problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
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|---|---|---|
Let $p,$ $q,$ and $r$ be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\]is true if and only if either $x < -6$ or $3 \le x \le 8.$ Given that $p < q,$ find the value of $p + 2q + 3r.$
|
1
| 0.5 |
Triangle $PQR$ has vertices $P(1, 6)$, $Q(3, -2)$, and $R(9, -2)$. A line through $R$ cuts the area of $\triangle PQR$ in half; find the sum of the slope and $y$-intercept of this line.
|
\frac{18}{7}
| 0.333333 |
In the diagram, $\angle ABC = 120^\circ$. What is the value of $y$?
[asy]
size(100);
draw((0,1)--(0,0)--(1,0));
draw((0,0)--(.6,.8));
draw((0,.1)--(.1,.1)--(.1,0));
label("$A$",(0,1),N); label("$B$",(0,0),SW); label("$C$",(1,0),E); label("$D$",(.6,.8),NE);
label("$3y^\circ$",(.15,.2)); label("$y^\circ$",(.32,-.02),N);
[/asy]
|
30
| 0.416667 |
Let $\mathbf{p} = \begin{pmatrix} -4 \\ 6 \\ 2 \end{pmatrix},$ $\mathbf{q} = \begin{pmatrix} 3 \\ e \\ -1 \end{pmatrix},$ and $\mathbf{r} = \begin{pmatrix} -1 \\ -3 \\ 5 \end{pmatrix}.$ Compute
\[(\mathbf{p} - \mathbf{q}) \cdot [(\mathbf{q} - \mathbf{r}) \times (\mathbf{r} - \mathbf{p})].\]
|
0
| 0.916667 |
Consider the sum
\[\text{cis } 20^\circ + \text{cis } 30^\circ + \text{cis } 40^\circ + \dots + \text{cis } 160^\circ.\]
Express this sum in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.
|
90^\circ
| 0.25 |
A triangle with side lengths in the ratio 5:12:13 is inscribed in a circle of radius 6.5. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth.
|
30
| 0.916667 |
Find the units digit of \(7^{6^5}\).
|
1
| 0.916667 |
Find the remainder when $x^4 - x^3 + 1$ is divided by $x^2 - 4x + 6.$
|
6x - 35
| 0.666667 |
Calculate the number of digits in the product $2^{15} \times 5^{10}$.
|
12
| 0.5 |
Evaluate $2022^3 - 2020 \cdot 2022^2 - 2020^2 \cdot 2022 + 2020^3$.
|
16168
| 0.833333 |
Determine the smallest positive integer \(n\) for which there exists positive real numbers \(a\) and \(b\) such that
\[(a + 3bi)^n = (a - 3bi)^n,\]
and compute \(\frac{b}{a}\).
|
\frac{\sqrt{3}}{3}
| 0.5 |
If $x^{3y} = 8$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.
|
1
| 0.5 |
If the least common multiple of two 7-digit integers has 11 digits, then their greatest common divisor has at most how many digits?
|
4
| 0.666667 |
Roberto now owns five pairs of trousers. He can wear any of his seven shirts with any outfit, but two specific pairs of his trousers must be paired with either one of two specific jackets from his collection of three jackets. How many different outfits can Roberto put together under these constraints?
|
91
| 0.5 |
Consider two lines: line \( l' \) parametrized as
\[
\begin{align*}
x &= 2 + 5t,\\
y &= 3 + 4t,
\end{align*}
\]
and line \( m' \) parametrized as
\[
\begin{align*}
x &=-7 + 5s,\\
y &= 8 + 4s.
\end{align*}
\]
Let \( A' \) be a point on line \( l' \), \( B' \) be a point on line \( m' \), and let \( P' \) be the foot of the perpendicular from \( A' \) to line \( m' \).
