problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]
|
200
| 0.666667 |
Find the sum of $312_4$, $31_4$, and $3_4$ in base $4$.
|
1012_4
| 0.666667 |
A girl has the following coins in her purse: $3$ pennies, $2$ nickels, $2$ dimes, and $2$ quarters. She draws two coins at random, notes their combined value, and then returns them to her purse. She repeats this process multiple times. What is the maximum number of different sums she can record?
|
10
| 0.916667 |
Determine the value of $n$ such that $5 \times 16 \times 2 \times n^2 = 8!$.
|
6\sqrt{7}
| 0.666667 |
In the diagram, what is the perimeter of polygon $ABCDE$? [asy]
import olympiad;
size(6cm);
pair a = (0, 8);
pair b = (4, 8);
pair c = (4, 4);
pair d = (0, 0);
pair e = (9, 0);
draw(a--b--c--e--d--cycle);
label("$A$", a, NW);
label("$B$", b, NE);
label("$C$", c, E + NE);
label("$D$", e, SE);
label("$E$", d, SW);
label("$8$", a / 2, W);
label("$4$", a + (b - a) / 2, 2 * N);
label("$9$", e / 2, S);
draw(rightanglemark(a, d, e));
draw(rightanglemark(d, a, b));
draw(rightanglemark(a, b, c));
add(pathticks(a--b, s=6));
add(pathticks(b--c, s=6));
[/asy]
|
25 + \sqrt{41}
| 0.083333 |
A right circular cone is sliced into five pieces by planes parallel to its base, and these pieces have the same height. Calculate the ratio of the volume of the second-largest piece to the volume of the largest piece.
|
\frac{37}{61}
| 0.25 |
If nine people decide to come to a baseball game, but five of them are only 1/3 sure that they will stay for the entire game (while the other four are sure they'll stay the whole time), what is the probability that at least seven people stayed for the entire game?
|
\frac{17}{81}
| 0.5 |
In a math test series, four of the following scores belong to Alex and the other two to Jamie: 75, 80, 85, 90, 92, 97. If Alex's mean score is 85.5, what is Jamie's mean score?
|
88.5
| 0.666667 |
Calculate the sum of the squares of the roots of the equation \[x^{12} + 7x^9 + 3x^3 + 500 = 0.\]
|
0
| 0.916667 |
A number \( n \) has \( 4 \) divisors. How many divisors does \( n^2 \) have?
|
9
| 0.916667 |
A system of inequalities defines a region on a coordinate plane as follows:
$$
\begin{cases}
x+y \leq 5 \\
3x+2y \geq 3 \\
x \geq 1 \\
y \geq 1
\end{cases}
$$
Determine the number of units in the length of the longest side of the quadrilateral formed by the region satisfying all these conditions. Express your answer in simplest radical form.
|
3\sqrt{2}
| 0.916667 |
How many different positive integers can be represented as a difference of two distinct members, where one member is from the set $\{1, 2, 3, \ldots, 9, 10\}$ and the other member is from the set $\{6, 7, 8, \ldots, 14, 15\}$?
|
14
| 0.166667 |
Seven distinct integers are picked at random from $\{1, 2, 3, \ldots, 12\}$. What is the probability that, among those selected, the third smallest is $5$?
|
\frac{35}{132}
| 0.666667 |
In how many ways can 8 distinct beads be placed on a necklace? (Note that two arrangements are the same if one can be rotated or reflected to produce the other.)
|
2520
| 0.916667 |
A bag contains 12 red marbles and 8 blue marbles. Three marbles are selected at random and without replacement. What is the probability that two marbles are red and one is blue? Express your answer as a common fraction.
|
\frac{44}{95}
| 0.916667 |
Calculate 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 |
Find the matrix $\mathbf{M}$ that rotates the elements of a $2 \times 2$ matrix by 90 degrees clockwise. In other words,
\[
\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & a \\ d & b \end{pmatrix}.
