problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
A stage play has two specific male roles, two specific female roles, and three roles that can be played by either gender. Only a man can be cast in a male role and only a woman can be cast in a female role. If four men and five women audition, in how many ways can the seven roles be assigned?
|
14400
| 0.666667 |
What is the smallest positive value of $m$ so that the equation $12x^2 - mx + 360 = 0$ has integral solutions?
|
132
| 0.916667 |
Triangle $XYZ$ has vertices $X(-1, 7)$, $Y(3, -1)$, and $Z(9, -1)$. A line through $Y$ cuts the area of $\triangle XYZ$ in half. Find the sum of the slope and the $y$-intercept of this line.
|
-9
| 0.666667 |
Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which
\[
\sqrt{1 + \sqrt{45 + 20 \sqrt{5}}} = \sqrt{a} + \sqrt{b}.
\]
|
(1,5)
| 0.75 |
Andrew's father's age is eight times Andrew's age. Andrew's grandfather's age is three times Andrew's father's age. If Andrew's grandfather was 72 years old when Andrew was born, how many years old is Andrew now?
|
\frac{72}{23}
| 0.5 |
Juan, Carlos, Manu, and Ana take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Ana will win? Express your answer as a common fraction.
|
\frac{1}{15}
| 0.75 |
The sum of the first $n$ terms in the infinite geometric sequence $\{1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \dots \}$ is $\frac{85}{64}$. Find $n$.
|
4
| 0.75 |
Compute the multiplicative inverse of $217$ modulo $397$. Express your answer as an integer from $0$ to $396$.
|
161
| 0.666667 |
What is the product of the digits in the base 6 representation of $7891_{10}$?
|
0
| 0.916667 |
What common fraction (in lowest terms) is equivalent to $0.4\overline{36}$?
|
\frac{24}{55}
| 0.833333 |
The operation $\#$ is redefined as $a \# b = a - \frac{b}{a}$. What is the value of $8 \# 4$?
|
7.5
| 0.166667 |
What is the units digit of the sum of the squares of the first 1013 odd, positive integers?
|
5
| 0.833333 |
Compute $2 \begin{pmatrix} 3 \\ -6 \end{pmatrix} + 4 \begin{pmatrix} -1 \\ 5 \end{pmatrix} - \begin{pmatrix} 5 \\ -20 \end{pmatrix}$.
|
\begin{pmatrix} -3 \\ 28 \end{pmatrix}
| 0.916667 |
A square has sides of length 8, and a circle centered at the center of the square has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.
|
144\pi
| 0.666667 |
Find the projection of the vector $\begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix}$ onto the line defined by $\frac{x}{3} = \frac{y}{-2} = \frac{z}{1}$.
|
\begin{pmatrix} \frac{51}{14} \\ -\frac{17}{7} \\ \frac{17}{14} \end{pmatrix}
| 0.75 |
A right triangle is to be constructed in the coordinate plane with its legs parallel to the axes, and the medians to the midpoints of the legs are required to lie on the lines $y = 2x + 1$ and $y = mx + 3$. Determine the number of different constants $m$ for which such a triangle exists.
|
1
| 0.75 |
Points $A(-1,3)$ and $B(9,12)$ are given in a coordinate plane. Through calculation, determine the area of a circle where point $A$ is one endpoint of a diameter and point $B$ is displaced 4 units horizontally toward the right. How many square units are in the area of this new circle, expressed in terms of $\pi$?
|
\frac{277\pi}{4}
| 0.583333 |
Ella has designed a rectangular garden divided into four quadrants by paths, with a circular flower bed in each quadrant. Each flower bed is tangent to the paths and its adjacent flower beds. If the garden paths create a rectangle that is 30 inches long and 24 inches wide, how many square inches are not occupied by flower beds?
|
720 - 144\pi \text{ square inches}
| 0.5 |
Find constants \( A \), \( B \), and \( C \) such that
\[ \frac{x^2 - 2x + 5}{x^3 - x} = \frac{A}{x} + \frac{Bx + C}{x^2 - 1} \]
Enter your answer as the ordered triplet \( (A, B, C) \).
