Find a nonzero $p$ such that $px^2 - 10x + 2 = 0$ has only one solution.


Each of the numbers $a_1,$ $a_2,$ $\dots,$ $a_{100}$ is $\pm 1.$  Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 100} a_i a_j.\]


A line is parameterized by a parameter $t$, such that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$, and the vector on the line at $t = 4$ is $\begin{pmatrix} 5 \\ -7 \end{pmatrix}$. Find the vector on the line at $t = -2$.

Solution: \boxed{\begin{pmatrix} -1 \\ 17 \end{pmatrix}}


Jackie and Phil each flip a set of three coins: one fair coin, one biased coin that comes up heads with probability $\frac{4}{7}$, and another biased coin with heads probability of $\frac{2}{3}$. They each flip the coins once. Compute the probability that Jackie and Phil get the same number of heads. Let this probability be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $m+n$.


Pentagon ABCDE has a vertical line of symmetry. What is the $y$-coordinate of vertex C so that the area of the pentagon is 50 square units? [asy]
unitsize(2mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), B=(0,5), C=(3,20), D=(6,5), E=(6,0);
pair[] dots={A,B,C,D,E};

draw(B--C--D--E);
dot(dots);

axes(Arrows(4));

label("A(0,0)",A,SW);
label("E(6,0)",E,SE);
label("D(6,5)",D,NE);
label("C",C,NE);
label("B(0,5)",B,NW);
[/asy]


What is the value of $x$ in the equation $\sqrt{\frac{72}{25}} = \sqrt[4]{\frac{x}{25}}$?


Let \(\mathbf{B} = \begin{pmatrix} b & 2 \\ -3 & e \end{pmatrix}\) for some real numbers \(b\) and \(e\). If
\[\mathbf{B} + \mathbf{B}^{-1} = \mathbf{I},\]
where \(\mathbf{I}\) is the identity matrix, then find \(\det \mathbf{B}\).


Given that $$(x+y+z)(xy+xz+yz)=45$$and that $$x^2(y+z) + y^2(x+z) + z^2(x+y) = 15 - xyz,$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


Triangle $PQR$ has vertices $P(0, 10)$, $Q(3, 0)$, and $R(10, 0)$. A line through $Q$ cuts the area of triangle $PQR$ in half. Find the sum of the slope and the $y$-intercept of this line.


What is the greatest integer less than 150 for which the greatest common factor of that integer and 30 is 5?


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 7x^2 + 10x = 12$.

Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.


The updated stem-and-leaf plot shows the duration of rides for each of the $21$ top-rated roller coasters. Each entry in the plot represents ride time, where, for example, $3 \ 05$ means $3$ minutes, $5$ seconds. Convert this time to seconds to find the median of the data set.

\begin{tabular}{c|cccccc}
0&28&28&50&55&&\\
1&00&02&10&&&\\
2&20&25&35&43&45&50\\
3&00&00&00&05&30&36\\
4&00&10&15&20&&\\
\end{tabular}


Find the matrix $\mathbf{M}$ that doubles the first row and triples the second row of a matrix. In other words,
\[
\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2a & 2b \\ 3c & 3d \end{pmatrix}.
\]
If no matrix $\mathbf{M}$ exists that satisfies these conditions, then use the zero matrix.

Solution: \boxed{\mathbf{M} = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}}


Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron with each edge measuring 1 meter. A bug, starting at vertex $A$, follows a rule: at each vertex, it randomly chooses one of the three edges with equal probability and crawls to the opposite vertex. Define $q = \frac m{2187}$ as the probability that the bug is at vertex $A$ after crawling exactly 8 meters. Find the value of $m$.


Given $ab+bc+cd+da = 42$ and $b+d = 5$, find $a+c$.


After Ben takes 30 shots, he has made $60\%$ of his shots. After he takes 10 more shots, he raises his percentage to $62\%$. How many of the last 10 shots did he make?


Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 3 \\ -3 \\ 2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{9}{22} & -\frac{9}{22} & \frac{3}{11} \\ -\frac{9}{22} & \frac{9}{22} & -\frac{3}{11} \\ \frac{3}{11} & -\frac{3}{11} & \frac{2}{11} \end{pmatrix}}


In a basketball league, the number of games won by six teams are displayed in a chart. The team names are not shown, but the following clues are provided:

1. The Hawks won more games than the Wolves.
2. The Rockets won more games than the Knicks, but fewer games than the Lakers.
3. The Knicks won at least 15 games.
4. The Clippers won fewer games than the Lakers.

How many games did the Rockets win if the games won are 15, 20, 25, 30, 35, and 40 games?


The polynomial
\[ax^4 + bx^3 + cx^2 + dx + e = 0\]
has coefficients that are all integers, and has roots $-3,$ $7,$ $11,$ and $-1/4.$ If $e$ is a positive integer, find its smallest possible value.


In right triangle $PQR$ where $\angle R = 90^\circ$, we have $$\frac{5}{4}\cos Q = \sin Q.$$ What is $\cos Q$?


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{6}$, what is the least possible value of $xy$?


Solution: \boxed{x^4 - 2x^2 - 4}


Given that $\begin{vmatrix} x & y \\ z & w \end{vmatrix} = 3$, calculate the determinant of the matrix
\[
\begin{vmatrix} x & 5x + 2y \\ z & 5z + 2w \end{vmatrix}.
\]


Find the largest prime divisor of $35^2 + 84^2$.


$200_{10}$ in base $b$ has exactly $4$ digits. What is the value of $b$?


Find the sum of the distinct prime factors of $7^7 - 7^4$.


Simplify $18 \cdot \frac{8}{15} \cdot \frac{2}{27}$.


A flagpole is braced by a wire that extends from the top of the pole to a point on the ground 4 meters from its base. When Tom stands 3 meters from the base of the pole towards the point where the wire is attached to the ground, his head just touches the wire. Tom is 1.8 meters tall. How tall is the flagpole?

Solution: \boxed{7.2\text{ meters}}


Let $\mathbf{B} = \begin{pmatrix} p & 3 \\ -4 & q \end{pmatrix}$ for some real numbers $p$ and $q.$  If
\[\mathbf{B} + 2\mathbf{B}^{-1} = \mathbf{0},\] then find $\det \mathbf{B}.$


Let \[f(x) = \left\{
\begin{array}{cl}
-x^2 - 4x - 4 & \text{if } x \le 2, \\
\frac{x^2}{4} - 1 & \text{if } x > 2.
\end{array}
\right.\]Find the sum of all values of $x$ such that $f(x) = 0$.


A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a heart, a spade, or a king?


Bertha has 8 daughters and no sons. Some of her daughters have 4 daughters, and the rest have none. Bertha has a total of 36 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


Factor $z^6 - 64$ as far as possible, where the factors are monic polynomials with real coefficients.


A 9x9 chessboard has its squares labeled such that the label of the square in the ith row and jth column is given by $\frac{1}{2 \times (i + j - 1)}$. We need to select one square from each row and each column. Find the minimum sum of the labels of the nine chosen squares.


The length of the segment between the points $(3b, b-5)$ and $(5, -2)$ is $3\sqrt{5}$ units. What is the product of all possible values for $b$?


Imagine a triangular array of squares, where there is one square in the first row, two in the second, continuing up to 10 squares in the tenth row. Except for the bottom row, each square rests on two squares immediately below it. We start by placing either a $0$ or $2$ in each square of the tenth row. The number in each subsequent square is the sum of the entries in the two squares directly below it. Determine how many initial distributions of $0$'s and $2$'s in the tenth row will make the number in the top square a multiple of $4$.


Solve the inequality
\[\frac{x - 5}{(x - 3)^2} < 0.\]
Enter your answer using interval notation.

Solution: \boxed{(-\infty, 3) \cup (3, 5)}


Let $\alpha$ and $\beta$ be real numbers. Find the minimum value of:
\[(3 \cos \alpha + 4 \sin \beta - 7)^2 + (3 \sin \alpha + 4 \cos \beta - 12)^2.\]

Solution: \boxed{242 - 14\sqrt{193}}


There are 18 ordered pairs of integers $(x, y)$ that satisfy the equation $(x-1)^2 + (y+2)^2 = 36$. What is the greatest possible sum $x+y$?


Define a function $g(x)$ such that $g(2) = 4$ and
\[g(x + y) = 5^y g(x) + 3^x g(y)\] for all real numbers $x$ and $y$. Find the function $g(x)$.

Solution: \boxed{\frac{5^x - 3^x}{4}}


The graphs of the function $f(x) = 4x + d$ and its inverse function $f^{-1}(x)$ intersect at the point $(7,m)$. Given that $d$ and $m$ are both integers, what is the value of $m$?


Evaluate the polynomial \[ x^3 - 3x^2 - 9x + 5, \] where \( x \) is the positive number such that \( x^2 - 3x - 9 = 0 \).


Moore's law is adapted for a specialized chip technology where the number of transistors doubles every three years. In 1992, a cutting-edge graphics processing unit (GPU) contained about 500,000 transistors. According to this adapted Moore's law, how many transistors would this GPU contain in the year 2011?


Let $a$, $b$, $c$, and $d$ be positive real numbers. What is the smallest possible value of $(a+b+c+d)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{a+d}+\frac{1}{b+c}+\frac{1}{b+d}+\frac{1}{c+d}\right)$?


In the diagram, $\triangle ABE$, $\triangle BCE$, and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=36$. [asy]
pair A, B, C, D, E;
A=(0,31.176);
B=(0,0);
C=(13.5,-7.794);
D=(20.25,-3.897);
E=(18,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B", B, W);
label("C", C, SW);
label("D", D, dir(0));
label("E", E, NE);
[/asy] Find the perimeter of quadrilateral $ABCD.$

Solution: \boxed{31.5\sqrt{3} + 40.5}


Convert \( \sqrt{2} e^{13\pi i/4} \) to rectangular form.

Solution: \boxed{-1 - i}


The polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
has coefficients that are all integers and has roots $-2$, $5$, $9$, and $-\frac{1}{2}$. If $e$ is a positive integer, then find its smallest possible value.


The graph of \( y = h(x) \) is derived from \( y = f(x) \) by reflecting it across the y-axis and then shifting it 3 units to the right. Provide the expression for \( y = h(x) \) in terms of \( f(x) \).

Solution: \boxed{h(x) = f(3 - x)}


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 160^\circ & -\sin 160^\circ \\ \sin 160^\circ & \cos 160^\circ \end{pmatrix}^n = \mathbf{I}.\]


A list of five positive integers has all of the following properties:

- The only integer in the list that occurs more than once is $7$,
- its median is $10$, and
- its average (mean) is $12$.

What is the largest possible integer that could appear in the list?


The equation $x^2 - 6x + 15 = 27$ has two solutions, $c$ and $d$, with $c \geq d$. What is the value of $3c + 2d$?

Solution: \boxed{15 + \sqrt{21}}


What is the value of \[\frac{x^1\cdot x^2\cdot x^3\cdots x^{20}}{x^2\cdot x^4 \cdot x^6 \cdots x^{30}}\]if $x=3$?