\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.
|
\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
| 0.75 |
Let $\mathcal{T}$ be the set $\lbrace 1, 2, 3, \ldots, 12 \rbrace$. Let $m$ be the number of sets consisting of two non-empty disjoint subsets of $\mathcal{T}$. Calculate the remainder when $m$ is divided by $1000$.
|
625
| 0.166667 |
If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 119$, what is the value of $x + y$?
|
20
| 0.666667 |
When a car's brakes are applied, it travels 9 feet less in each second than the previous second until it comes to a complete stop. The car goes 36 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops?
|
90
| 0.916667 |
Let \( a, b, c \) be positive real numbers such that \( a + b + c = 1 \). Find the minimum value of
\[
\frac{1}{a^2 + 2b^2} + \frac{1}{b^2 + 2c^2} + \frac{1}{c^2 + 2a^2}.
\]
|
9
| 0.25 |
Let $\alpha$ and $\beta$ be the roots of $x^2 + px + 4 = 0,$ and let $\gamma$ and $\delta$ be the roots of $x^2 + qx + 4 = 0.$ Express
\[(\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta)\]in terms of $p$ and $q.$
|
-4(p^2 - q^2)
| 0.333333 |
The functions $a(x),$ $b(x),$ and $c(x)$ are all invertible. Express the inverse of the function $g = a \circ c \circ b$ in terms of the inverses of $a,$ $b,$ and $c$.
|
b^{-1} \circ c^{-1} \circ a^{-1}
| 0.916667 |
Simplify $\tan \frac{\pi}{12} - \tan \frac{5\pi}{12}$.
|
-2\sqrt{3}
| 0.666667 |
In how many ways can the digits of the number $75,\!510$ be arranged to form a 5-digit number that does not begin with the digit '0'?
|
48
| 0.916667 |
A coach has a team of $30$ players. He needs to split the players into equal groups of at most $12$ players each. Additionally, one of the groups needs to be exactly half the size of another group. What is the least number of groups the coach needs?
|
3
| 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 |
Let $a$ and $b$ be constants. Suppose that the equation \[\frac{(x+a)(x+b)(x+10)}{(x+4)^2} = 0\] has exactly $3$ distinct roots, while the equation \[\frac{(x+2a)(x+4)(x+5)}{(x+b)(x+10)} = 0\] has exactly $1$ distinct root. Compute $100a + b.$
|
205
| 0.166667 |
Let $S$ be the sum of all positive integers $n$ such that $n^2 + 8n - 1225$ is a perfect square. Find the remainder when $S$ is divided by $100$.
|
58
| 0.25 |
Consider the given functions:
$$\begin{array}{ccc}
f(x) & = & 5x^3 - \frac{1}{x} + 3\\
g(x) & = & x^2 - kx + c
\end{array}$$
If $f(2) - g(2) = 2$, what are the values of $k$ and $c$ assuming $k=1$?
|
c = 38.5
| 0.083333 |
How many different four-digit numbers can be formed using the digits in 2025, given that each digit can only appear as many times as it does in 2025?
|
9
| 0.416667 |
The first three stages of a geometric pattern are shown, where each line segment of toothpick forms a growing square grid. Each stage adds layers to the grid in a pattern such that each successive stage squares the number of lines required for a complete layer. If the pattern continues such that at the first stage, one needs 8 toothpicks, that form a shape of a square, then each new stage requires that each side of the square increases by 4 toothpicks more than the extra toothpicks needed for each side in the previous stage, how many toothpicks are necessary to create the arrangement for the 20th stage?
|
3048
| 0.166667 |
On the game show $\text{\emph{Wheel of Fortune Redux}}$, you see the following spinner. Given that each region is of the same area, what is the probability that you will earn exactly $\$3200$ in your first three spins? The spinner includes the following sections: $"\$2000"$, $"\$300"$, $"\$700"$, $"\$1500"$, $"\$500"$, and "$Bankrupt"$. Express your answer as a common fraction.
|
\frac{1}{36}
| 0.833333 |
Point \(O\) is the center of an ellipse with major axis \(\overline{AB}\) and minor axis \(\overline{CD}\). Point \(F\) is one focus of the ellipse. If \(OF = 8\) and the diameter of the inscribed circle of triangle \(OCF\) is 4, compute the product \((AB)(CD)\).
|
240
| 0.75 |
Let $m = \underbrace{33333}_{\text{5 digits}}$ and $n = \underbrace{666666}_{\text{6 digits}}$.