|
(-5, 6, -2)
| 0.916667 |
The points \((2,10)\), \((14,19)\), and \((6,k)\), where \(k\) is an integer, are vertices of a triangle. What is the sum of the values of \(k\) for which the area of the triangle is a minimum?
|
26
| 0.166667 |
Determine the value of \(\frac{c}{b}\) when the quadratic \(x^2 - 2100x - 8400\) is expressed in the form \((x+b)^2 + c\), where \(b\) and \(c\) are constants.
|
1058
| 0.833333 |
What is the arithmetic mean of the integers from -6 through 7, inclusive?
|
0.5
| 0.75 |
In a triangle \( \triangle ABC \), for \( \angle A \) to be the largest angle, it must be that \( m < x < n \). The side lengths of the triangle are given by \( AB = x+5 \), \( AC = 2x+3 \), and \( BC = x+10 \). Determine the smallest possible value of \( n-m \), expressed as a common fraction.
|
6
| 0.916667 |
If $\frac{90}{2^4\cdot5^9}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?
|
3
| 0.25 |
Given $x, y,$ and $k$ are positive real numbers such that
\[4 = k^2\left(\frac{x^2}{y^2}+\frac{y^2}{x^2}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right),\]
find the maximum possible value of $k$.
|
1
| 0.916667 |
The average age of the four Smith children is 9 years. If the three younger children are 6 years old, 9 years old, and 12 years old, how many years old is the oldest child?
|
9
| 0.416667 |
Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$
|
1
| 0.083333 |
What is the arithmetic mean of the integers from -5 through 6, inclusive?
|
0.5
| 0.833333 |
Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as a quadratic polynomial \[Area(WXYZ) = \gamma \omega - \delta \omega^2.\] Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
|
229
| 0.166667 |
A bag contains 5 red marbles, 7 white marbles, and 8 blue marbles. Three marbles are drawn from the bag (without replacement). What is the probability that all three marbles drawn are the same color?
|
\frac{101}{1140}
| 0.916667 |
Find the remainder when $5x^4 - 13x^3 + 3x^2 - x + 15$ is divided by $3x - 9.$
|
93
| 0.583333 |
What is the value of \[\frac{x^3\cdot x^5\cdot x^7\cdots x^{21}}{x^4\cdot x^8\cdot x^{12} \cdots x^{24}}\] if \( x=3 \)?
|
3^{36}
| 0.916667 |
How many 3-digit squares are palindromes, considering only squares of numbers ending in $1$, $4$, or $9$?
|
1
| 0.333333 |
Suppose $f(x)$ and $g(x)$ are functions on $\mathbb{R}$ such that the range of $f$ is $[-4,3]$ and the range of $g$ is $[-3,2]$. Determine the maximum possible value of $b$ if the range of $f(x) \cdot g(x)$ is $[a,b]$.
|
12
| 0.916667 |
Given that \(b\) is a multiple of \(570\), find the greatest common divisor of \(5b^3 + 2b^2 + 5b + 95\) and \(b\).
|
95
| 0.916667 |
In a shooting competition, ten clay targets are arranged in three hanging columns with four targets in the first column, three in the second, and three in the third column. A marksman must break all the targets following these rules:
1) The marksman first chooses a column.
2) The marksman must then break the lowest unbroken target in the chosen column.
Determine the number of different sequences in which all ten targets can be broken.
|
4200
| 0.833333 |
Calculate both the arithmetic and harmonic mean of the reciprocals of the first four prime numbers.
|
\frac{4}{17}
| 0.25 |
What is the value of the expression $\frac{x^3 - 2x^2 - 21x + 36}{x - 6}$ for $x = 3$? Express your answer in simplest form.
|
6
| 0.416667 |
What is the length of the diagonal of a rectangle where the length is $100$ cm and the width is $100\sqrt{2}$ cm? Express your answer in simplest form.
|
100\sqrt{3}
| 0.916667 |
Determine the domain of the function $$g(t) = \frac{1}{(t-2)^3 + (t+2)^3}.$$
|
(-\infty, 0) \cup (0, \infty)
| 0.916667 |
A $6-8-10$ right triangle has vertices that are the centers of three mutually externally tangent circles. Furthermore, a smaller circle is inside the triangle, tangent to all three circles. What is the sum of the areas of the three larger circles?