Then \( \overrightarrow{P'A'} \) is the projection of \( \overrightarrow{B'A'} \) onto some vector \( \begin{pmatrix} v_1\\v_2\end{pmatrix} \) such that \( v_1 + v_2 = 7 \). Find \( \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \).
|
\begin{pmatrix} -28 \\ 35 \end{pmatrix}
| 0.25 |
When \( y \) is divided by each of \( 3 \), \( 7 \), and \( 8 \), remainders of \( 2 \), \( 6 \), and \( 7 \) (respectively) are obtained. What is the smallest possible positive integer value of \( y \)?
|
167
| 0.75 |
Factor the quadratic expression $16x^2 - 40x + 25$.
|
(4x - 5)^2
| 0.916667 |
Express 3125 as a sum of distinct powers of 2. What is the least possible sum of the exponents of these powers?
|
32
| 0.333333 |
Determine the remainder when the sum $1! + 2! + 3! + \cdots + 99! + 100!$ is divided by $30$.
|
3
| 0.916667 |
A grocer makes a display of cans where the top row has two cans and each lower row has three more cans than the row above it. If the display contains 225 cans, how many rows does it contain?
|
n = 12
| 0.916667 |
What is the modulo $7$ remainder of the sum $1+2+3+4+ \ldots + 152+153+154+155+156?$
|
3
| 0.416667 |
What is the smallest positive five-digit integer equivalent to 4 mod 9?
|
10003
| 0.916667 |
Suppose $a$, $b$, $c$, and $d$ are even integers satisfying: $a-b+c=8$, $b-c+d=10$, $c-d+a=4$, and $d-a+b=6$. What is the value of $a+b+c+d$?
|
28
| 0.583333 |
Let
\[
\mathbf{B} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.
\]
Compute \(\mathbf{B}^{106}\).
|
\begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}
| 0.916667 |
A sequence $b_1, b_2, b_3, \dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \ge 3,$
\[ b_k = \frac{1}{4} b_{k-1} + \frac{2}{5} b_{k-2}. \]
Evaluate $b_1 + b_2 + b_3 + \dotsb.$
|
\frac{85}{7}
| 0.416667 |
How many 3-digit numbers have the property that the units digit is at least three times the tens digit?
|
198
| 0.833333 |
For how many values of $c$ in the interval $[0, 2000]$ does the equation \[10 \lfloor x \rfloor + 3 \lceil x \rceil = c\] have a solution for $x$?
|
308
| 0.5 |
What are the last two digits in the sum of the factorials of the first 15 positive integers?
|
13
| 0.75 |
Find the remainder when $x^4 + x^2 + 1$ is divided by $x^2 - 2x + 4.$
|
-6x - 3
| 0.916667 |
Let $A = (2,0)$ and $B = (7, 6)$. Let $P$ be a point on the parabola $y^2 = 8x$. Find the smallest possible value of $AP + BP$.
|
9
| 0.083333 |
Expand the following expression: $(6x + 8 - 3y) \cdot (4x - 5y)$.
|
24x^2 - 42xy + 32x - 40y + 15y^2
| 0.916667 |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king?
|
\frac{1}{52}
| 0.5 |
The Cookie Monster encounters another cookie with the equation $x^2 + y^2 + 35 = 6x + 22y$. He needs help finding out the radius of this cookie to decide if it should be kept for dessert.
|
\sqrt{95}
| 0.5 |
Let $f(x) = 4x^2-3$ and $g(f(x)) = x^2 + x + 1$. Find the sum of all possible values of $g(35)$.
|
21
| 0.916667 |
A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For instance, July 19 is relatively prime since 7 and 19 have no common divisors other than 1. Identify the month with the fewest relatively prime dates when February has 29 days (leap year). How many relatively prime dates are in that month?
|
10
| 0.416667 |
A circle is tangent to the lines $3x + 4y = 24$ and $3x + 4y = -6$. The center of the circle lies on the line $3x - y = 0$. Find the center of the circle.
|
\left(\frac{3}{5}, \frac{9}{5}\right)
| 0.916667 |
Let $a,$ $b,$ $c$ be real numbers, all greater than 4, so that
\[\frac{(a + 3)^2}{b + c - 3} + \frac{(b + 5)^2}{c + a - 5} + \frac{(c + 7)^2}{a + b - 7} = 45.\]
Find the ordered triple $(a,b,c)$.