Let $\mathbf{a}$ and $\mathbf{b}$ be orthogonal vectors. If $\operatorname{proj}_{\mathbf{a}} \begin{pmatrix} 4 \\ -2 \end{pmatrix} = \begin{pmatrix} -\frac{4}{5} \\ -\frac{8}{5} \end{pmatrix},$ then find $\operatorname{proj}_{\mathbf{b}} \begin{pmatrix} 4 \\ -2 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{24}{5} \\ -\frac{2}{5} \end{pmatrix}}


Seven couples are at a conference. If each person shakes hands exactly once with everyone else except their spouse and the last person they shook hands with, how many handshakes were exchanged?


Let \[g(x) =
\begin{cases}
7x+3 &\text{if }x\text{ is an integer}, \\
\lfloor{x}\rfloor+3x &\text{if }x\text{ is not an integer}.
\end{cases}
\]
Find $g(\sqrt{50})$.

Solution: \boxed{7 + 15\sqrt{2}}


Let $Q$ be the point on line segment $\overline{CD}$ such that $CQ:QD = 3:5$. Then
\[\overrightarrow{Q} = r \overrightarrow{C} + s \overrightarrow{D}\] for some constants $r$ and $s$. Find the ordered pair $(r, s)$.

Solution: \boxed{\left(\frac{5}{8}, \frac{3}{8}\right)}


Consider a \(9 \times 9\) checkerboard. The board will have multiple horizontal and vertical lines. Calculate the total number of rectangles that can be formed using these lines, and determine the number of these rectangles that are squares. The problem asks for the simplification of the ratio of the number of squares to the number of rectangles, given as a fraction \(m/n\) where m and n are relatively prime, then, find \(m + n\).


On Tuesday, I worked $2t-7$ hours and earned $3t-4$ dollars per hour. My colleague Bob worked $3t-9$ hours but only earned $2t-6$ dollars an hour. We discovered that we both earned the same amount of money. What is the value of $t$?


A basketball court has a rectangular painting area where the width $x$ of the rectangle must satisfy the equation $16x^2 + 32x - 40 = 0$. The coach wants to rewrite this equation in the form $(ax + b)^2 = c$, where $a$, $b$, and $c$ are integers and $a > 0$. Find the value of $a + b + c$.


Juan rolls a fair regular decagonal die marked with numbers from 1 to 10. Then Amal rolls a fair eight-sided die marked with numbers from 1 to 8. What is the probability that the product of the two rolls is a multiple of 4?


Given that the point $(3, -2)$ is on the graph of $y = f(x)$, determine a point that must be on the graph of $4y = 2f(2x) - 6$. What is the sum of the coordinates of that point?


The numbers $a$ and $b$ are inversely proportional. When the sum of $a$ and $b$ is 60, $a$ is three times $b$. What is the value of $b$ when $a = -12$?


What is the only integer whose square is less than three times itself?

Solution: \boxed{1 \text{ and } 2}


Let $a^2=\frac{9}{14}$ and $b^2=\frac{(3+\sqrt{7})^2}{14}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the form $\frac{x\sqrt{y}}{z}$, where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$?


When the base-16 number $ABCDEF_{16}$ is written in base 2, how many base-2 digits (bits) does it have?


Find the smallest value of \( k \) such that \( 3 \) is in the range of \( g(x) = x^2 - 7x + k \).


There are 50 students in a science class. 25 of them have cool dads, 30 of them have cool moms, and 15 of them have both cool dads and cool moms. Additionally, 10 students have cool siblings. How many students have parents and siblings who are all uncool?


The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$?


Find the $3 \times 3$ matrix $\mathbf{P}$ such that for a $3 \times 3$ matrix $\mathbf{Q},$ $\mathbf{P} \mathbf{Q}$ results in the swapping the second row and third row of $\mathbf{Q},$ and tripling the first row of $\mathbf{Q}.$ In other words,
\[
\mathbf{P} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} 3a & 3b & 3c \\ g & h & i \\ d & e & f \end{pmatrix}.
\]

Solution: \boxed{\begin{pmatrix} 3 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}}


Find the value of $k$ if the roots of the quadratic $5x^2 + 7x + k$ are $\frac{-7 \pm i\sqrt{171}}{10}$.


In a promotional event for a new sports league, ten players from four different teams are to be seated in a row for a photo session. The teams consist of three from the Dragons, two from the Lions, three from the Unicorns, and two from the Griffins. If players from the same team insist on sitting together, how many different ways can these ten players be seated?


Calculate the sum of the smaller angles formed by the hour and minute hands of a clock at both 9:20 and 2:50. Express your answer as a decimal to the nearest tenth.


Sasha has $\$2.10$ in U.S. coins. She has the same number of quarters and dimes. What is the greatest number of quarters she could have?


In right triangle $ABC$, we have $\angle A = 40^\circ$, $\angle B = 90^\circ$, and $BC = 12$. Find $AB$ to the nearest tenth. You may use a calculator for this problem.


Consider a hexagonal bipyramid, which is formed by two hexagonal pyramids glued together at their bases. An ant starts at the top vertex and randomly walks to one of the six adjacent vertices in the middle ring, naming it vertex A. From vertex A, the ant then proceeds to randomly walk to one of the six adjacent vertices, naming it vertex B. Calculate the probability that vertex B will be the bottom vertex of the hexagonal bipyramid. Express your answer as a common fraction.


In a pet shelter with 150 dogs, 30 dogs like watermelon, 70 dogs like salmon, and 15 dogs like chicken. 10 dogs like both watermelon and salmon, 7 like both salmon and chicken, 5 like both watermelon and chicken, and 3 like all three. How many dogs in the shelter will not eat any of these foods?


A buoy was carefully placed in a river and tethered to the riverbed. When the river froze and the buoy was removed, it left a hole 30 cm across at the surface and 10 cm deep. What is the radius of the buoy (in centimeters)?


The set of vectors \(\mathbf{v}\) such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ -1 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} \frac{3}{2} \\ -\frac{1}{2} \end{pmatrix}\] lie on a line. Enter the equation of this line in the form "$y = mx + b$".

Solution: \boxed{y = 3x - 5}


Jorge sells four hotdogs and five burgers for $\$3.75$ and he sells five hotdogs and three burgers for $\$3.45$. Assuming a fixed price per item, what is the cost, in dollars, of six hotdogs and seven burgers? Express your answer as a decimal to the nearest hundredth.


In a coordinate plane, consider a larger circle with radius $R$ and a smaller circle with radius $r$, where $R > r$. Both circles are concentric at the origin $(0,0)$. Let $\overline{OA}$ be the radius of the larger circle extending to point $A=(R,0)$. Draw a line from $A$ to a point $B$ on the $y-axis$ such that $B=(0,y)$ and line segment $\overline{AB}$ is perpendicular to the $y-axis$. Let the length of $AB$ equal $s$. Also assume that $R^2 - r^2 = s^2$. Find the area of the annulus formed between these two circles.

Solution: \boxed{\pi s^2}


Points $A$, $B$, $C$, $D$, and $E$ are located on a line segment $\overline{AB}$ such that $AB = 4AD = 8BE = 2BC$. What is the probability that a point randomly chosen on $\overline{AB}$ lies between $B$ and $E$?

[asy]
draw((0,0)--(16,0.000000001));
dot((0,0)); dot((4,0)); dot((8,0)); dot((12,0)); dot((16,0));
label("$A$", (0,0), S);
label("$D$", (4,0), S);
label("$E$", (8,0), S);
label("$B$", (12,0), S);
label("$C$", (16,0), S);
[/asy]


A circle is tangent to the lines $3x - 4y = 12$ and $3x - 4y = -48$. The center of the circle lies on the line $x - 2y = 0$. Find the center of the circle.

Solution: \boxed{(-18, -9)}


In a race, each participant starts at point $A$, must touch any part of the 1200-meter wall, and finish at point $B$. The vertical distance from $A$ to the bottom of the wall is 300 meters, and from $B$ to the top of the wall is 600 meters. Calculate the minimum distance a runner must cover, rounding off to the nearest meter. Assume a flat terrain.
[asy]
import olympiad; import geometry; size(250);
defaultpen(linewidth(0.8));
draw((0,3)--origin--(12,0)--(12,6));
label("300 m",(0,3)--origin,W); label("1200 m",(0,0)--(12,0),S); label("600 m",(12,0)--(12,6),E);
draw((0,3)--(6,0)--(12,6),linetype("3 3")+linewidth(0.7));
label("$A$",(0,3),N); label("$B$",(12,6),N);
[/asy]


The expression $512x^3 + 27$ can be written as $(ax+b)(cx^2+dx+e)$. Find $a + b + c + d + e$.


Let $x$ and $y$ be real numbers. Consider the following five statements:

1. $\frac{1}{x} > \frac{1}{y}$
2. $x^2 < y^2$
3. $x > y$
4. $x > 0$
5. $y > 0$

What is the maximum number of these statements that can be true for any values of $x$ and $y$?


The table below is partially filled in, representing results from a survey conducted by the online streaming service StreamNow. Determine the number of non-binary individuals who listen to this service.

\begin{tabular}{| l | c | r |}
\hline
& listen & don't listen \\
\hline
males & 85 & \\ \hline
females & & 95 \\ \hline
non-binary & & 45 \\ \hline
total & 250 & 230 \\
\hline
\end{tabular}


Two real numbers $x$ and $y$ satisfy $x - y = 3$ and $x^3 - y^3 = 27$. Compute $xy$.

Solution: \boxed{xy = 0}


Find the integer $m$, $-180 \le m \le 180$, such that $\cos m^\circ = \sin 318^\circ$.


A running track is formed by two concentric circles. If the difference in the circumferences of the two circles is $24\pi$ feet, determine both the width of the track and the area enclosed between these two circles.


Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit, and the third outcome is the units digit, what is the probability that you will end up with a three-digit number that is divisible by 5? Express your answer as a common fraction.

[asy]
draw(Circle((0,0),10));
draw((0,0)--(8.7,-5));
draw((0,0)--(-8.7,-5));
draw((0,0)--(0,10));
label("1",(7,7),SW);
label("3",(-7,7),SE);
label("5",(0,-2),S);
draw((0,0)--(3,5),Arrow);
[/asy]


At a national conference, 30 participants are assigned rooms numbered from 1 to 30 in a hotel. Each participant has his/her own room. If the participants allocated rooms 16, 17, and 18 did not show up, what is the median room number of the remaining 27 participants?


Add $96.23$ to $47.849$ and round your answer to the nearest tenth.


Jessica drives to a nearby town to visit her cousin. Her car's odometer reads 32,430 miles when she starts, and 32,464 miles when she arrives. Her car consumes fuel at a rate of 32 miles per gallon, and the current price of gasoline is $\$3.85$ per gallon. Calculate the cost of gasoline for Jessica's trip.


Eight distinct points, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, and $P_8$, are evenly spaced around a circle. If four points are chosen at random from these eight points to form two chords, what is the probability that the chord formed by the first two points chosen intersects the chord formed by the last two points chosen?