|
56\pi
| 0.833333 |
Find the minimum value of the expression
\[x^2 + 2xy + 3y^2 + 2xz + 3z^2\]
over all real numbers \(x\), \(y\), and \(z\).
|
0
| 0.916667 |
Add the numbers $254_{9}$, $367_{9}$, and $142_9$. Express your answer in base $9$.
|
774_9
| 0.916667 |
Simplify $\frac{2-i}{1+4i}$, where $i^2 = -1$.
|
-\frac{2}{17} - \frac{9}{17}i
| 0.833333 |
Triangle $ABC$ has vertices $A(1,3)$, $B(3,7)$, and $C(5,3)$. The triangle is first reflected across the $y$-axis to form $A'B'C'$, and then $A'B'C'$ is reflected across the line $y = x - 2$. Find the coordinates of $C''$, the image of $C'$ after both reflections.
|
(5, -7)
| 0.5 |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\beta,$ and the angle between $\mathbf{c}$ and $\mathbf{a} \times \mathbf{b}$ is $\beta.$ If $\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{\sqrt{3}}{4},$ find the smallest possible value of $\beta,$ in degrees.
|
30^\circ
| 0.666667 |
Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^4}{9^4 - 1} + \frac{3^8}{9^8 - 1} + \cdots.$$
|
\frac{1}{2}
| 0.333333 |
The numbers $a,$ $b,$ $c,$ $d$ are equal to -1, 1, 2, 3, in some order. Find the largest possible value of
\[
ab + bc + cd + da.
\]
|
6
| 0.666667 |
Solve the inequality
\[\left| \frac{2x - 1}{x - 1} \right| > 3.\]
|
\left( \frac{4}{5}, 1 \right) \cup (1, 2)
| 0.916667 |
Suppose the probability that a baby born in a certain hospital will speak in the next day is 1/3. What is the probability that at least 3 babies out of a cluster of 7 babies will speak tomorrow?
|
\frac{939}{2187}
| 0.333333 |
In Mrs. Reed's English class, students are currently reading an $840$-page novel. Two friends, Mia and Leo, are sharing the reading workload. Mia reads a page in $60$ seconds, while Leo reads a page in $40$ seconds. They decide that Mia will read from page $1$ to a specific page, and Leo will read from the next page through page $840$. Determine the last page that Mia should read so that both spend equal time reading.
|
336
| 0.583333 |
In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 7$. Find $GH$ to the nearest tenth. You may use a calculator for this problem.
|
8.3
| 0.583333 |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y = |x^2 - 8x + 15|$ and $y = 8 - x$.
|
\frac{7 + \sqrt{21}}{2} + \frac{7 - \sqrt{21}}{2} = 7
| 0.833333 |
Solve for \(x\) in the quadratic equation \(x^2 - 5x + 12 = 2x + 60\) and find the positive difference between the solutions.
|
\sqrt{241}
| 0.166667 |
Evaluate the determinant:
\[
\begin{vmatrix} \cos(\alpha + \frac{\pi}{4})\cos(\beta + \frac{\pi}{4}) & \cos(\alpha + \frac{\pi}{4})\sin(\beta + \frac{\pi}{4}) & -\sin(\alpha + \frac{\pi}{4}) \\ -\sin(\beta + \frac{\pi}{4}) & \cos(\beta + \frac{\pi}{4}) & 0 \\ \sin(\alpha + \frac{\pi}{4})\cos(\beta + \frac{\pi}{4}) & \sin(\alpha + \frac{\pi}{4})\sin(\beta + \frac{\pi}{4}) & \cos(\alpha + \frac{\pi}{4}) \end{vmatrix}
\]
|
1
| 0.916667 |
A rectangular picture frame is made from two-inch-wide pieces of wood. The area of just the frame is $36$ square inches, and one of the outer edges of the frame is $7$ inches long. What is the sum of the lengths of the four interior edges of the frame?
|
10
| 0.916667 |
If \(k\) and \(\ell\) are positive 4-digit integers such that \(\gcd(k,\ell)=5\), what is the smallest possible value for \(\mathop{\text{lcm}}[k,\ell]\)?
|
201,000
| 0.5 |
In the diagram, $ABCD$ is a square piece of paper with a side length of 8 cm. Corner $A$ is folded such that it coincides with $G$, the midpoint of $\overline{BC}$. If $\overline{HF}$ represents the crease created by the fold, with $F$ on $AB$, what is the length of $\overline{FB}$? Describe your solution using common fractions.
|
3 \text{ cm}
| 0.75 |
Eight students participate in an apple eating contest. The graph shows the number of apples eaten by each participating student. Adam ate the most apples and Zoe ate the fewest. Calculate how many more apples Adam ate compared to Zoe and determine the total number of apples eaten by all participants.