|
(12,10,8)
| 0.25 |
What is the greatest common divisor of $7!$ and $\frac{12!}{5!}?$ Express your answer as an integer.
|
5040
| 0.916667 |
My five friends and I gather every weekend for a game night. Every time, three of us set up the games and events, while the remaining three handle cleaning and arranging the food. How many different ways can we choose the groups for setup and others for food and cleaning?
|
20
| 0.083333 |
The Grunters play the Screamers 6 times. The Grunters are a strong team with an $80\%$ probability of winning any given game. What is the probability that the Grunters will win exactly 5 out of these 6 games? Express your answer as a common fraction.
|
\frac{6144}{15625}
| 0.583333 |
A lattice is structured such that each row contains 7 consecutive integers. If the pattern continues and the lattice has 10 rows, what is the fifth number in the 10th row?
|
68
| 0.166667 |
Eleven positive integers from a list include $5, 6, 7, 2, 4, 8, 3$. What is the largest possible value of the median of this list of eleven positive integers if one of the remaining numbers must be at least 10?
|
7
| 0.416667 |
On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming cars are 5 meters long and can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye placed beside the road in one hour in one direction. Find $N$ divided by $10$.
|
200
| 0.166667 |
What is the domain of the function $g(x) = \frac{x - 2}{\sqrt{x^2 - 5x + 6}}$?
|
(-\infty, 2) \cup (3, \infty)
| 0.916667 |
Evaluate $\log_{16}2$ and $\log_{16}8$.
|
\frac{3}{4}
| 0.333333 |
Find $y$ such that $\log_y 243 = \log_3 81$.
|
3^{5/4}
| 0.916667 |
The circles given by the equations \(x^2 + y^2 - 6x + 4y - 20 = 0\) and \(x^2 + y^2 - 8x + 18y + 40 = 0\) intersect at points \(C\) and \(D\). Determine the slope of \(\overline{CD}\).
|
\frac{1}{7}
| 0.833333 |
Jasmine hiked the Sierra Crest Trail last week. It took five days to complete the trip. The first three days she hiked a total of 36 miles. The second and fourth days she averaged 15 miles per day. The last two days she hiked a total of 38 miles. The total hike for the first and fourth days was 32 miles. How many miles long was the trail?
|
74
| 0.833333 |
Simplify $5\cdot\frac{14}{3}\cdot\frac{9}{-42}$.
|
-5
| 0.833333 |
Six identical rectangles are arranged to form a larger rectangle $PQRS$, where the width of each smaller rectangle is $\frac{2}{5}$ of the length of the larger rectangle. If the area of $PQRS$ is $6000$, what is the length $x$, rounded off to the nearest integer?
|
50
| 0.25 |
Find the matrix $\mathbf{X}$ such that
\[
\mathbf{X} \begin{pmatrix} 2 & -1 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix} = \mathbf{I}.
\]
|
\begin{pmatrix} \frac{5}{7} & \frac{1}{7} & 0 \\ \frac{3}{7} & \frac{2}{7} & 0 \\ 0 & 0 & \frac{1}{2} \end{pmatrix}
| 0.833333 |
In the diagram, $ABCD$ is a trapezoid with an area of $30.$ $CD$ is three times the length of $AB.$ Determine the area of $\triangle ABC.$
[asy]
draw((0,0)--(1,4)--(9,4)--(16,0)--cycle);
draw((9,4)--(0,0));
label("$C$",(0,0),W);
label("$A$",(1,4),NW);
label("$B$",(9,4),NE);
label("$D$",(16,0),E);
[/asy]
|
7.5
| 0.5 |
Find $3^{\frac{1}{2}} \cdot 9^{\frac{1}{4}} \cdot 27^{\frac{1}{8}} \cdot 81^{\frac{1}{16}} \dotsm.$
|
9
| 0.833333 |
Find the greatest common divisor of 9242, 13863, and 34657.
|
1
| 0.666667 |
Define a $\textit{great word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many nine-letter great words are there?