$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 5$ and $AE = 2$. Compute $(1 - \cos \angle B)(1 - \cos \angle ACE).$


Given that $\begin{vmatrix} p & q \\ r & s \end{vmatrix} = 3,$ find the determinant of the matrix $\begin{vmatrix} p & 5p + 4q \\ r & 5r + 4s \end{vmatrix}.$


Determine the number of non-zero digits to the right of the decimal point when the fraction $\frac{90}{3^2 \cdot 2^5}$ is expressed as a decimal.


Completely factor the following expression: \[(20x^3 + 45x^2 - 10) - (-5x^3 + 15x^2 - 5).\]

Solution: \boxed{5(5x^3 + 6x^2 - 1)}


Regular decagon \( Q_1 Q_2 \dotsb Q_{10} \) is drawn in the coordinate plane with \( Q_1 \) at \((2,0)\) and \( Q_6 \) at \((4,0)\). If \( Q_n \) is the point \((x_n, y_n)\), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).
\]


What is the fourth power of the cube of the third smallest prime number?


Triangle $ABC$ has $AB=15$, $AC=17$, and $BC=8$. Points $F$ and $G$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{FG}$ is parallel to $\overline{BC}$ and contains the centroid of triangle $ABC$. Compute the length of $\overline{FG}$.


Determine an expression for the area of $\triangle QCA$ in terms of $p$ and $x$. Given points $Q(0, 12)$, $A(x, 12)$, and $C(0, p)$ where $x > 0$ and $p < 12$. Ensure your answer is simplified.


Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smallest number divides both other numbers? Express your answer as a common fraction.


In $\triangle ABC,$ $AB = AC = 17$ and $BC = 16.$ Points $D, E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?


Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1.$ Compute
\[
\omega^{10} + \omega^{12} + \omega^{14} + \dots + \omega^{30}.
\]


In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 13$, and $AC = 5$. What is $\tan B$?


Find the difference between $1234_7$ and $652_7$ in base $7$.


Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $1729$, what is the sum of the numbers written on the faces?


A bakery sells five types of bread. How many different combinations of breads could Lucy purchase if she buys a total of eight breads, ensuring that she buys at least two of each of two specific types (let's say type A and B)?


At a regional football match, the local team provided the following incentives to its supporters:
- Every 25th supporter who entered the stadium received a free t-shirt.
- Every 40th supporter got a free hat.
- Every 90th supporter received a free scarf.
The stadium has a seating capacity of 5000 and it was fully occupied at this match. How many of the supporters received all three items?


What is the sum of the first 20 odd integers greater than 10?


We are allowed to remove exactly one integer from the list $$-2,-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,12$$, and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?


A right cylinder with a base radius of 4 units and a height of 10 units is inscribed in a sphere of radius 6 units. A right cone with the same base radius as the cylinder and height equal to half the cylinder's height is attached to one base of the cylinder. Calculate the total volume, in cubic units, of the space inside the sphere and outside the combined cylinder and cone. Give your answer in the form of $X\pi$, where $X$ is a common fraction.


Solve the equation $|y-4| + 3y^2 = 16$ for $y$.

Solution: \boxed{y = \frac{-1 + \sqrt{241}}{6}, y = \frac{1 - \sqrt{145}}{6}}


In a workshop, there are 24 participants, and each participant must pair up with another for a team activity. If the pairs are formed randomly, what is the probability that John pairs up with his coworker, Alice?


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. Both the square and the circle have equal area. What is \( \frac{a}{b} \)?


Below is a portion of the graph of a function, $y = v(x)$:

[asy]
import graph; size(5.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4,xmax=4,ymin=-4,ymax=4;

pen cqcqcq=rgb(0.75,0.75,0.75);

/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return x+2*cos(x*pi/2);}
draw(graph(f2,-4,4),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]

What is the exact value of $v(-3.14) + v(-1.57) + v(1.57) + v(3.14)$?


A large hall is setting up seating for a conference. They initially set up $144$ chairs, arranged in rows such that each row contains exactly $12$ chairs. The organizers expect $100$ participants and want to ensure nearly full occupancy with minimal empty seats while maintaining complete rows. How many chairs should be removed to meet this requirement?


Let $T$ be the set of integers between $1$ and $2^{50}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $T,$ the probability that it is divisible by $11$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$


Let $x$ and $y$ be positive real numbers such that
\[\frac{1}{x + 3} + \frac{1}{y + 3} = \frac{1}{4}.\]
Find the minimum value of $x + 3y.$

Solution: \boxed{4 + 8 \sqrt{3}}


If $q(x) = x^4 - 4x + 3$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^3$.


For how many integer values of $n$ between 1 and 499 inclusive does the decimal representation of $\frac{n}{500}$ terminate?


What is the greatest 3-digit base 7 positive integer that is divisible by 7? (Express your answer in base 7.)


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 10 and 25 respectively. 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 the trapezoid $ABCD$? Express your answer as a common fraction.


The seven digits in Jane's phone number and the four digits in her house number have the same sum. The four digits in her house number are distinct, and her phone number is 745-3281. What is the largest possible value of Jane's house number?


Evaluate $\lfloor -6 -.5 \rfloor \cdot \lceil 6 +.5 \rceil \cdot \lfloor -5 -.5 \rfloor \cdot \lceil 5 +.5 \rceil \cdot \dots \cdot \lfloor -.5 \rfloor \cdot \lceil .5 \rceil$.


What is the units digit of the sum of the eleven terms of the sequence $1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 10! + 10, \, 11! + 11$?


What is the sum of the fifth and sixth elements in Row 20 of Pascal's triangle?


Solve the congruence \(8n \equiv 5 \pmod{29}\) as a residue modulo 29. (Give an answer between 0 and 28.)


Two circles, one with radius 4 and the other with radius 5, are externally tangent to each other and are circumscribed by a third circle. Calculate the area of the shaded region formed between these three circles. Express your answer in terms of $\pi$.


A cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth.


When a biased coin is flipped four times, the probability of getting heads exactly once equals the probability of getting heads exactly twice. Find the probability, in simplest fraction form, that the coin comes up heads in exactly 3 out of 4 flips.


Find the smallest positive integer $b$ for which $x^2 + bx + 1764$ factors into a product of two polynomials, each having integer coefficients.


A bicycle travels due north at $\frac{1}{2}$ mile per minute on a straight path. Simultaneously, a circular fog bank with a radius of $40$ miles moves northeast at $\frac{1}{3}\sqrt{2}$ mile per minute. At time $t=0$, the center of the fog is $100$ miles due east of the bicycle. Determine $\frac{1}{2}(t_1+t_2)$, where $t_1$ and $t_2$ are the times when the bicycle enters and leaves the fog bank, respectively.


Two parallelograms have a fixed side length of 36 cm and a perimeter of 144 cm. What is the greatest possible difference between the areas of these two parallelograms?


Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{2}$ if $b_{n-1}$ is even and $b_n=3b_{n-1}+1$ if $b_{n-1}$ is odd. For how many positive integers $b_1 \le 1000$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$?


In right triangle $ABC$, we have $\angle A = 30^\circ$, $\angle B = 90^\circ$, and $BC = 12$. Find $AB$ to the nearest tenth.


For $-1 < r < 1$, let $T(r)$ denote the sum of the geometric series \[18 + 18r + 18r^2 + 18r^3 + \cdots .\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b) = 2916$. Find $T(b) + T(-b)$.


There are 25 people in a club. 10 of them are left-handed. 18 of them like rock music. 3 of them are right-handed and do not like rock music. How many club members are left-handed and like rock music? Assume people are either left-handed or right-handed, but not both.


Four tiles are marked X and three other tiles are marked O. The seven tiles are randomly arranged in a row. What is the probability that the arrangement starts with XOX and ends with OXO?


In a recent basketball game, Jason attempted only three-point shots, two-point shots, and free throws. He was successful on $25\%$ of his three-point shots, $40\%$ of his two-point shots, and $80\%$ of his free throws. Jason attempted a total of $40$ shots, of which $10$ were free throws. How many points did he score?


Henry's Hamburger Heaven has expanded its menu to include an additional condiment option: jalapeños, and also now allows customers to choose between two types of bread: white or whole wheat. The restaurant still offers the original eight condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, three, or four meat patties, and any combination of condiments. How many different kinds of hamburgers can be ordered now?


Each pack of hot dogs contains 12 hot dogs (no buns), but each pack of hot dog buns contains 10 buns. After a barbecue, where an equal number of hot dogs and buns were used, Joel finds that he has 6 hot dogs left over, but all buns were used. What is the SECOND smallest number of packs of hot dogs he could have bought?


For how many integers $n$ between 1 and 14 (inclusive) is $\frac{n}{18}$ a repeating decimal?


An ellipse has its foci at $(1, 1)$ and $(1, 3)$. Given that it passes through the point $(6, 2)$, its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$


A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction.


Each of the numbers \( b_1, b_2, \dots, b_{83} \) is \( \pm 1 \). Find the smallest possible positive value of
\[
\sum_{1 \le i < j \le 83} b_i b_j.
\]


Without using a calculator, find the largest prime factor of \(17^4 + 2 \times 17^2 + 1 - 16^4\).


In a math quiz, eight students participated and their scores are shown on a bar graph. Emily scored the highest and Tim scored the lowest. Determine how many more points Emily scored than Tim.

[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[11];
for( int i = 0 ; i < 11 ; ++i ){

yaxis[i] = (0,i);

draw(yaxis[i]--yaxis[i]+(17,0));
}
draw((0,0)--(0,10));
draw((17,10)--(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)--(5,9)--(6,9)--(6,0)--cycle,grey);
fill((7,0)--(7,7)--(8,7)--(8,0)--cycle,grey);
fill((9,0)--(9,6)--(10,6)--(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)--(15,8)--(16,8)--(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("10",yaxis[10],W);
label("Students/Participants",(8.5,0),S);
label("Results of a Math Quiz",(8.5,10),N);
label(rotate(90)*"$\#$ of Points Scored",(-1,5),W);
[/asy]


The sum of three different numbers is 77. The two larger numbers differ by 9 and the two smaller numbers differ by 5. What is the value of the largest number?


Determine the product of the roots of the equation \[(3x^3 + 2x^2 - 8x + 20)(4x^4 - 25x^2 + 19) = 0.\]


What is $(11111111_2 + 11111_2)$ when first converted to base 8 and then to base 10?


Suppose 1988 points are evenly distributed on a circle. Four distinct points, $P$, $Q$, $R$, and $S$, are to be selected from these 1988 points, with all quadruples being equally likely to be chosen. Compute the probability that the chord $\overline{PQ}$ intersects the chord $\overline{RS}$.


A can is in the shape of a right circular cylinder. The circumference of the base of the can is 15 inches, and the height of the can is 9 inches. A spiral strip is painted on the can such that it wraps around the can exactly once, starting from the bottom and reaching the top directly above its starting point. Find the length in inches of the stripe.


A car travels 180 miles from $C$ to $D$ at 90 miles per hour, stops at $D$ for 1 hour, and then returns to $C$ on the same road at a different constant speed. If the average rate of the round trip, excluding the stop, is 60 miles per hour, what is the rate, in miles per hour, of the car traveling back from $D$ to $C$?