[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[10];
for( int i = 0 ; i < 10 ; ++i ){
yaxis[i] = (0,i);
draw(yaxis[i]--yaxis[i]+(17,0));
}
draw((0,0)--(0,9));
draw((17,9)--(17,0));
fill((1,0)--(1,3)--(2,3)--(2,0)--cycle,grey);
fill((3,0)--(3,5)--(4,5)--(4,0)--cycle,grey);
fill((5,0)--(5,8)--(6,8)--(6,0)--cycle,grey);
fill((7,0)--(7,6)--(8,6)--(8,0)--cycle,grey);
fill((9,0)--(9,7)--(10,7)--(10,0)--cycle,grey);
fill((11,0)--(11,4)--(12,4)--(12,0)--cycle,grey);
fill((13,0)--(13,2)--(14,2)--(14,0)--cycle,grey);
fill((15,0)--(15,5)--(16,5)--(16,0)--cycle,grey);
label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("8",yaxis[8],W);
label("9",yaxis[9],W);
label("Students/Participants",(8.5,0),S);
label("Results of an Apple Eating Contest",(8.5,9),N);
label(rotate(90)*"$\#$ of Apples Eaten",(-1,4.5),W);
[/asy]
|
40
| 0.083333 |
The graph of \( y = g(x) \) is depicted, with \( 1 \) unit between grid lines. Assume \( g(x) \) is defined only on the domain shown. Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly \( 4 \) solutions.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-6,6,-7,7);
real g(real x) {return (x-4)*(x-2)*(x+2)*(x+4)/48-2;}
draw(graph(g,-4.5,4.5,operator ..), blue);
[/asy]
|
-3
| 0.083333 |
Consider the sequence starting at $8820$, formed by repeatedly dividing by 5. How many integers are in this sequence?
|
2
| 0.75 |
Let \( a \), \( b \), and \( c \) be solutions of the equation \( x^3 - 6x^2 + 11x = 6 \).
Compute \( \frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b} \).
|
\frac{49}{6}
| 0.833333 |
Let
\[
f(x) = 3x + 4, \quad g(x) = (\sqrt{f(x)} - 3)^2, \quad h(x) = f(g(x))
\]
Determine the value of $h(3)$.
|
70 - 18\sqrt{13}
| 0.916667 |
The sum of the first 1000 terms of a geometric sequence is 100. The sum of the first 2000 terms is 190. Find the sum of the first 3000 terms.
|
271
| 0.583333 |
What is the greatest common factor of 90, 135, and 225?
|
45
| 0.75 |
The polynomial $x^{106} + Cx + D$ is divisible by $x^2 + x + 1$ for some real numbers $C$ and $D$. Find $C + D$.
|
-1
| 0.75 |
In a chess tournament each participant played exactly once against all others. The winner of each game received $1$ point, the loser $0$ points, and each scored $\frac{1}{2}$ point if the game was a draw. At the end of the tournament, it was found that exactly half of the points each player obtained came from playing against the twelve players with the lowest scores. Notably, each of these twelve lowest scoring participants got half of their points from games among themselves. Determine the total number of participants in the tournament.
|
24
| 0.083333 |
The equation $y = -16t^2 + 100t$ describes the height (in feet) of a projectile launched from the ground at 100 feet per second. At what $t$ will the projectile reach 50 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth.
|
0.5
| 0.916667 |
When $\sqrt[4]{2^9 \cdot 5^2}$ is fully simplified, the result is $\alpha\sqrt[4]{\beta}$, where $\alpha$ and $\beta$ are positive integers. What is $\alpha + \beta$?
|
54
| 0.916667 |
Let $Q$ be the plane passing through the origin with normal vector $\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$. Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u}$, $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto plane $Q$.