What is $\gcd(m, n)$?
|
3
| 0.416667 |
Compute $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.
|
20000
| 0.916667 |
Calculate $0.25 \cdot 0.75 \cdot 0.1$.
|
0.01875
| 0.916667 |
The side length of square $C$ is 24 cm. The side length of square $D$ is 54 cm. What is the ratio of the area of square $C$ to the area of square $D$? Express your answer as a common fraction.
|
\frac{16}{81}
| 0.833333 |
Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac{1}{6}$ the measure of $\angle 2$. What is the degree measure of $\angle 5$? Refer to the original geometry setup.
|
\frac{180^\circ}{7}
| 0.75 |
Brand Z juice claims, "We offer 30% more juice than Brand W at a price that is 15% less." What is the ratio of the unit price of Brand Z juice to the unit price of Brand W juice? Express your answer as a common fraction.
|
\frac{17}{26}
| 0.916667 |
Let $x$, $y$, and $z$ be positive real numbers. What is the smallest possible value of $(x+y+z)\left(\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\right)$?
|
\frac{9}{2}
| 0.833333 |
Triangle $DEF$ has a perimeter of 398. The sides have lengths that are all integer values with $DE < EF \leq FD$. What is the smallest possible value of $EF - DE$?
|
1
| 0.5 |
How many zeros are in the expansion of $999,\!999,\!999,\!98^2$?
|
10
| 0.083333 |
A whole number is defined as "7-light" if the remainder when the number is divided by 7 is less than 3. What is the least three-digit "7-light" whole number?
|
100
| 0.833333 |
A function $g$ from the integers to the integers is defined as follows:
\[g(n) = \left\{
\begin{array}{cl}
n - 2 & \text{if $n$ is even}, \\
3n & \text{if $n$ is odd}.
\end{array}
\right.\]Suppose $m$ is even and $g(g(g(m))) = 54.$ Find $m.$
|
60
| 0.666667 |
At a school cafeteria, Noah wants to buy a meal consisting of one main course, one beverage, and one snack. The table below lists Noah's available choices in the cafeteria. Additionally, Noah avoids having soda with pizza. How many distinct possible meals can Noah buy from these options?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Pizza & Soda & Apple \\ \hline
Burger & Juice & Banana \\ \hline
Pasta & & Cookie \\ \hline
\end{tabular}
|
15
| 0.416667 |
Find the equation of the directrix of the parabola $y = 16x^2 + 4.$
|
y = \frac{255}{64}
| 0.916667 |
Evaluate 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 |
Let $g(x) = x^2 + 4x + d$, where $d$ is a real number. For what values of $d$ does $g(g(x))$ have exactly 3 distinct real roots?
|
d = 0
| 0.25 |
Simplify the expression:
\[
\frac{4 + 2i}{4 - 2i} + \frac{4 - 2i}{4 + 2i} + \frac{4i}{4 - 2i} - \frac{4i}{4 + 2i}.
\]
|
\frac{2}{5}
| 0.666667 |
What is the sum of the last two digits of $7^{15} + 13^{15}$?
|
0
| 0.583333 |
The number \( 975 \) can be written as \( 23q + r \) where \( q \) and \( r \) are positive integers. What is the greatest possible value of \( q - r \)?
|
33
| 0.916667 |
For what real value of \(v\) is \(\frac{-15-\sqrt{409}}{12}\) a root of \(6x^2 + 15x + v\)?
|
-\frac{23}{3}
| 0.75 |
A point \((x,y)\) is randomly selected from inside the rectangle with vertices \((0,0)\), \((4,0)\), \((4,3)\), and \((0,3)\). What is the probability that \(x+1 < y\)?
|
\frac{1}{6}
| 0.333333 |
Ninety percent of adults drink coffee and eighty percent drink tea, while seventy percent drink soda. What is the smallest possible percentage of adults who drink both coffee and tea, but not soda?