Suppose we roll a standard fair 12-sided die. What is the probability that a perfect square less than 10 is rolled?


A right cylinder with a height of 8 inches has a radius of 3 inches. A cone with the same radius but a height of 5 inches shares the base with the cylinder. What is the total surface area exposed to the outside (including the cylinder's curved surface, the cone's curved surface, and the base of the cone)?

Solution: \boxed{57\pi + 3\pi \sqrt{34}}


Six partners in a technology firm decide to split the profits of their company in the ratio 1:2:2:3:4:5. If the profit one year is \$51,000, what is the largest number of dollars received by any of the four partners?


Solve for $x$ in the following equation: $5 + 2ix = 3 - 4ix$.


Find the length of the diagonal $DA$ of a regular hexagon where each side has a length of 6 units. After finding the length, calculate the perimeter of the hexagon. Express your answer in simplest radical form.

Solution: \boxed{36 \text{ units}}


A repair crew uses two planks to reach the center $C$ of a circular pool. The first plank is 20 feet long, extending from $A$ to $B$, and the second is 12 feet long from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. Determine the area of the circular pool. Express your answer in terms of $\pi$.

Solution: \boxed{244 \pi \text{ square feet}}


The letters of the alphabet are reassigned numeric values based on a modified pattern, using values $-3$, $-2$, $-1$, $0$, $1$, $2$, and $3$. Starting from A through Z, the new value assignment pattern is as follows:
$$
-3, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2
$$
Calculate the sum of the numeric values of the letters in the word "marvel".


The six faces of a four-inch wooden cube are each painted red. The cube is then cut into one-inch cubes. How many of the one-inch cubes have red paint on at least two faces?


In how many ways can 6 people be seated around a round table, if two seatings are considered the same if one is a rotation of the other, and one person always wants to sit next to another specific person?


Using the digits 2, 3, 3, 6, 0, and 5, when randomly arranged, what is the probability that the resulting six-digit positive integer is divisible by 30? Express your answer as a common fraction.

Solution: \boxed{\frac{0}{720} = 0}


Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. 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 the quadratic polynomial
\[ \text{Area}(WXYZ) = \gamma \omega - \delta \omega^2. \]
Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


At a mixed doubles badminton tournament, there were four teams. Two teams consist of two players each, and two teams consist of three players each. After the tournament, each player shook hands only once with each of the other players except their own teammates. How many handshakes occurred?


Triangle $PQR$ has a perimeter of 2021 units. The sides have lengths that are all integer values with $PQ < QR \leq PR$. What is the smallest possible value of $QR - PQ$?


In the diagram, triangles $ABC$ and $CDE$ are isosceles. The perimeter of $\triangle CDE$ is $25$, the perimeter of $\triangle ABC$ is $24$, and the length of $CE$ is $9$. Additionally, points $A$, $B$, $C$, $D$, and $E$ form a pentagon $ABCDE$ with $AB = DE$. What is the length of $AB$?


$PQRS$ is a trapezoid where base $\overline{PQ}$ is three times the length of base $\overline{RS}$. The diagonals intersect at point $T$. The length of diagonal $\overline{PR}$ is 15. Find the length of segment $\overline{RT}$. Express your answer as a common fraction.


How many real numbers $x$ are solutions to the following equation?
\[ |x-2| = |x-4| + |x-6| + 1 \]


A company is creating a series of unique employee IDs. Each ID consists of five characters chosen from the three letters in CAT and the digits in 2021. No character may appear in an ID more times than it appears in CAT or 2021. A set of IDs in which each possible sequence appears exactly once contains $M$ IDs. Calculate $\frac{M}{10}$.


Simplify $\displaystyle\frac{2-i}{1+4i}$, where $i^2 = -1.$

Solution: \boxed{-\frac{2}{17} - \frac{9}{17}i}


My friend and I planned to eat at a local diner. I didn't realize earlier that she's vegan and allergic to nuts. The diner offers six dishes that do not contain animal products, which is just one-sixth of the entire menu. From those vegan dishes, five include nuts. What fraction of the menu items at this diner can my friend eat?


Let $0 \le x, y, z, w \le 1$. Find the possible values of the expression
\[
\sqrt{x^2 + (1 - y)^2} + \sqrt{y^2 + (1 - z)^2} + \sqrt{z^2 + (1 - w)^2} + \sqrt{w^2 + (1 - x)^2}.
\]

Solution: \boxed{[2 \sqrt{2}, 4]}


Carla rotated point $A$ $480^\circ$ clockwise about point $B$ and it landed at point $C$. Devon rotated the original point $A$ $y^\circ$ counterclockwise about point $B$ after a $30^\circ$ clockwise rotation, and it also landed at point $C$. If $y<360$, what is the value of $y$?


Let $\theta$ be an angle such that $\cos 2\theta = \frac{1}{5}$. Compute $\sin^6 \theta + \cos^6 \theta$.


Simplify the sum: \(i^0 + i^1 + \cdots + i^{2031}.\)


My school's math club consists of 5 boys and 10 girls. I need to select a team of 6 for the state math competition. The team must include at least 3 girls. In how many ways can I select the team?


The function $f(x)$ satisfies
\[ f(xy) = 3xf(y) \]
for all real numbers $x$ and $y$. If $f(1) = 10$, find $f(5)$.


In triangle $DEF$, we have $\angle D = 90^\circ$ and $\tan E = \frac{3}{4}$. Find $\sin F$.


The graph of the function $g(x) = px^3 + qx^2 + rx + s$ passes through the points $(-2, 0)$, $(0, -3)$, and $(2, 0)$. Find the value of $q$.


In quadrilateral $ABCD$, $AB = 7$, $BC = 10$, and $CD = 18$ units. Angle $B$ and angle $C$ are both right angles. What is the length of segment $AD$?


Determine the number of possible multisets \( S = \{r_1, r_2, \dots, r_{12}\} \) where each \( r_i \) is an integer root of the polynomial
\[ a_{12} x^{12} + a_{11} x^{11} + \dots + a_1 x + a_0 = 0 \]
with integer coefficients and non-zero \( a_{12} \) and \( a_0 \). It is given that the roots \( r_1, r_2, \dots, r_{12} \) of the polynomial
\[ a_0 x^{12} + a_1 x^{11} + \dots + a_{11} x + a_{12} = 0 \]
are also integers.


Mr. Johnson has 8 children. Assuming that the gender of each child is determined independently and with an equal likelihood of male and female, what is the probability that Mr. Johnson has at least twice as many daughters as sons?


If $q(x) = x^4 - 4x + 5$, find the coefficient of the $x^3$ term when $(q(x))^2$ is expanded.


The image of the point with coordinates $(-2, 0)$ under the reflection across the line $y = mx + c$ is the point with coordinates $(6, 4)$. Find $m+c$.


If $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ are vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{a} + \mathbf{b}\| = 2,$ and
\[
\mathbf{c} - 2\mathbf{a} - \mathbf{b} = 4(\mathbf{a} \times \mathbf{b}),
\]
find $\mathbf{b} \cdot \mathbf{c}$.


Isosceles triangle $ABE$ of area 200 square inches is cut by $\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 150 square inches. If the altitude of triangle $ABE$ from $A$ is 25 inches, what is the number of inches in the length of $\overline{CD}$?

Solution: \boxed{8 \text{ inches}}


I have math homework that I complete at a rate of $p$ problems per hour, taking $t$ hours. My friend finishes the same homework at a rate of $2p-10$ problems per hour and it only takes him $t-1$ hours. It's known that $p$ and $t$ are positive whole numbers, and I solve at least 15 problems per hour. How many problems did I complete?


For how many integers $n$ is it true that $\sqrt{n+2} \le \sqrt{3n - 5}$ and $\sqrt{3n - 5} < \sqrt{2n + 10}$?


On a Cartesian coordinate plane, points $(1,-2)$ and $(4, 1)$ are adjacent vertices of a square. Calculate the area of this square.


Let $x = (3 + \sqrt{5})^{500}$, let $n = \lfloor x \rfloor$, and let $f = x - n$. Find
\[x(1 - f).\]


A right-angled isosceles triangle shares a common side with a rectangle as shown in a diagram (not displayed). If angle $B$ is the right angle in the triangle, find the number of degrees in $m\angle CDB$, where $C$ and $D$ are points on the rectangle, and $B$ is a point on the triangle.


Determine the matrix $\mathbf{N}$ such that
\[\mathbf{N} \mathbf{u} = \begin{pmatrix} 7 \\ -3 \\ 4 \end{pmatrix} \times \mathbf{u}\] for all vectors $\mathbf{u}$.

Solution: \boxed{\begin{pmatrix} 0 & -4 & -3 \\ 4 & 0 & -7 \\ 3 & 7 & 0 \end{pmatrix}}


The volume of a certain rectangular solid is $512\text{ cm}^3$, its total surface area is $448\text{ cm}^2$, and the three dimensions are such that the quotient between consecutive dimensions increases by 1. If the sequence of dimensions is $x$, $x(d+1)$, and $x(d+1)^2$, find the sum of the lengths in cm of all the edges of this solid.


Find the smallest positive four-digit number that is divisible by each of the five smallest prime numbers.


Six standard six-sided dice are rolled. We are told that there is a triplet and a pair among these dice, but no four-of-a-kind. The triplet and the pair are set aside, and the remaining die is re-rolled. What is the probability that after re-rolling this single die, there is at least a four-of-a-kind or a full house?


Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+mn$ for all positive integers $m$ and $n.$ Find $b_{12}$.


Find the number of positive integers $n$ that satisfy
\[(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.\]


Find the number of complex solutions to
\[\frac{z^4 - 1}{z^3 - 3z + 2} = 0.\]


Suppose $x, y, z$ are three positive numbers that satisfy the equations $xyz = 1, x + \frac{1}{z} = 6$, and $y + \frac{1}{x} = 15$. Find $z + \frac{1}{y}$.


Find the interval of all $x$ such that both $3x$ and $4x$ are in the interval $(2,5)$.

Solution: \boxed{\left(\frac{2}{3}, \frac{5}{4}\right)}


Evaluate the sum of $1001101_2$ and $111000_2$, and then add the decimal equivalent of $1010_2$. Write your final answer in base $10$.


The function $f : \mathbb{R} \to \mathbb{R}$ satisfies
\[ f(x) + 3f(1 - x) = 4x^2 \]
for all $x$. Find $f(3).


What is the coefficient of $x^2$ when $-3x^3 - 6x^2 - 8x + 2$ is multiplied by $2x^2 - 8x + 3$ and the resulting terms are combined?


There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are seven different colors of triangles from which to choose, how many distinguishable large equilateral triangles can be constructed?


Compute $\sqrt{72} \cdot \sqrt{18} \cdot \sqrt{8}$.


Let $\mathbf{a} = \begin{pmatrix} 7 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$. Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$


A repair crew needs to reach the center $C$ of a circular fountain, so they place a 20-foot plank from $A$ to $B$ and then a 12-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. What is the area of the circular base of the fountain? Express your answer in terms of $\pi$.

Solution: \boxed{244 \pi \text{ square feet}}


Twelve people sit down at random seats around a round table. Five of them are math majors, four others are physics majors, and the remaining three are biology majors. What is the probability that all five math majors sit in consecutive seats?


