| 0.916667 |
In the diagram below, we have \(\sin \angle APB = \frac{3}{5}\). What is \(\sin \angle APC\)?
[asy]
pair A,P,B,C;
C = (-2,0);
P = (0,0);
B = (2,0);
A = rotate(aSin(3/5))*(1.5,0);
dot("$C$",C,S);
dot("$B$",B,S);
dot("$A$",A,N);
dot("$P$",P,S);
draw(B--C);
draw(P--A);
[/asy]
|
\frac{3}{5}
| 0.833333 |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 24$ and $X$ is an integer, what is the smallest possible value of $X$?
|
4625
| 0.25 |
At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. Initially, a nitrogen gas is in a 3.4 liter container at a pressure of 7 kPa. If I transfer all the nitrogen to a 4.25 liter container while maintaining the same temperature, what will be the new pressure?
|
5.6 \, \text{kPa}
| 0.333333 |
What is the sum of all the positive divisors of 154?
|
288
| 0.916667 |
The sum of 22 consecutive positive integers is a perfect cube. What is the smallest possible value of this sum?
|
1331
| 0.75 |
If the lengths of two sides of a right triangle are 7 and 24 units, what is the least possible length, in units, of the third side? Express your answer in simplest radical form.
|
\sqrt{527}
| 0.916667 |
Consider a modified triangular array where each number is obtained similarly by summing two adjacent numbers from the previous row, but now, the numbers on the sides start from 0 and increase by 2, i.e., 0, 2, 4, ..., etc. Calculate the sum of the numbers in the 50th row, assuming each row's sum follows the recursion pattern $f(n) = 2f(n-1) + 4$ with $f(1)=0$.
|
2^{51} - 4
| 0.833333 |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x + f(y)) = x + y + k \]
for all real numbers $x$ and $y$, where $k$ is a real constant.
|
1
| 0.833333 |
Each term in the sequence $4096, 1024, 256, x, y, 4, 1, \frac{1}{4},...$ is obtained by multiplying the previous term by a constant. What is the value of $x + y$?
|
80
| 0.916667 |
Three dice with faces numbered 1 through 6 are stacked as shown. Eight of the eighteen faces are visible, leaving 10 faces hidden (back, bottom, in between). The visible numbers are 1, 2, 3, 4, 4, 5, 6, and 6. What is the total number of dots NOT visible in this view?
|
32
| 0.916667 |
Solve the inequality
\[\frac{(x - 1)(x - 4)(x - 5)}{(x - 2)(x - 6)(x - 7)} > 0.\]
|
(-\infty, 1) \cup (2, 4) \cup (5, 6) \cup (7, \infty)
| 0.916667 |
Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{v},$ $\mathbf{Q} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}.$
|
\begin{pmatrix} \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \end{pmatrix}
| 0.916667 |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 15$ and $X$ is an integer, what is the smallest possible value of $X$?
|
74
| 0.916667 |
Solve the equation:
\[\frac{1}{x + 10} + \frac{1}{x + 8} = \frac{1}{x + 11} + \frac{1}{x + 7}.\]
|
-9
| 0.916667 |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at the same time?
|
42
| 0.833333 |
Yan is situated between his home and the stadium. To get to the stadium, Yan can either walk directly there or return home first and then ride his bicycle to the stadium. He now rides 10 times as fast as he walks, and both routes take the same amount of time. Determine the ratio of the distance from Yan's current position to his home and to the stadium.
|
\frac{9}{11}
| 0.5 |
How many values of $x$, $-30<x<120$, satisfy $\cos^2 x + 3\sin^2 x = 1?$ (Note: $x$ is measured in radians.)
|
48
| 0.5 |
If the lengths of two sides of a right triangle are 7 and 24 units, what is the least possible length, in units, of the third side? Express your answer in simplest radical form.
|
\sqrt{527}
| 0.5 |
What is the largest integer $n$ for which $\binom{10}{3} + \binom{10}{4} = \binom{11}{n}$?