|
0\%
| 0.25 |
Real numbers $x$ and $y$ satisfy the equations $2^x=16^{y+1}$ and $27^y=3^{x-2}$. Find the product $xy$.
|
8
| 0.916667 |
What is the result when you compute the sum $$2^3 + 4^3 + 6^3 + \ldots + 198^3 + 200^3$$ and $$(-2)^3 + (-4)^3 + (-6)^3 + \ldots + (-198)^3 + (-200)^3, $$ and then add these two results?
|
0
| 0.916667 |
Let $f(x) = |x|$ for $-4\leq x \leq 4$. Determine the range of $f(x)-x$ within this interval.
|
[0, 8]
| 0.916667 |
If $x$, $y$, and $z$ are positive with $xy=30$, $xz = 60$, and $yz=90$, what is the value of $x+y+z$?
|
11\sqrt{5}
| 0.333333 |
If the current time is 3:00 p.m., what time will it be 1567 minutes later?
|
5\!:\!07 \text{ p.m.}
| 0.833333 |
In a classroom, there are 40 students. 18 of them have cool dads, 22 of them have cool moms, and 10 of them have both cool dads and cool moms. Additionally, 8 students have cool siblings. How many students do not have cool parents but have cool siblings?
|
8
| 0.583333 |
A steer initially weighs 300 kilograms. After feeding and care, its weight increases by 15%. With approximately 0.4536 kilograms in a pound, to the nearest whole pound, how much does the steer weigh after the weight increase?
|
761\ \text{pounds}
| 0.166667 |
How many integers $m \neq 0, -1, 1$ satisfy the inequality $\frac{1}{|m|} \geq \frac{1}{6}$?
|
10
| 0.833333 |
Find the integer $n$, $0 \le n \le 10$, such that \[n \equiv 123456 \pmod{11}.\]
|
3
| 0.583333 |
Find the value of $k$ so that the line $4x + 6y + k = 0$ is tangent to the parabola $y^2 = 32x.$
|
72
| 0.833333 |
Given that 12 is the arithmetic mean of the set \(\{8, 15, 24, 9, 12, y\}\), what is the value of \(y\)?
|
4
| 0.833333 |
How many two-digit positive integers are congruent to 2 (mod 4)?
|
23
| 0.833333 |
Find the greatest common divisor of 1975 and 2625.
|
25
| 0.916667 |
Given that $$(x+y+z)(xy+xz+yz)=27$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=9$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?
|
6
| 0.916667 |
Determine the total number of pieces needed to create a seven-row triangle using the pattern where each row's rods increase by three and connectors form a triangle with one extra row than the triangle's rows.
|
120
| 0.166667 |
Emma has a quadratic of the form $x^2 + bx + 88$, where $b$ is a specific positive number. Using her knowledge of how to complete the square, Emma reshapes this quadratic into the form $(x+n)^2 + 16$. Determine the value of $b$.
|
12\sqrt{2}
| 0.916667 |
A triangle is formed with one vertex at the vertex of the parabola \( y = x^2 + 2 \) and the other two vertices at the intersections of the line \( y = r \) and the parabola. If the area of the triangle is between \( 10 \) and \( 50 \) inclusive, find all possible values of \( r \). Express your answer in interval notation.
|
[10^{2/3} + 2, 50^{2/3} + 2]
| 0.083333 |
Let $G$ be the centroid of triangle $XYZ$. If $GX^2 + GY^2 + GZ^2 = 90,$ find the value of $XY^2 + XZ^2 + YZ^2$.
|
270
| 0.916667 |
Consider the circle given by the equation $x^2 - 6y - 4 = -y^2 + 6x + 16$. Determine the center $(a, b)$ and radius $r$ of the circle and find the value of $a+b+r$.
|
6+\sqrt{38}
| 0.666667 |
Let $p,$ $q,$ $r,$ $s$ be nonzero integers such that
\[\begin{pmatrix} p & q \\ r & s \end{pmatrix}^2 = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}.\]Find the smallest possible value of $|p| + |q| + |r| + |s|$.
|
8
| 0.25 |
If 3 cards from a standard deck are selected randomly, what is the probability that either three kings or at least 2 aces occur? (There are 4 Aces, 4 kings, and 52 total cards in a standard deck.)