What is the largest integer $n$ for which $\binom{9}{4} + \binom{9}{5} = \binom{10}{n}$?


What is the sum of the last two digits of this modified portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+55!$?

Solution: \boxed{5 + 0 = 5}


Find the intersection of the lines \(5x - 3y = 19\) and \(6x + 2y = 14\). Express your answer as an ordered pair \((x, y)\).

Solution: \boxed{\left(\frac{20}{7}, -\frac{11}{7}\right)}


Find $\frac{8}{15} - \frac{11}{20}$. Express your answer as a fraction in simplest form.


Find all real numbers \(x\) which satisfy \[\frac{x+2}{x-4} \ge 3.\] (Give your answer in interval notation.)

Solution: \boxed{(4, 7]}


Define $f(x)$ to be the minimum value of the three numbers $3x + 1,$ $-\frac{4}{3}x + 3,$ and $\frac{1}{3} x + 9$ for each value of $x.$ Find the maximum value of $f(x).$


How many three-digit positive integers \(x\) satisfy \(2895x + 547 \equiv 1613 \pmod{17}\)?


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 10 dips, and 2 dips are equivalent to 1 dirp, how many daps are equivalent to 20 dirps?

Solution: \boxed{15\text{ daps}}


If $C\,841\,F\,25$ is divisible by $9$, where $C$ and $F$ are single digit integers, what is the product of all possible values of $C \cdot F$?


Two of the roots of the cubic equation \[ax^3 + bx^2 + cx + d = 0\] are \(4\) and \(-3\). Given that \(a \neq 0\), compute \(\frac{b+c}{a}\).


In Jeremy's garden, the ratio of tulips to daisies is 3:4. He currently has 32 daisies. If he plans to add 8 more daisies and wants to maintain the same ratio, how many additional tulips does he need to add, and what will be the total number of tulips?

Solution: \boxed{30 \text{ tulips}}


A deck of fifty-two cards consists of four Aces, four 2's, ..., and four Kings. A matching pair (two cards with the same rank) is removed from the deck. Determine the probability that two randomly selected cards from the remaining deck also form a pair. Express the probability in the form $m/n$, where $m$ and $n$ are coprime integers, and find the sum $m + n$.


What is the remainder when $3001 \cdot 3002 \cdot 3003 \cdot 3004 \cdot 3005$ is divided by 17?


What is the sum of the two solutions to the equation $40 - 20x - 2x^2 = 0$?


Let $z$ be a complex number such that
\[ |z^2 + 16| = |z(z + 4i)|. \]
Find the smallest possible value of $|z + i|$.


Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \]


When the expression $3(x^2 - 3x + 3) - 8(x^3 - 2x^2 + 4x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?


What is the ratio of the area of the shaded square to the area of the large square given that the large square is divided into a grid of $25$ smaller squares each of side $1$ unit? The shaded square consists of a pentagon formed by connecting the centers of adjacent half-squares on the grid. The shaded region is not provided in a diagram but is described as occupying $5$ half-squares in the center of the large square.


Two cross sections of a right pentagonal pyramid are obtained by slicing the pyramid with planes parallel to the pentagonal base. The areas of these cross sections are $100\sqrt{3}$ square feet and $225\sqrt{3}$ square feet, respectively. The two planes are $5$ feet apart. Determine the distance from the apex of the pyramid to the larger cross section.

Solution: \boxed{15 \text{ feet}}


Calculate the value of \(0.\overline{864} - 0.\overline{579} - 0.\overline{135}\). Express your answer as a fraction in lowest terms.


Let \( \triangle ABC \) be a right triangle with \( B \) as a right angle. A circle with diameter \( BC \) intersects side \( AC \) at point \( D \). If the area of \( \triangle ABC \) is \( 180 \) and \( AC = 30 \), what is the length of \( BD \)?


How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit?


Expand and simplify the expression $2x(3x^2 - 4x + 5) - (x^2 - 3x)(4x + 5)$.

Solution: \boxed{2x^3 - x^2 + 25x}


On a balance scale, $4$ red balls balance $8$ blue balls, $3$ orange balls balance $7.5$ blue balls, and $8$ blue balls balance $6$ purple balls. How many blue balls are needed to balance $5$ red, $3$ orange, and $3$ purple balls?


Let \[f(x) = \left\{
\begin{array}{cl}
-x + 4 & \text{if } x < 0, \\
3x - 6 & \text{if } x \geq 0.
\end{array}
\right.\]How many solutions does the equation $f(f(x)) = 6$ have?


If $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ are unit vectors in space, then find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\]


In a sequence of four positive integers, the first three terms form a geometric progression, the last three terms form an arithmetic progression, and the first and fourth terms differ by $50.$ Find the sum of the four terms.


Find the greatest common divisor of 10,010 and 15,015.


Find the $y$-coordinate of the vertex of the parabola given by the equation $y = 2x^2 + 16x + 29$.


In the coordinate plane, the curve $xy = 2$ intersects a circle at four points, three of which are $\left(4, \frac{1}{2}\right),$ $\left(-6, -\frac{1}{3}\right),$ and $\left(\frac{1}{4}, 8\right).$ Find the fourth point of intersection.

Solution: \boxed{\left(-\frac{2}{3}, -3\right)}


The six edges of a tetrahedron $EFGH$ measure $10, 14, 19, 28, 37$ and $42$ units. If the length of edge $EF$ is $42$, determine the length of edge $GH$.

Solution: \boxed{GH = 14}


A function $f$ is defined by $f(z) = i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. Determine how many values of $z$ satisfy both $|z| = 7$ and $f(z) = z$.


What is the smallest abundant number that is not a multiple of 5?


The function \(g\) satisfies
\[
g(x) + g(3x+y) + 7xy = g(4x - y) + 3x^2 + 2
\]
for all real numbers \(x, y\). Determine the value of \(g(10)\).


The sum of the length and width of a rectangle is such that twice this sum equals three times the area of the rectangle. Find the dimensions of the rectangle if the length is twice the width.


What is the sum of the positive odd divisors of $90$ that are greater than $3$?


Find the positive value of $t$ that satisfies $ab = t - 3 + 5i$ given $|a| = 3$ and $|b| = 5$.

Solution: \boxed{3 + 10\sqrt{2}}


The U.S. produces about 7 million tons of apples each year. Of the total, $25\%$ is exported, $60\%$ of the remainder is used for apple juice, and the remaining $40\%$ of that subset is sold fresh. However, before processing for apple juice, $5\%$ of the apples intended for juice are lost due to spoilage. How many million tons of apples are actually used for apple juice? Express your answer as a decimal to the nearest tenth.


The sum of a negative integer $N$ and twice its square is 12. What is the value of $N$?


Let $b_n = \frac{8^n - 1}{7}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on?


Find the minimum value of
\[
\begin{vmatrix}
1 & 1 & 1 \\
1 & 1 + \tan \theta & 1 \\
1 + \cot \theta & 1 & 1
\end{vmatrix},
\]
as $\theta$ ranges over all angles where $\tan \theta$ and $\cot \theta$ are defined.


Determine the maximum value of
\[
\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_1,
\]
for all real numbers $\theta_1, \theta_2, \theta_3, \theta_4$.


In $\triangle XYZ$, $\angle X = 90^\circ$ and $\tan Z = \sqrt{3}$. If $YZ = 150$, what is $XY$?


Calculate the least number of digits in the repeating block of the decimal expansion of $7/13$.


Using only the digits 5, 6, and 7, how many positive five-digit integers can be made that are palindromes and contain the digit 5 at least once?


Factor completely: $x^6 - 5x^4 + 8x^2 - 4$.


What is the greatest common divisor of $765432$ and $654321$?


Let $x$, $y$, and $z$ be nonzero complex numbers such that $x + y + z = 30$ and
\[(x - y)^2 + (x - z)^2 + (y - z)^2 = 2xyz.\]
Find $\frac{x^3 + y^3 + z^3}{xyz}$.


A series of five concentric circles are drawn with radii of 2, 4, 6, 8, and 10. Starting from the innermost circle which is painted black, the next ring is white, followed by a black ring, then white, and the outermost ring is painted black. Calculate the ratio of the black area to the white area. Express your answer as a common fraction.


Given the numbers $a,$ $b,$ $c,$ $d$ are equal to 1, 2, 4, 5, in some order. Find the largest possible value of
\[ab + bc + cd + da\]


Solution: \boxed{27a^{12} + 18a^6b^2 - 42a^2b^5 - 343b^9}


What is the smallest positive integer $n$ such that $5n$ is a perfect square and $3n$ is a perfect cube?


If $x@y = x^2y - 3x$, what is the value of $(5@3) - (3@5)$?


What is the perimeter, in cm, of quadrilateral $EFGH$ if $\overline{EF} \perp \overline{FG}$, $\overline{HG} \perp \overline{FG}$, $EF=12$ cm, $HG=3$ cm, and $FG=16$ cm?

Solution: \boxed{31 + \sqrt{337} \text{ cm}}


I want to choose a license plate that is 4 characters long, where the first and last two characters are letters, and the middle two characters are digits. Furthermore, I want there to be at least one pair of adjacent characters that are the same. How many ways are there for me to choose a license plate with these restrictions?


For how many different negative values of $x$ is $\sqrt{x +169}$ a positive integer?


Find the greatest integer that satisfies the inequality: $6x^2 + 5x - 8 < 3x^2 - 4x + 1$.


Joann ate a total of 150 lollipops in seven days. Each day after the first day, she ate four more lollipops than she had on the previous day. How many lollipops did she eat on the fifth day?


Laura drove from her office to her friend's house. The odometer read 85,432 miles when she started, and it read 85,470 miles when she arrived. Laura's car consumes fuel at a rate of 25 miles per gallon. If the price of one gallon of gas is $\$3.85$, what was the cost of the gas used for Laura to drive from her office to her friend’s house? (Express your answer in dollars and round to the nearest cent.)


What is the smallest five-digit number that is both a perfect square and a perfect cube?


An ellipse has its foci at $(1, 1)$ and $(1, 3)$. Given that it passes through the point $(-4, 2)$, its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$


The first four terms in a geometric sequence are $x+y, x-y, x^2y,$ and $xy^2,$ in that order. What is the fifth term?


Hermione, Draco, and Luna decide to join Harry, Ron, and Neville in a broomstick race. If there are no ties, in how many different possible orders can they finish?


It can be shown that for any positive integer \( n \),
\[
\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}^n = \begin{pmatrix} G_{n+1} & 2G_n \\ 2G_n & G_{n-1} \end{pmatrix},
\]
where \( G_n \) denotes the \( n \)th modified Fibonacci number, defined by the recurrence relation \( G_{n} = 3G_{n-1} + 2G_{n-2} \) with initial conditions \( G_0 = 0 \) and \( G_1 = 1 \).

Compute \( G_{784} G_{786} - 4G_{785}^2 \).


Find the product of all positive integral values of $x$ such that $x^2 - 40x + 399 = q$ for some prime number $q$. Note that there must be at least one such $x$.