|
7
| 0.75 |
The sequence \( 9720, 3240, 1080, \ldots \) is created by repeatedly dividing by 3. How many integers are in this sequence?
|
6
| 0.833333 |
Eight people can paint a house in four hours. If six people start painting the house and work for two hours, how many more people are needed to finish painting the house in the next two hours?
|
4
| 0.833333 |
Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{2}{5} \\ -\frac{4}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$
|
\begin{pmatrix} \frac{22}{5} \\ -\frac{6}{5} \end{pmatrix}
| 0.75 |
A triangle has sides of lengths 6, 8, and 10 and is also a right triangle. Each vertex of this triangle is the center of a circle, and each circle is externally tangent to the other two circles. Determine the sum of the areas of these circles.
|
56\pi
| 0.916667 |
If \( g(x) = \frac{3x + 2}{x - 2} \), find the value of \( g(8) \).
|
\frac{13}{3}
| 0.916667 |
Linda has 8 daughters and no sons. Some of her daughters have 5 daughters each, and the rest have none. Linda has a total of 43 daughters and granddaughters, and no great-granddaughters. How many of Linda's daughters and granddaughters have no daughters?
|
36
| 0.916667 |
Find the smallest positive integer $b$ for which $x^2 + bx + 1800$ factors into a product of two binomials, each having integer coefficients.
|
85
| 0.416667 |
Let the sequence \(b_1, b_2, b_3, \dots\) be defined such that \(b_1 = 24\), \(b_{12} = 150\), and for all \(n \geq 3\), \(b_n\) is the arithmetic mean of the first \(n - 1\) terms. Find \(b_2\).
|
276
| 0.833333 |
A park has two parallel paths 60 feet apart. A crosswalk bounded by two parallel lines crosses the paths at an angle. The length of the path between the lines is 20 feet and each line is 75 feet long. Find the distance, in feet, between the lines.
|
16
| 0.083333 |
Let \[f(x) =
\begin{cases}
2x^2 + 5 & \text{if } x \leq 4, \\
bx + 3 & \text{if } 4 < x \leq 6, \\
cx^2 - 2x + 9 & \text{if } x > 6.
\end{cases}
\]
Find values of $b$ and $c$ such that the graph of $y = f(x)$ is continuous.
|
b = 8.5, c = \frac{19}{12}
| 0.416667 |
In the diagram, there are two circles. The larger circle has center $O$ and an area of $100\pi$. The smaller circle has its center on the circumference of the larger circle and touches the larger circle internally at one point. Each circle is divided into three equal areas, and we want to find the total area of the shaded regions which cover two of the three parts for each circle.
```asy
size(120);
import graph;
fill(Arc((0,0),2,240,480)--cycle,mediumgray);
fill(Arc((1,0),1,60,300)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((1,0),1));
dot((0,0)); label("$O$", (0,0), W);
draw((-2,0)--(2,0));
draw((1,-1)--(1,1));
```
|
\frac{250\pi}{3}
| 0.833333 |
The roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{73}$. Find the greatest possible value of $k$.
|
\sqrt{105}
| 0.916667 |
Find the smallest two-digit prime number with 2 as the tens digit such that reversing the digits of the number produces a composite number.
|
23
| 0.916667 |
On a 6 by 6 square grid where each dot is 1 cm apart from its nearest horizontal and vertical neighbors, consider a right triangle with vertices labeled $P$, $Q$, and $R$. The vertices $P$, $Q$, and $R$ are located at coordinates $(1, 5)$, $(5, 5)$, and $(1, 1)$, respectively. What is the product of the area of triangle $PQR$ and the perimeter of triangle $PQR$? Express your answer in simplest form.
|
64 + 32\sqrt{2} \text{ cm}^3
| 0.916667 |
A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form a square, and the piece of length $b$ is bent to form a circle. The square and the circle have equal area. What is $\frac{a}{b}$?