|
\frac{74}{5525}
| 0.916667 |
Consider the modified Lucas sequence starting from 2, 3, where the first term is 2, the second term is 3 and each term after that is the sum of the previous two terms. What is the remainder when the $105^{\text{th}}$ term of the sequence is divided by 9?
|
8
| 0.416667 |
Find the number of real solutions to the equation
\[\frac{6x}{x^2 + 2x + 5} + \frac{7x}{x^2 - 7x + 5} = -1.\]
|
2
| 0.416667 |
Starting with the number 200, Shaffiq repeatedly halves the number and then takes the greatest integer less than or equal to that number. He stops when the number goes below 3. How many times must he do this?
|
7
| 0.75 |
Find the greatest possible sum of $x + y$ for integer points $(x, y)$ that lie in the first quadrant and satisfy the equation $x^2 + y^2 = 64$.
|
8
| 0.166667 |
A right pyramid has a rectangular base with dimensions 8 cm by 15 cm. Its peak is 10 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges. Round your answer to the nearest whole number.
|
99
| 0.083333 |
Factor the following expression: $74a^2 + 222a + 148a^3$.
|
74a(2a^2 + a + 3)
| 0.333333 |
Sarah rolls 8 fair 6-sided dice. What is the probability that at least two dice show the same number?
|
1
| 0.833333 |
In square $XYZW$, points $P$ and $S$ lie on $\overline{XZ}$ and $\overline{XW}$, respectively, such that $XP=XS=\sqrt{3}$. Points $Q$ and $R$ lie on $\overline{YZ}$ and $\overline{YW}$, respectively, and points $T$ and $U$ lie on $\overline{PS}$ so that $\overline{QT} \perp \overline{PS}$ and $\overline{RU} \perp \overline{PS}$. If triangle $XPS$, quadrilateral $YQTP$, quadrilateral $WSUR$, and pentagon $YRUTQ$ each has an area of $1.5$, find $QT^2$.
[asy]
real x = 3;
real y = 3*sqrt(3)/2;
real z = 3/sqrt(3);
pair X, Y, Z, W, P, Q, R, S, T, U;
X = (0,0);
Y = (4,0);
Z = (4,4);
W = (0,4);
P = (x,0);
Q = (4,y);
R = (y,4);
S = (0,x);
T = Q + z * dir(225);
U = R + z * dir(225);
draw(X--Y--Z--W--X);
draw(P--S);
draw(T--Q^^U--R);
draw(rightanglemark(Q, T, P), linewidth(.5));
draw(rightanglemark(R, U, S), linewidth(.5));
dot("$X$", X, SW);
dot("$Y$", Y, S);
dot("$Z$", Z, N);
dot("$W$", W, NW);
dot("$P$", P, S);
dot("$Q$", Q, E);
dot("$R$", R, N);
dot("$S$", S, W);
dot("$T$", T, SW);
dot("$U$", U, SW);
[/asy]
|
3
| 0.666667 |
Let $\mathbf{c}$ and $\mathbf{d}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{c}} \begin{pmatrix} 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{d}} \begin{pmatrix} 4 \\ 2 \end{pmatrix}.$
|
\begin{pmatrix} 3 \\ 0 \end{pmatrix}
| 0.583333 |
The points $(1,3)$ and $(-4,6)$ are adjacent vertices of a square. What is the area of the square?
|
34
| 0.916667 |
Find the quadratic polynomial $q(x)$ such that $q(-1) = 6,$ $q(2) = 1,$ and $q(4) = 20.$
|
\frac{67}{30}x^2 - \frac{39}{10}x - \frac{2}{15}
| 0.916667 |
Roger has exactly one of each of the first 30 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. A graph shows the number of states that joined the union in each decade. What fraction of Roger's 30 coins represents states that joined the union during the decade 1800 through 1809? Express your answer as a common fraction.
(note: every space represents 2 states.)