|
\begin{pmatrix} \frac{5}{7} & \frac{1}{7} & -\frac{3}{7} \\ \frac{1}{7} & \frac{13}{14} & \frac{3}{14} \\ -\frac{3}{7} & \frac{3}{14} & \frac{5}{14} \end{pmatrix}
| 0.916667 |
Suppose that there are two congruent triangles $\triangle ABC$ and $\triangle ACD$ such that $AB = AC = AD.$ If $\angle BAC = 40^\circ,$ determine what $\angle BDC$ is.
|
\angle BDC = 20^\circ
| 0.083333 |
The base $8$ representation of a positive integer is $AC$ and its base $6$ representation is $CA$. What is the integer expressed in base $10$?
|
47
| 0.666667 |
A wizard is preparing a magical elixir. He has four different magical roots and six different mystical minerals to choose from. However, two of the minerals are incompatible with one of the roots, and one mineral is incompatible with two other roots. How many valid combinations can the wizard use to prepare his elixir?
|
20
| 0.833333 |
Let $f(x) = \frac{2x^2 + 4x + 7}{x^2 - 2x + 5}$ and $g(x) = x - 2$. Find $f(g(x)) + g(f(x))$, evaluated when $x = 2$.
|
4
| 0.833333 |
A stock investment increased by 30% in the first year. In the following year, it further increased by 10%. What percent would it need to decrease in the third year to return to its original price at the beginning of the first year?
|
30.07\%
| 0.583333 |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three mutually orthogonal unit vectors, such that
\[\mathbf{a} = p (\mathbf{b} \times \mathbf{c}) + q (\mathbf{c} \times \mathbf{a}) + r (\mathbf{a} \times \mathbf{b})\] for some scalars $p,$ $q,$ and $r,$ and $\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = 1.$ Find $p + q + r.$
|
1
| 0.916667 |
A new license plate in a particular region consists of 4 digits and 3 letters. The digits are not necessarily distinct and the same applies to the letters. The three letters must appear consecutively. How many distinct license plates are possible?
|
878,800,000
| 0.416667 |
In a cross country meet between 2 teams of 6 runners each, a runner who finishes in the $n^{th}$ position contributes $n$ points to his team's score. The team with the lower score wins. If no ties occur among the runners, how many different winning scores are possible?
|
18
| 0.75 |
My friend Sara only likes numbers that are divisible by 8, such as 24, or 8,016. How many different ones digits are possible in numbers that Sara likes?
|
5
| 0.75 |
In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written?
|
19
| 0.916667 |
Suppose the function $g(x)$ is defined on the domain $\{u_1, u_2, u_3\}$, and the graph of $y=g(x)$ consists of just three points. Suppose those three points form a triangle of area $50$ square units.
The graph of $y = 3g\left(\frac{x}{4}\right)$ also consists of just three points. What is the area of the triangle formed by these three points?
|
600
| 0.833333 |
The graph below represents the total distance Lisa drove from 8 a.m. to 1 p.m. How many miles per hour is the car's average speed for the period from 8 a.m. to 1 p.m.?
[asy]
unitsize(0.2inch);
draw((0,0)--(6,0));
draw((0,0)--(0,10));
draw((1,0)--(1,10));
draw((2,0)--(2,10));
draw((3,0)--(3,10));
draw((4,0)--(4,10));
draw((5,0)--(5,10));
draw((0,1)--(6,1));
draw((0,2)--(6,2));
draw((0,4)--(6,4));
draw((0,6)--(6,6));
draw((0,8)--(6,8));
draw((0,10)--(6,10));
draw((0,0)--(1,1)--(2,3)--(3,6)--(4,8)--(5,10));
dot((0,0));
dot((1,1));
dot((2,3));
dot((3,6));
dot((4,8));
dot((5,10));
label("8",(0,-0.5),S);
label("9",(1,-0.5),S);
label("10",(2,-0.5),S);
label("11",(3,-0.5),S);
label("12",(4,-0.5),S);
label("1",(5,-0.5),S);
label("0",(-0.5,0),W);
label("20",(-0.5,2),W);
label("40",(-0.5,4),W);
label("60",(-0.5,6),W);
label("80",(-0.5,8),W);
label("100",(-0.5,10),W);
label("Time of Day (a.m.)",(2.7,-2),S);
label("Total distance",(-0.5,11),N);
[/asy]
|
20
| 0.916667 |
Evaluate $|z^2 + 8z + 65|$ if $z = 7 + 3i$.