|
\frac{2}{\sqrt{\pi}}
| 0.333333 |
Find the reflection of $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ over the vector $\begin{pmatrix} 2 \\ -1 \end{pmatrix}.$
|
\begin{pmatrix} \frac{17}{5} \\ -\frac{6}{5} \end{pmatrix}
| 0.75 |
In right triangle $DEF$, $DE=15$, $DF=9$ and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?
|
7.5
| 0.083333 |
What is the remainder when \(5x^8 - 3x^7 + 2x^6 - 8x^4 + 3x^3 - 5\) is divided by \(3x - 6\)?
|
915
| 0.333333 |
Find the remainder when $7^{1985}$ is divided by $17$.
|
7
| 0.833333 |
A biased die has probabilities of rolling a 1, 2, 3, 4, 5, or 6 as \( \frac{1}{12}, \frac{1}{12}, \frac{1}{12}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \) respectively. If rolling a 1, 2, or 3 wins you \$4, but rolling a 4, 5, or 6 loses you \$3, what is the expected value, in dollars, of one roll?
|
-\$1.25
| 0.833333 |
At a regional science fair, 25 participants each have their own room in the same hotel, with room numbers from 1 to 25. All participants have arrived except those assigned to rooms 14 and 20. What is the median room number of the other 23 participants?
|
12
| 0.916667 |
Find $101^{-1} \pmod{102}$, as a residue modulo 102.
|
101
| 0.916667 |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{4} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]
Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Find $\sin \theta.$
|
\frac{\sqrt{15}}{4}
| 0.833333 |
From a circular piece of paper with radius $BC = 16$ cm, Jeff removes an unshaded sector. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 15 centimeters and of volume $675\pi$ cubic centimeters. Determine the number of degrees in the measure of angle $ABC$ of the sector that is not used.
|
22.5^\circ
| 0.916667 |
A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the car enters and leaves the storm respectively.
|
\frac{880}{29}
| 0.333333 |
The function $g(x)$ satisfies
\[ g(xy) = \frac{g(x)}{y} \]
for all positive real numbers $x$ and $y$. If $g(45) = 15$, find $g(60)$.
|
11.25
| 0.25 |
Given that $(x - y)^2 = 49$ and $xy = -8$, find the value of $x^2 + y^2$.
|
33
| 0.916667 |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 8x + 12|$ and $y = 6 - x$.
|
10
| 0.5 |
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{1}{8}
| 0.5 |
A box contains 7 white balls and 8 black balls. Three balls are drawn out of the box at random. What is the probability that all three are white?
|
\frac{1}{13}
| 0.916667 |
The number of gallons of coffee a scientist drinks in a day is inversely proportional to the number of hours he sleeps the previous night. On Wednesday, he slept for 8 hours and consumed 3 gallons of coffee. On Thursday, he slept for 4 hours. On Friday, he slept for 10 hours. How many gallons of coffee did he drink on Thursday and Friday, and what was his average coffee consumption over these three days?
|
3.8
| 0.333333 |
Solve $\log_3 x + \log_9 x^3 = 9$.
|
3^{\frac{18}{5}}
| 0.666667 |
How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^3)(x-2^3)(x-3^3)\dotsm(x-50^3)} = 0\] have for \(x\)?
|
47
| 0.333333 |
In triangle PQR below, $\cos Q = \frac{5}{13}$, where PQ = 13. What is QR?
[asy]
pair P,Q,R;
P = (0,12);
Q = (0,0);
R = (5,0);
draw(P--Q--R--cycle);
label("$P$",P,NW);
label("$Q$",Q,SW);
label("$R$",R,SE);
label("$13$",(P+Q)/2,W);
[/asy]
|
5
| 0.333333 |
Simplify $$\frac{13!}{10! + 3\cdot 9!}$$
|
1320
| 0.833333 |
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