|
26244
| 0.666667 |
What is the value of $n$ in the equation $n + (n + 1) + (n + 2) + (n + 3) = 20$?
|
3.5
| 0.75 |
Let $g(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. If $n$ is a multiple of 24, what is the smallest value of $n$ such that $g(n) > 24$?
|
n = 696
| 0.166667 |
On a certain day, I worked \( t+2 \) hours and earned \( 4t-4 \) dollars per hour. My friend Bob worked \( 4t-7 \) hours but only earned \( t+3 \) dollars an hour. At the end of the day, I had earned three dollars more than Bob. What is the value of \( t \)?
|
10
| 0.75 |
Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of
\[
\frac{(x^2 + 4x + 1)(y^2 + 4y + 1)(z^2 + 4z + 1)}{xyz}.
\]
|
216
| 0.75 |
A regular dodecagon \( Q_1 Q_2 \ldots Q_{12} \) is drawn in the coordinate plane with \( Q_1 \) at \( (2,0) \) and \( Q_7 \) at \( (4,0) \). If \( Q_n \) is the point \( (x_n, y_n) \), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \ldots (x_{12} + y_{12} i).
\]
|
531440
| 0.166667 |
Tom has a red marble, a green marble, a blue marble, three identical yellow marbles, and two identical white marbles. How many different groups of two marbles can Tom choose?
|
12
| 0.166667 |
Define $a * b$ as $2a - b^2$. If $a * 7 = 16$, what is the value of $a$?
|
32.5
| 0.416667 |
A school has eight identical copies of a specific textbook. Every day, some number of these textbooks are available in the classroom and some are taken by students for homework. Find the number of ways in which this distribution can occur if at least two books must be available in the classroom and at least three books must be taken by students.
|
4
| 0.916667 |
In $\triangle ABC$, the sides are such that $AC:CB = 4:5$. The bisector of the interior angle at $C$ intersects $AB$ at point $Q$. Determine the ratio $AQ:QB$.
|
4:5
| 0.166667 |
Factor the expression $5y(y-2) + 10(y-2) - 15(y-2)$.
|
5(y-2)(y-1)
| 0.916667 |
A sea turtle claims that the mysteriously inscribed number on an ancient coral reef is $732_{8}$. First, determine the number of years this represents in base sixteen and then convert it to base ten.
|
474_{10}
| 0.083333 |
Find the sum $m + n$ where $m$ and $n$ are integers, such that the positive difference between the two roots of the quadratic equation $2x^2 - 5x - 12 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, and $m$ is not divisible by the square of any prime number.
|
123
| 0.916667 |
If $\sec x + \tan x = \frac{7}{3},$ then find $\sec x - \tan x.$
|
\frac{3}{7}
| 0.833333 |
How many integers \( n \neq 0 \) satisfy the inequality \( \frac{1}{|n|} \geq \frac{1}{5} \)?
|
10
| 0.916667 |
In how many ways can the digits of $36,\!720$ be arranged to form a 5-digit number, where numbers cannot begin with $0$?
|
96
| 0.916667 |
The sides of the parallelogram are given as \(5\), \(11\), \(3y+2\), and \(4x-1\) respectively. Determine the value of \(x+y\).
\[ \text{[asy]} \draw((0,0)--(22,0)--(27,20)--(5,20)--cycle); \label("$3y+2$",(11,0),S); \label("11",(26,10),E); \label("5",(16,20),N); \label("$4x-1$",(2.5,10),W); \text{[/asy]} \]
|
4
| 0.666667 |
Find the sum of the distinct prime factors of $7^7 - 7^4$.
|
31
| 0.583333 |
Each day, Jenny ate $25\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the third day, 27 jellybeans remained. How many jellybeans were in the jar originally?