[asy]size(200);
label("1780",(6,0),S);
label("1800",(12,0),S);
label("1820",(18,-12),S);
label("1840",(24,0),S);
label("1860",(30,-12),S);
label("1880",(36,0),S);
label("1900",(42,-12),S);
label("1950",(48,0),S);
label("to",(6,-4),S);
label("to",(12,-4),S);
label("to",(18,-16),S);
label("to",(24,-4),S);
label("to",(30,-16),S);
label("to",(36,-4),S);
label("to",(42,-16),S);
label("to",(48,-4),S);
label("1789",(6,-8),S);
label("1809",(12,-8),S);
label("1829",(18,-20),S);
label("1849",(24,-8),S);
label("1869",(30,-20),S);
label("1889",(36,-8),S);
label("1909",(42,-20),S);
label("1959",(48,-8),S);
draw((0,0)--(50,0));
draw((0,2)--(50,2));
draw((0,4)--(50,4));
draw((0,6)--(50,6));
draw((0,8)--(50,8));
draw((0,10)--(50,10));
draw((0,12)--(50,12));
draw((0,14)--(50,14));
draw((0,16)--(50,16));
draw((0,18)--(50,18));
fill((4,0)--(8,0)--(8,12)--(4,12)--cycle,gray(0.8));
fill((10,0)--(14,0)--(14,5)--(10,5)--cycle,gray(0.8));
fill((16,0)--(20,0)--(20,7)--(16,7)--cycle,gray(0.8));
fill((22,0)--(26,0)--(26,6)--(22,6)--cycle,gray(0.8));
fill((28,0)--(32,0)--(32,7)--(28,7)--cycle,gray(0.8));
fill((34,0)--(38,0)--(38,5)--(34,5)--cycle,gray(0.8));
fill((40,0)--(44,0)--(44,4)--(40,4)--cycle,gray(0.8));
[/asy]
|
\frac{1}{6}
| 0.166667 |
Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AL) + \log_{10} (AM) &= 3, \\
\log_{10} (ML) + \log_{10} (MR) &= 4, \\
\log_{10} (RA) + \log_{10} (RL) &= 5.
\end{align*}
Compute the value of the product $ARML.$
|
10,000
| 0.916667 |
Convert the decimal number $93_{10}$ to its base $2$ representation.
|
1011101_2
| 0.25 |
A club consists of 24 members, split evenly with 12 boys and 12 girls. There are also two classes, each containing 6 boys and 6 girls. In how many ways can we choose a president and a vice-president if they must be of the same gender and from different classes?
|
144
| 0.833333 |
Carla rotated point $A$ 750 degrees clockwise about point $B$, and it landed at point $C$. Devon then rotated the original point $A$ $x$ degrees counterclockwise about point $B$ and also landed at point $C$. Afterwards, Devon rotated point $C$ another $45^\circ$ counterclockwise about point $B$. If $x < 360$, what is the new position of $A$ after all rotations, if it does not land back at point $C$?
|
15^\circ \text{ counterclockwise from its original position}
| 0.666667 |
An $\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number itself. Determine how many numbers less than 50 are abundant numbers.
|
9
| 0.75 |
What is the value of the least positive base ten number that requires seven digits for its binary representation?
|
64
| 0.833333 |
What is the largest integer that is a divisor of
\[
(n+3)(n+5)(n+7)(n+9)(n+11)
\]
for all positive even integers \( n \)?
|
15
| 0.5 |
Find the largest negative integer $x$ which satisfies the congruence $50x + 14 \equiv 10 \pmod {24}$.
|
-2
| 0.583333 |
A line \( y = -\frac{2}{3}x + 6 \) crosses the \( x \)-axis at \( P \) and the \( y \)-axis at \( Q \). Point \( T(r,s) \) is on the line segment \( PQ \). If the area of \( \triangle POQ \) is four times the area of \( \triangle TOP \), what is the value of \( r+s \)?
|
8.25
| 0.166667 |
Four circles of radius 2 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.
|
2 + 2\sqrt{2}
| 0.666667 |
Evaluate $\left(\dfrac{5}{3}\right)^6$.
|
\dfrac{15625}{729}
| 0.833333 |
Evaluate $|(7 - 4i)(3 + 10i)|$.
|
\sqrt{7085}
| 0.916667 |
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