Find the range of the function \[
f(x) = \frac{x^2}{x^2-x+1},
\]
where $x$ is any real number. (Give your answer in interval notation.)

Solution: \boxed{[0, \frac{4}{3}]}


The Garcia family consists of two adults (Mr. and Mrs. Garcia), their two children with the eldest being responsible enough to drive, and their grandmother who does not drive. When driving somewhere together in their car, two people sit in the front including the driver, while the three others sit in the back. The driver must be either Mr. Garcia, Mrs. Garcia, or the eldest child. How many seating arrangements are possible?


Calculate the inverse of the matrix
\[
\begin{pmatrix}
5 & 2 \\
3 & 1
\end{pmatrix}.
\]
If the inverse does not exist, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} -1 & 2 \\ 3 & -5 \end{pmatrix}}


A $150$-gon $Q_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $150$ vertices equals 3010. The midpoints of the sides of $Q_1$ form a second $150$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $150$-gon, $Q_3$. Find the sum of the $x$-coordinates of the vertices of $Q_3$.


The measures of angles $A$ and $B$ are both positive integer numbers of degrees. The measure of angle $A$ is a multiple of the measure of angle $B$, and angles $A$ and $B$ are supplementary angles. Determine how many measures are possible for angle $A$.


Let \(a\) and \(b\) be positive real numbers such that \(a + 3b = 1.\) Find the minimum value of
\[
\frac{1}{a} + \frac{3}{b}.
\]


Suppose $a,$ $b,$ and $c$ are three positive numbers such that $abc = 1,$ $a + \frac {1}{c} = 7,$ and $b + \frac {1}{a} = 12.$ Find $c + \frac {1}{b}.$


There are four points that are $6$ units from the line $y = 15$ and $14$ units from the point $(6,15)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


How many integers are common solutions to these three inequalities?
\[
\begin{array}{cccc}
(1) & -3y + 1 & \geq & y - 9 \\
(2) & -2y + 2 & \leq & 14 \\
(3) & -5y + 1 & \geq & 3y + 20
\end{array}
\]


A cinema is setting up seats in rows for a special screening. Each complete row consists of $8$ seats, and the cinema currently has $240$ seats arranged. The cinema wishes to minimize the number of unoccupied seats, ensuring every row is filled. If $150$ people are expected to attend the screening, how many seats should be removed?


What is the ratio of the volume of a cube with edge length nine inches to the volume of a cube with edge length three feet?


Define positive real numbers $x$ and $y$ such that $x^2 + y^2 = 2$ and $x^4 + y^4 = \frac{15}{8}$. Find $xy$.


Define a **valid sequence** 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 — where $A$ 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 eight-letter valid sequences are there?


Find all \( y \in [0, 2\pi] \) for which \[\cos(x+y) \geq \cos(x) - \cos(y)\] when \( x = \frac{\pi}{2} \).

Solution: \boxed{[0, \frac{\pi}{4}] \cup [\frac{5\pi}{4}, 2\pi]}


The measure of angle $ACB$ is 50 degrees. If ray $CA$ is rotated 450 degrees about point $C$ in a clockwise direction, and then reflected over line $CB$, what will be the positive measure of the new acute angle $ACB$, in degrees?


Given a prime \(p = 11\), determine the sum of all integers in the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) that are primitive roots modulo \(11\).


The admission fee for a museum is $ \$20$ per adult and $ \$10$ per child. On a certain day, the museum collected a total of $ \$1600$ in admission fees, when they had at least one adult and at least one child visiting. Among all possible ratios of adults to children on that day, which one is closest to $ 1$?


In a bag, Sam has 9 blue marbles and 6 red marbles. He draws a marble at random, notes its color, returns it to the bag, and repeats the process until he has drawn 8 marbles. What is the probability that exactly five of the marbles he draws are blue? Express your answer as a decimal rounded to the nearest thousandth.


In pentagon $ABCDE$, interior angles $\angle A = 70^\circ$ and $\angle B = 90^\circ$. Angles $\angle C$ and $\angle D$ are equal. The fifth angle $\angle E$ is $3x-10^\circ$ where $x$ is the measure of angles $C$ and $D$. Find the measure of the largest angle.


An ellipse has an area of $M \text{ cm}^2$ and a perimeter approximation of $N \text{ cm}$ (using \( N = \pi (a + b) \) as a rough estimate for a circle-like perimeter). If \( \frac{M}{N} = 10 \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis, calculate \( a + b \).


A circle of radius 2 is surrounded by 4 circles of radius \( r \) as shown in an analogous arrangement to the original problem, using the same formation of the squares and triangles. Calculate the radius \( r \).

Solution: \boxed{r = 2(\sqrt{2} + 1)}


Determine the greatest common divisor of $765432$ and $654321$.


A stock investment increased by $30\%$ in the first year. At this new value, what percentage decrease would be required in the following year for the stock to return to $90\%$ of its original value?


A rectangle has a perimeter of 120 inches and each side has an even integer length. How many non-congruent rectangles meet these criteria?


Let \( y = 2^{3}3^{3}4^{4}5^{5}6^{6}7^{7}8^{8}11^{3} \). Find the smallest positive integer that, when multiplied with \( y \), produces a product that is a perfect square.


A larger square fully contains two non-overlapping shapes: a small square and a right triangle. The side length of the small square is 4 units and the side length of the larger square is 10 units. The base and the height of the right triangle, which are both entirely contained within the larger square, measure 3 units each. What is the total area in square units of the regions that are not covered by the smaller square or the triangle within the larger square?

Solution: \boxed{79.5 \text{ square units}}


How many divisors of $9!$ are multiples of 4?


Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3$.


Consider the equation of a circle in the form \[r = 4 \cos \theta + 3 \sin \theta.\] Determine both the area and the circumference of the circle.

Solution: \boxed{5 \pi}


Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15}\dotsm\frac{3n+6}{3n}\dotsm\frac{3003}{2997}.\]


For how many different negative values of $x$ is $\sqrt{x + 200}$ a positive integer?


Find the sum of all the solutions to the equation \(2^{|x|} + 3|x| + x^2 = 24.\)


Find the units digit of $8 \cdot 19 \cdot 1978 - 8^3$.


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 48 dips?

Solution: \boxed{22.5 \text{ daps}}


Stuart has drawn a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the smaller circle. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$?


If $a$ is a constant such that $9x^2 + 27x + a$ is the square of a binomial, then what is $a$?


A sphere is inscribed in a cylinder which is itself inscribed in a cube. What are the ratios of the volume of the inscribed sphere and the cylinder to the volume of the cube? Express your answers as common fractions in terms of $\pi$.


A student must choose a program of five courses from a list consisting of English, Algebra, Geometry, History, Art, Latin, and Science. This program must contain English and at least two mathematics courses. In how many ways can this program be chosen?


Suppose a cube has a side length of $8$. Its vertices are alternately colored black and green. What is the volume of the tetrahedron whose vertices are the green vertices of the cube?


What is the area enclosed by the graph of $|2x| + |5y| = 10$?


Let $\Omega$ be a nonreal root of $z^4 = 1$. Find the number of ordered pairs $(a,b)$ of integers such that $|a\Omega + b| = 2$.


How many even four-digit integers have the property that their digits, read left to right, are in strictly increasing order?


Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ and $D$ such that
\[5 \overrightarrow{OA} - 3 \overrightarrow{OB} + 4 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0},\] the four points $A,$ $B,$ $C,$ and $D$ are collinear. Find $k.$


In the diagram, $CP$ and $CQ$ trisect $\angle ACB$. $CM$ bisects $\angle PCQ$. Find the ratio of the measure of $\angle MCQ$ to the measure of $\angle ACQ$.


How many perfect squares are factors of 450?


A curious scientist is investigating a circular petri dish whose boundary is described by the equation $x^2 + y^2 + 10 = 6x + 12y$. Determine the radius of the petri dish to assess if it is suitable for an experiment.


Rationalize the denominator of $\frac{3}{4\sqrt{7} + 3\sqrt{13}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?


Square EFGH has its center at $(5, -5)$ and has an area of 16 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at $(0,0)$ and a scale factor of 3. What are the coordinates of the vertex of the dilated image of square EFGH that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(21, -21)}


How many ways are there to arrange the letters of the word $\text{B}_1\text{B}_2\text{A}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3$, in which the three A's, two B's, and the two N's are considered different?


Jane and her brother each spin a spinner once. The spinner has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.


A triangle is constructed with wooden sticks of lengths 7, 24, and 25 inches joined end-to-end. Determine the length of the smallest piece that can be cut from each of the three sticks so that the remaining pieces no longer form a triangle.


The Central Mathematics University is welcoming its new batch of students. The total number of incoming students is less than 600. When the students arrange themselves in rows of 26, there are 25 students in the last row. When they arrange themselves in rows of 24, there are 15 students in the last row. Determine the number of students in the incoming batch.


What is the slope of the line determined by any two solutions to the equation $\frac{4}{x} + \frac{6}{y} = 0$? Express your answer as a common fraction.


Let $S = \{8^k : k \text{ is an integer}, 0 \leq k \leq 3000\}$. Given that $8^{3000}$ has 2713 digits and that its first (leftmost) digit is 8, how many elements of $S$ have 8 as their leftmost digit?


A right circular cone is cut into five pieces by four planes parallel to its base, each piece having equal height. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece.


Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a heart and the second card is a king?


Find the units digit of the product of 7 and the sum, $$ 1! + 2! + 3! + \cdots + 2023!. $$


Let $a,$ $b,$ $c$ be the roots of the cubic polynomial $x^3 - 2x - 2 = 0.$  Find
\[a(b - c)^2 + b(c - a)^2 + c(a - b)^2.\]


A hexagon is obtained by joining, in order, the points $(0,1)$, $(1,2)$, $(2,2)$, $(2,1)$, $(3,0)$, $(1,0)$, and $(0,1)$. The perimeter of the hexagon can be written in the form $a+b\sqrt{2}+c\sqrt{3}+d\sqrt{5}$, where $a$, $b$, $c$, and $d$ are whole numbers. Find $a+b+c+d$.


My friend Alex has a peculiar fondness for numbers that are divisible by 6. How many different last digits are possible in numbers that Alex likes?


Determine the number of ways to arrange the letters of the word RADAR.


Triangle $PQR$ is a right triangle with $PQ = 6,$ $QR = 8,$ and the right angle at $R.$ Point $S$ is the midpoint of $PQ,$ and $T$ is on the same side of line $PQ$ as $R$ so that $PT = QT = 10.$ Given that the area of triangle $TRS$ may be expressed as $\frac {x\sqrt {y}}{z},$ where $x,$ $y,$ and $z$ are positive integers, $x$ and $z$ are relatively prime, and $y$ is not divisible by the square of any prime, find $x + y + z.$


Find all real numbers $x$ such that \[4 \le \frac{2x}{3x-7} < 9.\] (Give your answer in interval notation.)

Solution: \boxed{(\frac{63}{25}, 2.8]}


How many positive divisors does 8! have?


A $\textit{composite number}$ is a number that has at least two distinct positive divisors aside from one and itself. The number 93 can be expressed as the sum of two composite numbers. Find the minimum positive difference between any two such composite numbers that sum up to 93.