|
64
| 0.833333 |
In the diagram, $ABCD$ is a square with side length $8,$ and $WXYZ$ is a rectangle with $ZY=12$ and $XY=8.$ Also, $AD$ and $WX$ are perpendicular. If the shaded area is equal to one-third of the area of $WXYZ,$ what is the length of $AP?$
[asy]
draw((0,0)--(12,0)--(12,8)--(0,8)--cycle,black+linewidth(1));
draw((2,2)--(10,2)--(10,10)--(2,10)--cycle,black+linewidth(1));
filldraw((2,2)--(10,2)--(10,8)--(2,8)--cycle,gray,black+linewidth(1));
label("$W$",(0,8),NW);
label("$X$",(12,8),NE);
label("$Y$",(12,0),SE);
label("$Z$",(0,0),SW);
label("$A$",(2,10),NW);
label("$B$",(10,10),NE);
label("$C$",(10,2),E);
label("$D$",(2,2),W);
label("$P$",(2,8),SW);
label("8",(2,10)--(10,10),N);
label("8",(12,0)--(12,8),E);
label("12",(0,0)--(12,0),S);
[/asy]
|
4
| 0.583333 |
In a right-angled triangle, one leg is twice the length of the other leg. The sum of the squares of the three side lengths is 1450. What is the length of the hypotenuse of this triangle?
|
5 \sqrt{29}
| 0.833333 |
In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively. Determine the sum of the $x$ and $y$ coordinates of $F$, the point of intersection of $\overline{AE}$ and $\overline{CD}$. Given:
- Point $A$ is at $(0,8)$
- Point $B$ is at $(0,0)$
- Point $C$ is at $(10,0)$ [asy]
size(200); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,8);
B=(0,0);
C=(10,0);
D=(0,4);
E=(5,0);
F=(10/3,16/3);
draw(E--A--C--D);
draw(D--B, dashed);
draw((-1,0)--(12,0), EndArrow);
draw((0,-1)--(0,9), EndArrow);
label("$A(0,8)$", A, NW);
label("$B(0,0)$", B, SW);
label("$C(10,0)$", C, SE);
label("$D$", D, W);
label("$E$", E, S);
label("$F$", F, NE);
label("$x$", (12,0), dir(0));
label("$y$", (0,9), dir(90));
[/asy]
|
\frac{10}{3} + \frac{8}{3} = \frac{18}{3} = 6
| 0.583333 |
Find the range of $g(x) = \sin^4 x + \cos^4 x + 2\sin x \cos x$.
|
\left[ -\frac{1}{2}, \frac{3}{2} \right]
| 0.833333 |
Given the matrix
\[\mathbf{B} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix},\]
compute $\mathbf{B}^{101}$.
|
\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}
| 0.833333 |
Find the matrix $\mathbf{N}$ that halves the second column of a matrix. In other words,
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & \frac{b}{2} \\ c & \frac{d}{2} \end{pmatrix}.\] If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.
|
\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
| 0.833333 |
Given values \(f\), \(g\), \(h\), and \(j\) correspond to numbers 3, 4, 5, and 6, but not necessarily in that order. Find the largest possible sum of the four products \(fg\), \(gh\), \(hj\), and \(fj\).
|
81
| 0.583333 |
What is the smallest four-digit palindrome that is divisible by 5?
|
5005
| 0.833333 |
Billy chose a positive integer less than 500 that is a multiple of 30, and Bobbi selected a positive integer less than 500 that is a multiple of 45. What is the probability that they selected the same number? Express your answer as a common fraction.
|
\frac{5}{176}
| 0.583333 |
Rationalize the denominator of the fraction $\frac{\sqrt{50}}{\sqrt{25} - 2\sqrt{2}}$. Write your answer in the form $\frac{A\sqrt{B} + C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Find the minimum possible value of $A+B+C+D$.
|
64
| 0.666667 |
How many distinct four-digit numbers can be written with the digits $1$, $2$, $3$, $4$, and $5$ if each digit can be used only once and the number must include the digit $5$?
|
96
| 0.916667 |
I have 6 shirts, 5 pairs of pants, and 6 hats. Each piece of clothing comes in six possible colors: tan, black, blue, gray, white, and yellow. I refuse to wear an outfit in which all 3 items are the same color, and I also refuse to wear pants and hats of the same color. How many outfit selections, consisting of one shirt, one pair of pants, and one hat, do I have?