|
\sqrt{30277}
| 0.666667 |
Evaluate $|\omega^2 + 4\omega + 40|$ if $\omega = 5 + 3i$.
|
2\sqrt{1885}
| 0.916667 |
Let $Q$ be the point on line segment $\overline{CD}$ such that $CQ:QD = 3:5.$ Assume that $\overline{CD}$ is defined in three-dimensional space with $\overrightarrow{C}$ and $\overrightarrow{D}$ given. Then
\[\overrightarrow{Q} = p \overrightarrow{C} + q \overrightarrow{D}\]
for some constants $p$ and $q.$ Determine the ordered pair $(p, q).$
|
\left(\frac{3}{8}, \frac{5}{8}\right)
| 0.083333 |
How many two-digit numbers have a difference of exactly two between the tens digit and the ones digit?
|
15
| 0.75 |
What is the remainder when $2021 \cdot 2022 \cdot 2023 \cdot 2024 \cdot 2025$ is divided by 23?
|
0
| 0.5 |
The arithmetic mean of seven numbers is 45. If three numbers $x$, $y$, and $z$ are added to the list, the mean of the ten-member list becomes 58. What is the mean of $x$, $y$, and $z$?
|
\frac{265}{3}
| 0.75 |
How many positive integer multiples of $143$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers, and $0 \leq i < j \leq 99$?
|
784
| 0.333333 |
Find the area of triangle $ABC$ given below:
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$1$",(B+C)/2,NE);
label("$45^\circ$",(0,0.75),E);
[/asy]
|
\frac{1}{4}
| 0.75 |
Calculate $\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}^4.$
|
\begin{pmatrix} -8 & 8\sqrt{3} \\ -8\sqrt{3} & -8 \end{pmatrix}
| 0.416667 |
Determine the remainder when \( (x^5 - 1)(x^3 - 1) \) is divided by \(x^2 + x + 1\).
|
0
| 0.916667 |
What is the smallest possible median for the five-number set $\{x, 2x, 9, 7, 4\}$ if $x$ can be any positive integer?
|
4
| 0.666667 |
A stack contains three layers of square blocks, each block having sides of $10 \text{ cm}$. The stack is arranged in a stepped pyramid fashion where the top layer contains one block, the middle layer contains two blocks, and the bottom layer contains three blocks. Find the total height, $h$, of this pyramidal stack of blocks.
|
30 \text{ cm}
| 0.75 |
What is the perimeter, in cm, of quadrilateral $EFGH$ if $\overline{EF} \perp \overline{FG}$, $\overline{GH} \perp \overline{FG}$, $EF=12$ cm, $GH=7$ cm, and $FG=15$ cm?
|
P = 34 + 5\sqrt{10} \text{ cm}
| 0.25 |
Forty slips are placed into a hat, each bearing a number 1 through 8, with each number entered on five slips. Four slips are drawn from the hat at random and without replacement. Let $p'$ be the probability that all four slips bear the same number. Let $q'$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b\ne a$. What is the value of $q'/p'$?
|
70
| 0.583333 |
What is the smallest natural number that can be added to 25,751 to create a palindrome?
|
1
| 0.416667 |
The angles of quadrilateral $PQRS$ satisfy $\angle P = 3\angle Q = 4\angle R = 6\angle S$. What is the degree measure of $\angle P$, rounded to the nearest whole number?
|
206
| 0.75 |
Let $C$ be a point not on line $AE$, and $D$ a point on line $AE$ such that $CD \perp AE$. Also, $B$ is a point on line $CE$ such that $AB \perp CE$. Introduce a point $F$ on line $AE$ such that $BF \perp AE$. If $AB = 5$, $CD = 10$, $BF = 3$, and $AE = 6$, what is the length of $CE$?
|
12
| 0.166667 |
The arithmetic mean, geometric mean, and harmonic mean of $x$, $y$, and $z$ are $10$, $6$, and $4$, respectively. What is the value of $x^2+y^2+z^2$?
|
576
| 0.916667 |
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