At a national conference, 25 scientists are accommodated at a hotel. Each scientist is assigned one unique room, with room numbers ranging from 1 to 25. If the scientists assigned to rooms 10, 11, and 20 are absent, what is the median room number of the rooms occupied by the remaining 22 scientists?


If \( p, q, \) and \( r \) are three non-zero integers such that \( p + q + r = 36 \) and\[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{540}{pqr} = 1, \] compute \( pqr \).


Positive real numbers $r,s$ satisfy the equations $r^2 + s^2 = 2$ and $r^4 + s^4 = \frac{15}{8}$. Find $rs$.


Convert the point \( (3, -3\sqrt{3}, 2) \) in rectangular coordinates to cylindrical coordinates. Enter your answer in the form \( (r,\theta,z) \), where \( r > 0 \) and \( 0 \le \theta < 2\pi \).

Solution: \boxed{(6, \frac{5\pi}{3}, 2)}


In a group of 25 people, 10 are left-handed. 18 of them enjoy classical music. 3 of them are right-handed and do not enjoy classical music. How many group members are left-handed and enjoy classical music? Assume people are either left-handed or right-handed.


The quadratic equation $ax^2 + 10x + c = 0$ has exactly one solution. If $a + c = 11$, and $a < c$, find the ordered pair $(a, c)$.

Solution: \boxed{\left(\frac{11 - \sqrt{21}}{2}, \frac{11 + \sqrt{21}}{2}\right)}


Let $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ 0 \\ -2 \end{pmatrix}.$  Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$

Solution: \boxed{\begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix}}


Consider a rectangular parallelepiped (box) with dimensions 2 by 3 by 4 units. Determine the total volume of space inside or within one unit of this box, expressed as \(\frac{m + n\pi}{p}\), where \(m\), \(n\), and \(p\) are positive integers with \(n\) and \(p\) relatively prime. Find \(m+n+p\).

Solution: \boxed{m+n+p = 228 + 31 + 3 = 262}


For how many integers $n$ between 1 and 17 (inclusive) is $\frac{n}{18}$ a repeated decimal?


Given vectors $\mathbf{u}$ and $\mathbf{v}$ such that $\|\mathbf{u}\| = 5$ and $\|\mathbf{v}\| = 13$, and the angle $\theta$ between them is $60^\circ$, find the value of $\mathbf{u} \cdot \mathbf{v}$.


Let \( A, B \) be the points on the coordinate plane with coordinates \( (t-3, 0) \) and \( (-1, t+2) \), respectively. The square of the distance between the midpoint of \( \overline{AB} \) and an endpoint of \( \overline{AB} \) is \( t^2 + 1 \). What is the value of \( t \)?

Solution: \boxed{\pm \sqrt{2}}


A plot of land is shaped as a trapezoid and is represented on a map using a scale where 1 cm equals 3 miles. The shorter base of the trapezoid is 12 cm, the longer base is 18 cm, and the height is 15 cm. Given that one square mile is equal to 640 acres, calculate the actual size of the plot in acres.

Solution: \boxed{1,296,000 \text{ acres}}


Using the digits 1, 2, 3, 4, 5, how many odd three-digit numbers less than 400 can be formed if each digit can be used more than once?


What is the smallest positive integer $n$ such that $629n \equiv 1181n \pmod{35}?$


A collection of five positive integers has mean 4.6, unique mode 3, and median 4. If a 9 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth.


Find the degree measure of the least positive angle $\theta$ for which
\[\cos 15^\circ = \sin 45^\circ + \sin \theta.\]


The expression \(4y^2 - 5y - 21\) can be written as \((4y + a)(y + b)\), where \(a\) and \(b\) are integers. Find \(a - b\).


How many five-digit positive integers are there with the ten-thousands digit $3$?


When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 24. One of the integers is 60. Determine the smallest possible value for the other integer, provided this integer must be divisible by 5.


Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that
\[
f(xf(y) - x) = xy - f(x)
\]
for all \( x, y \). Determine the number \( n \) of possible values of \( f(3) \), and let \( s \) be the sum of all possible values of \( f(3) \). Find \( n \times s \).


April has five different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?


What is the largest integer less than $\log_2 \frac{2}{1} + \log_2 \frac{3}{2} + \cdots + \log_2 \frac{10009}{10008} + \log_2 \frac{10010}{10009}$?


Integers \(a\) and \(b\) with \(a>b>0\) satisfy \(a+b+ab=152\). What is \(a\)?

Solution: \boxed{a = 50}


A zealous botanist is organizing a competition where participants must guess the age of an ancient tree. It's known that the age of the tree consists of six digits: 1, 1, 1, 3, 7, and 9, with the stipulation that the age must start with a digit greater than 5.


Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2010 \left( \frac{1}{x} + \frac{1}{y} + 2009 \right)\]
for all $x,$ $y > 0.$

Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$


Triangle $PQR$ is a right triangle with legs $PQ$ and $PR$. Points $U$ and $V$ are on legs $PQ$ and $PR$, respectively so that $PU:UQ = PV:VR = 1:3$. If $QU = 18$ units, and $RV = 45$ units, what is the length of hypotenuse $PQ$? Express your answer in simplest radical form.


The graph of \( y=f(x) \) for \(-4\le x\le 4\) is shown below. On this interval, what is the range of \( f(x) - x \)? Express your answer in interval notation.

[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(-4,4,-4,4);
draw((-4,-4)--(4,4),green+1);
draw((-4,-4)--(-3,-4),red+1);
draw((-3,-2)--(-2,-2),red+1);
draw((-2,-1)--(-1,-1),red+1);
draw((-1,0)--(0,0),red+1);
draw((0,1)--(1,1),red+1);
draw((1,2)--(2,2),red+1);
draw((2,3)--(3,3),red+1);
draw((3,4)--(4,4),red+1);
dot((-4,-4),red); dot((-3,-2),red); dot((-2,-1),red); dot((-1,0),red); dot((0,1),red); dot((1,2),red); dot((2,3),red); dot((3,4),red); dot((4,4),red);
[/asy]

Solution: \boxed{[0, 1]}


Fully factor the following expression: $3y^2 - 75$.


Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2500$ and let $N$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $N$?


Find the smallest positive integer \(b\) for which \(x^2 + bx + 2016\) factors into a product of two polynomials, each having integer coefficients.


How many different counting numbers will each leave a remainder of 11 when divided into 61?


Find the largest prime factor of $21^3 + 14^4 - 7^5$.


Let $s$ be a real number, $|s| < 3,$ and let $w$ be a complex number such that
\[w + \frac{2}{w} = s.\]Find $|w|.$


The number $6\,31G\,309\,4H2$ is divisible by $72.$ If $G$ and $H$ each represent a single digit, what is the sum of all distinct possible values of the product $GH$? (Count each possible value of $GH$ only once, even if it results from multiple $G,$ $H$ pairs.)


For what values of $x$ is $x^2 + 2x \le -1$? Express your solution in interval notation.


Each of the numbers $b_1,$ $b_2,$ $\dots,$ $b_{100}$ is $\pm 1.$  Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 100} b_i b_j.\]


In triangle $DEF$, the perimeter is 1801 units. Each side length is an integer, and $DE < EF \leq FD$. What is the smallest possible value of $EF - DE$?


An equilateral triangle has vertices at $(1,6)$ and $(9,6)$. If the third vertex is within the first quadrant, what could be its possible y-coordinate(s)? Express your answer in simplest radical form.

Solution: \boxed{6 + 4\sqrt{3}}


The sum of the squares of three consecutive even integers is 2450. What is the sum of the fourth powers of these integers?


How many integers between $200$ and $250$ have three different digits in increasing order? One such integer is $234$.


How many two-digit integers are there in which each of the two digits is prime, and the first digit is always greater than the second?


Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 30$ and
\[(x - y)^2 + (x - z)^2 + (y - z)^2 = 2xyz.\] Find $\frac{x^3 + y^3 + z^3}{xyz}.$


Express the number 2023 as a sum of distinct powers of 2. What would be the least possible sum of the exponents of these powers?


The following line is parameterized, so that its direction vector is of the form $\begin{pmatrix} 2 \\ b \end{pmatrix}.$  Find $b$ given that the line passes through points $\begin{pmatrix} -5 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ 5 \end{pmatrix}$.


A fair 8-sided die is rolled. If I roll $n$, then I win $(n+1)^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.


A regular octahedron is constructed by connecting the centers of adjacent faces of a cube with side length 2x. What is the ratio of the volume of the octahedron to the volume of this cube?


Let $g(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be denoted as $s_1, s_2, \dots, s_{2010}$. It is known that there are exactly 1008 distinct values among the magnitudes $|s_1|, |s_2|, \dots, |s_{2010}|$. Determine the minimum number of real roots that $g(x)$ can have.


Of the originally proposed 20 provinces, 15 needed to agree on a new amendment for it to be ratified. What is the ratio of 15 to 20, and what is it when rounded to the nearest tenth?


The graph of the parabola $x = 3y^2 - 9y + 5$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.


What is the largest four-digit number whose digits add up to 20?


Compute the sixth power of the matrix $\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 64 & 0 \\ 0 & 64 \end{pmatrix}}


How many integers $-9 \leq n \leq 13$ satisfy $(n-1)(n+5)(n+11)<0$?


Consider the following equations of five different lines:
\[
y = 4x + 2, \quad 3y = 9x + 3, \quad 2y = -8x + 10, \quad 3y = 8x + 5, \quad y = -\frac{1}{4}x + 6.
\]
Determine how many pairs of these lines are "good", where a pair of lines is considered "good" if they are either parallel or perpendicular to each other.


Two sides of a right triangle have the lengths 8 and 15. What is the sum of the possible lengths of the third side? Express the sum as a decimal rounded to the nearest tenth.


Determine the number of positive integer values for \(a\) such that \(x = 3\) is the only positive integer solution to the following system of inequalities:
$$
\begin{cases}
3x > 4x - 4\\
4x - a > -8
\end{cases}
$$


In a cylinder with a height of 15 cm and a radius of 9 cm, a cone and a sphere are placed inside. The cone has the same radius as the cylinder and a height of 5 cm. The sphere has a diameter equal to the cylinder's radius. Find the ratio of the combined volume of the cone and sphere to the volume of the cylinder.


A school has between 180 and 250 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into eight distinct sections of students. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


Find the greatest common divisor of 1043 and 2295.


A tree triples its height each year under good conditions, but only doubles its height in bad conditions. The tree ends up being 1458 feet tall after a total of six years, with the first three years being good growth years, and the last three years being bad growth years. What was the height of the tree at the end of the third year?

Solution: \boxed{182.25 \text{ feet}}


If \( g(x) = \dfrac{3x + 2}{x - 2} \), find the value of \( g(8) \).


Suppose that \(a\) varies inversely with \(b^2\). If \(a = 9\) when \(b = 2\), find the value of \(a\) when \(b = 4\).


Determine the resultant quotient and remainder when the polynomial $x^5 + 3x^4 - 28x^3 + 45x^2 - 58x + 24$ is divided by $x - 3$.