|
150
| 0.083333 |
Find $y$ such that $\log_y 243 = \log_3 81$.
|
3^{5/4}
| 0.916667 |
A team is divided into two subgroups of 3 members each. In how many ways can a President and a Vice-President be chosen from this team of 6, such that they are not from the same subgroup?
|
18
| 0.833333 |
The polynomial $x^4 - ax^3 + bx^2 - cx + 2520$ has four positive integer roots. What is the smallest possible value of $a$?
|
29
| 0.416667 |
A rectangle $ABCD$ contains four small squares within it. Three squares are shaded and have non-overlapping interiors, as shown in the diagram. The side length of each smaller shaded square is $2$ inches. Calculate the area of rectangle $ABCD$.
[asy]
size(150);
pair A = (0,0), D = (6,0), C = (6,4), B = (0,4);
draw(A--B--C--D--cycle);
draw(A--(2,0)--(2,2)--(0,2)--cycle);
filldraw(B--(2,4)--(2,2)--(0,2)--cycle, gray(0.7), black);
filldraw((2,0)--(4,0)--(4,2)--(2,2)--cycle, gray(0.7), black);
filldraw((4,0)--(6,0)--(6,2)--(4,2)--cycle, gray(0.7), black);
label("$A$",A,WSW);
label("$B$",B,WNW);
label("$C$",C,ENE);
label("$D$",D,ESE);
[/asy]
|
24
| 0.916667 |
The statue of Liberty is 305 feet tall. A scale model of the statue in a museum is 10 inches tall. How many feet of the statue does one inch of the model represent?
|
30.5
| 0.916667 |
In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 10 and 15 respectively, and the height (distance between the bases) is 6. 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}{5}
| 0.333333 |
If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 48 dips?
|
22.5 \text{ daps}
| 0.916667 |
Given the areas of three squares in the diagram, find the area of the triangle formed. The triangle shares one side with each of two squares and the hypotenuse with the third square.
[asy]
/* Modified AMC8-like Problem */
draw((0,0)--(10,0)--(10,10)--cycle);
draw((10,0)--(20,0)--(20,10)--(10,10));
draw((0,0)--(0,-10)--(10,-10)--(10,0));
draw((0,0)--(-10,10)--(0,20)--(10,10));
draw((9,0)--(9,1)--(10,1));
label("100", (5, 5));
label("64", (15, 5));
label("100", (5, -5));
[/asy]
Assume the triangle is a right isosceles triangle.
|
50
| 0.083333 |
Rosie can make three pies out of 12 apples and 6 pears. How many pies can she make if she has 36 apples and 18 pears?
|
9
| 0.833333 |
In square $PQRS$, point $X$ is the midpoint of side $PQ$ and point $Y$ is the midpoint of side $QR$. If point $Z$ is the midpoint of segment $PY$, find the ratio of the area of triangle $PXZ$ to the area of square $PQRS$.
|
\frac{1}{16}
| 0.916667 |
Two distinct positive integers \( a \) and \( b \) are factors of 48. If \( a \cdot b \) is not a factor of 48, what is the smallest possible value of \( a \cdot b \)?
|
18
| 0.083333 |
Determine the value of $b$ for which $\frac{1}{\log_3 b} + \frac{1}{\log_5 b} + \frac{1}{\log_6 b} = 1$.
|
90
| 0.833333 |
A math teacher asks Liam to undertake a challenging homework regimen. For earning each of the first seven homework points, Liam needs to complete three homework assignments; for each of the next seven points, he needs to complete four homework assignments; and so on, such that the number of homework assignments increases by one for every new set of seven points. Calculate the smallest number of homework assignments that Liam needs to complete in order to earn exactly 40 homework points.
|
215
| 0.083333 |
If two distinct numbers are selected at random from the first nine prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction.
|
\frac{7}{9}
| 0.833333 |
Find the modular inverse of $34$, modulo $35$.
Express your answer as an integer from $0$ to $34$, inclusive.
|
34
| 0.916667 |
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