A wheel is divided into six equal sectors numbered from 1 to 6. Alice and Bob each spin the wheel once. Alice wins if the absolute difference between their numbers is less than 4. Otherwise, Bob wins. What is the probability that Alice wins? Express your answer as a common fraction.


How many different positive values of \( x \) will make this statement true: there are exactly 3 three-digit multiples of \( x \)?


Points $P$, $Q$, $R$, and $S$ are located on $\overline{PQ}$ such that $PQ = 4PS = 8QR$. If a point is selected at random on $\overline{PQ}$, what is the probability that it is between $R$ and $S$?


How many different four-letter arrangements can be formed using the seven letters $A, B, C, D, E, F, G$, if the first letter must be $D$, one of the other letters must be $E$, and no letter can be used more than once in the arrangement?


How many of the divisors of $10!$ are larger than $9!$?


A teacher challenges her class with a new problem: calculate the value of $\left(7^2 - 5^2\right)^4$.


In a right-angled triangle, the sum of the squares of the three side lengths is 2500. What is the length of the hypotenuse of this triangle?


What are the last two digits in the sum of factorials of the first 50 positive integers?


Six friends decide to have a picnic every month. For each picnic, they want to assign three of them to prepare the food and three to set up the picnic area. How many different ways can they divide these tasks among themselves?


Find the point in the plane \(2x + 3y - z = 15\) that is closest to the point \((2, -1, 4)\).

Solution: \boxed{\left(\frac{32}{7}, \frac{20}{7}, \frac{19}{7}\right)}


In the equation $w^3+x^3+y^3=z^3$, $w^3$, $x^3$, $y^3$, and $z^3$ are distinct, consecutive even positive perfect cubes listed in ascending order. What is the smallest possible value of $z$?


In triangle $DEF$, we have $\angle D = 90^\circ$ and $\sin E = \frac{3}{5}$. Find $\cos F$.


The sides of a triangle have lengths $13, 17,$ and $m,$ where $m$ is a positive integer. For how many values of $m$ is the triangle obtuse?


Find the sum of the infinite series $1 + 2\left(\frac{1}{999}\right) + 3\left(\frac{1}{999}\right)^2 + 4\left(\frac{1}{999}\right)^3 + \dotsb$.


For the circle with diameter $\overline{AB}$ equal to 36, point $C$ is another point on the circle, distinct from $A$ and $B$. $C$ traces the entire circle around $A$ and $B$. As $C$ moves, the centroid of $\triangle ABC$ also traces a path. Find the closest integer value for the area enclosed by the path that the centroid traces.


Triangle $PQR$ has vertices with coordinates $P(1,2),$ $Q(5,7),$ and $R(-2,5)$. The triangle is reflected about line $M$. The image points are $P'(-1,2),$ $Q'(-5,7),$ and $R'(2,5)$. What is the equation of line $M$?

Solution: \boxed{x = 0}


Find a constant $k$ such that for any chord $\overline{AB}$ of the circle $x^2 + y^2 = 1$ passing through the point $P = (0, \frac{1}{2})$, the value
\[ k = \frac{1}{PA} + \frac{1}{PB} \]
remains constant.


Ten women of different heights are at a gathering. Each woman decides to only shake hands with women taller than herself. How many handshakes occur?


A wizard is preparing a magical elixir. He needs to choose one of four magical herbs and one of six mystical crystals. However, two of the crystals are incompatible with three of the herbs. How many valid combinations can the wizard use to create his elixir?


What is the average of all the integer values of $N$ such that $\frac{N}{70}$ is strictly between $\frac{2}{5}$ and $\frac{4}{7}$?


Compute the sum of the squares of the roots of the equation \[x^{10} + 20x^7 + 8x^3 + 800 = 0.\]


The sides of triangle $PQR$ are in the ratio $3:4:5$. Segment $QS$ is the angle bisector drawn to the longest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the shorter subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express your answer as a common fraction.


Let $AB$ be a diameter of a circle centered at $O$. Let $F$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $F$ and $AF$ at $G$ and $H$, respectively. If $\angle BAF = 30^\circ$, find $\angle CGH$, in degrees.


In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$.


In a new walking challenge, I decide to make a move for each integer from 1 to 30. On the first move, I stand still. For subsequent moves $n$ (2 ≤ $n$ ≤ 30), I will take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After completing all 30 moves, I walk straight back to my starting point. How many steps is my return journey?


What is the product of all real numbers that are tripled when added to their reciprocals?


What is the units digit of the sum of the sequence \(1! + 1^2, \, 2! + 2^2, \, 3! + 3^2, \, \ldots , \, 9! + 9^2\)?


Consider a square with sides of length $4/\pi$, on which semicircular arcs are constructed on each side. What is the total perimeter of the region bounded by these arcs?


What is the least three-digit whole number, the product of whose digits is 18?


When $\frac{5}{480}$ is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?


Solution: \boxed{2x^3 - x^2 + 23x - 3}


You recently bought more than 150 eggs. The eggs are sorted in containers that can store exactly 15 eggs. However, upon inspecting the containers, you realize that three containers each hold only 14 eggs, while two containers each hold 16 eggs. Find the smallest number of eggs you could have right now.


Calculate the product \(\left(\frac{4}{8}\right)\left(\frac{8}{12}\right)\left(\frac{12}{16}\right)\cdots\left(\frac{2004}{2008}\right)\). Express your answer as a common fraction.


Add $528_{12} + 274_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.


If $x, y, z$ satisfy the system of equations
\begin{align*}
y + z &= 15 - 2x \\
x + z &= -10 - 2y \\
x + y &= 4 - 2z,
\end{align*}
what is $2x + 2y + 2z$?


A circle is tangent to the lines $5x - 2y = 40$ and $5x - 2y = -20.$ The center of the circle lies on the line $x + 3y = 0.$ Find the center of the circle.

Solution: \boxed{\left(\frac{30}{17}, -\frac{10}{17}\right)}


The side length of square $C$ is 45 cm, and the side length of square $D$ is 60 cm. Calculate the ratio of the area of square $C$ to that of square $D$ and determine the difference in their areas.

Solution: \boxed{1575 \text{ cm}^2}


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q-p$?


If 38,759,863 candies are packaged in sets of 6, how many will be left over?


Evaluate the expression $\sqrt{8} \times 2^{\frac{3}{2}} + 18 \div 3 \times 3 - 6^{\frac{5}{2}}$.

Solution: \boxed{26 - 36 \sqrt{6}}


Determine how many integer values of $x$ satisfy $\lceil{\sqrt{x}}\rceil=17$.


Find the smallest positive integer $Y$ such that a number $U$, made only of digits 0s and 1s, is divisible by 15, and $U = 15Y$.


Find the integer $n$, $-90 < n < 90$, such that $\tan n^\circ = \tan 850^\circ$.


The average of the numbers $1, 2, 3, \dots, 97, 98,$ and $x$ is $50x$. What is $x$?

Solution: \boxed{x = \frac{4851}{4949}}


At a team press conference, there are eight players from three teams to be seated: four from the Cubs, three from the Red Sox, and one coach. The Cubs insist on sitting together, the Red Sox also wish to sit together, and the coach needs to sit at one end of the row. How many ways can they be seated?


For $1 \le n \le 150$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal?


For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

[asy]
draw((0,0)--(6,6)--(13,2)--(8,-7)--cycle,linewidth(0.7));
draw((0,0)--(13,2),dashed);
label("9",(3,3),NW);
label("11",(9.5,4),NE);
label("15",(10.5, -2.5),SE);
label("13",(4,-2.5),SW);
[/asy]


Let $p$, $q$, and $r$ be the roots of the equation $x^3 - 6x^2 + 11x - 6 = 0$. Compute $\frac{pq}{r} + \frac{pr}{q} + \frac{qr}{p}$.


If only one third of the earth's surface is not covered by bodies of water, and only one third of that exposed land area is inhabitable for humans (because of natural barriers like poles and deserts), and further technological advancements make an additional half of the non-inhabitable land viable for human habitation, what fraction of the earth's surface can humans eventually live on?


Four people sit in a straight line, and each person will roll a standard twelve-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction.


Given the expression $(5x+3y-4)(2x-3y+6)$, determine the sum of the coefficients of the terms containing a nonzero power of $y$ when fully expanded.


A $\textit{composite number}$ is a number that has more than one prime factor. Find the minimum positive difference between two composite numbers whose sum is 93.


Mr. Long has 12 grandchildren. Assuming that the gender of each child is determined independently and with equal likelihood of male and female, what is the probability that Mr. Long has more grandsons than granddaughters or more granddaughters than grandsons?


Two vertices of a regular cube are chosen at random. What is the probability that these vertices will be on the same face of the cube?


Contractor Alice agreed to complete a task in 40 days. After 10 days, she found that the 12 people assigned had completed $\frac{2}{5}$ of the work. Assuming everyone works at the same rate, what is the minimum number of people she must keep on the job to ensure that the task is completed on time?


Add $528_{12} + 273_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.


Let $T$ be the set of positive integer divisors of $12^7$. Three numbers are chosen independently and at random with replacement from the set $T$ and labeled $a_1$, $a_2$, and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p$.


Find the distance between the points $(-3, 4)$ and $(6, -2)$.


The arithmetic mean of an even number of consecutive even integers is $y$. Additionally, the sum of every third integer starting from the first one in this sequence is $x$. Find the sum of the smallest and largest integers in terms of $y$ and $x$.


What is the largest possible median for the five-number set $\{y, 3y, 6, 4, 8\}$ if $y$ can be any integer?


How many five-digit numbers $N$ have the property that the four-digit number obtained by removing the leftmost digit is one eleventh of $N$?


Given 45 feet of fencing, what is the greatest possible number of square feet in the area of an equilateral triangular pen enclosed by the fencing?


Given that $z$ is a complex number such that $z + \frac{1}{z} = 2\cos 5^\circ$, find $z^{600} + \frac{1}{z^{600}}$.


Let $x$ be a real number. Consider the following five statements:

1. $0 < x^2 < 2$
2. $x^2 > 2$
3. $-2 < x < 0$
4. $0 < x < 2$
5. $0 < x - x^2 < 2$

What is the maximum number of these statements that can be true for any value of $x$?


Let's find the projection of the vector \(\begin{pmatrix} 2 \\ 0 \\ 3 \end{pmatrix}\) onto the vector \(\mathbf{u}\) if the projection of \(\begin{pmatrix} 1 \\ 2 \\ 6 \end{pmatrix}\) onto \(\mathbf{u}\) results in \(\begin{pmatrix} -1 \\ -2 \\ -6 \end{pmatrix}\).

Solution: \boxed{\begin{pmatrix} \frac{20}{41} \\ \frac{40}{41} \\ \frac{120}{41} \end{pmatrix}}


How many lattice points lie on the graph of the equation \(x^2 - y^2 = 51\)?


Find a matrix $\mathbf{N}$ that not only swaps the columns of a matrix but also multiplies the first column by 2. In other words,
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2b & a \\ 2d & c \end{pmatrix}.\]If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}}


Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of \[\frac{x + y}{xyz}.\]