The surface area of a sphere is $256\pi\text{ cm}^2$. Find the volume of the sphere in terms of $\pi$.

Solution: \boxed{\frac{2048}{3}\pi \text{ cm}^3}


A trapezoid is situated on a sloping hill, making the vertical elevations from the base to a midpoint in height unequal. The lower base of the trapezoid is four times the overall height $h$, while the upper base is five times as large. Moreover, the height to the midpoint from the lower base is $h/2$. Find the total area of this trapezoid considering a uniform slope.


A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?


Let \(a\) and \(b\) be positive real numbers such that each of the equations \(x^2 + 3ax + 4b = 0\) and \(x^2 + 4bx + 3a = 0\) has real roots. Find the smallest possible value of \(a + b\).


Find the integer $n$, $0 \le n \le 10$, such that \[n \equiv 99999 \pmod{11}.\]


For how many positive integers $n$ does $\frac{1}{n}$ yield a terminating decimal with a non-zero thousandths digit, where $n$ is less than or equal to 200?


Let \(x, y, \) and \(z\) be nonnegative real numbers such that \(x + y + z = 3.\) Find the maximum value of \(x + y^4 + z^5.\)


Let \( p \) and \( q \) be real numbers so that the roots of
\[ z^2 + (6 + pi) z + (13 + qi) = 0 \]
are complex conjugates. Find the ordered pair \( (p,q) \).


If a 4'' by 4'' square is added at each successive stage, what will be the area of the rectangle at Stage 8?


The number $0.6375$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?


Let $x^2+cx+d = 0$ be a quadratic whose roots are each three less than the roots of $5x^2-3x-8$. What is $d$?


Determine the domain of the real-valued function $$g(x) = \sqrt{3 - \sqrt{7 - \sqrt{x}}}.$$

Solution: \boxed{[0, 49]}


In hexagon $ABCDEF$, two interior angles $A$ and $B$ are $90^\circ$ and $120^\circ$ respectively. Two other angles, $C$ and $D$, are equal. The fifth angle, $E$, is half of the sum of angles $C$ and $D$, and the last angle $F$ is $30^\circ$ less than twice angle $C$. Determine the measure of the largest angle.


On a Cartesian coordinate plane, points $(1,1)$ and $(4, 5)$ are diagonally opposite corners of a square. What is the area of the square?


Every week, Bob goes to the bookstore and buys the following: 4 notebooks at $2 each, 3 magazines at $5 each, and 2 books at $15 each. This week, books are on a 20% discount. Additionally, Bob has a $10 off coupon for purchases of $50 or more. How much does Bob spend in total this week?


Simplify $(3x^5)^4$ while multiplying $x$ by a coefficient, say 2 (i.e., simplify $(3(2x)^5)^4$).


The total amount of money in John’s wallet is $6.80$, consisting of only dimes and quarters. There are four more dimes than quarters. How many dimes does John have?


Consider the geometric sequence starting with $4$, followed by $\dfrac{16}{3}$, $\dfrac{64}{9}$, and so forth. Find the tenth and fifth terms of this sequence, expressing your answers as common fractions.


Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\frac{\sqrt{(x^2 + y^2)(4x^2 + y^2)}}{xy}.\]


Given that the point $(8,5)$ is on the graph of $y=g(x)$, determine one point that must be on the graph of $3y = \frac{g(3x)}{3} + 3$. What is the sum of the coordinates of that point?


Let $a,$ $b,$ and $c$ be the roots of the polynomial
\[x^3 - 5x + 7 = 0.\]
Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$

Solution: \boxed{x^3 - 9x^2 + 22x - 5}


A circle with center $O$ has radius $10$ units, and circle $P$ has radius $4$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in simplest radical form.


Place each of the digits 4, 5, 8, and 9 in exactly one square so you end up with two two-digit numbers. Then, multiply these two numbers, subtract the smaller of the two initial two-digit numbers from the product for the smallest possible result. [asy]draw((0,.5)--(10,.5),linewidth(1));
draw((4,1)--(6,1)--(6,3)--(4,3)--(4,1),linewidth(1));
draw((7,1)--(9,1)--(9,3)--(7,3)--(7,1),linewidth(1));
draw((7,4)--(9,4)--(9,6)--(7,6)--(7,4),linewidth(1));
draw((4,4)--(6,4)--(6,6)--(4,6)--(4,4),linewidth(1));
draw((1,3)--(2,4),linewidth(1));
draw((1,4)--(2,3),linewidth(1)); [/asy]


Sandy's daughter has upgraded her playhouse in the backyard. She plans to cover one exterior wall measuring 10 feet wide by 7 feet high and the two identical sloped roof faces, each measuring 10 feet by 6 feet, with a special siding to enhance durability. The siding is now available only in 10-foot by 15-foot sections costing $\$35$ each. How much will Sandy spend on siding if she can cut the siding at home?


The function \(g(x)\) satisfies
\[g(xy) = g(x) g(y)\]
for all real numbers \(x\) and \(y\), and \(g(1) \neq 0.\) Find \(g(5)\).


A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located and crawls along a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers, find \( p + q \).


A circle passes through the point $(1, 2)$ and is tangent to the parabola $y = x^2$ at $(1, 1)$. Find the center of the circle.

Solution: \boxed{\left( 0, \frac{3}{2} \right)}


Determine how many numbers exist in the list $$ 3.5, 7.5, 11.5, 15.5, \ldots, 51.5, 55.5. $$


Suppose the base-10 numbers 348 and 76 are multiplied. The product is then written in base-8. What is the units digit of the base-8 representation?


How many integers 1-9 are divisors of the five-digit number 31,824?


For how many integers $n$ between 1 and 150 is the greatest common divisor of 21 and $n$ equal to 3?


Determine the polynomial $h(x)$ if:
\[8x^4 - 4x^2 + 2 + h(x) = 2x^3 - 6x + 4.\]

Solution: \boxed{-8x^4 + 2x^3 + 4x^2 - 6x + 2}


Suppose I have 7 shirts, 5 ties, 4 pairs of pants, and two types of shoes: formal and casual. An outfit consists of a shirt, pants, optionally a tie (which can be one of the five tie options or no tie at all), and shoes that must match the choice of wearing a tie or not (formal shoes must be worn if a tie is worn; casual shoes are worn without a tie). How many different outfits can I make?


Let \(a, b,\) and \(c\) be nonnegative real numbers such that \(a^2 + b^2 + c^2 = 1.\) Find the maximum value of
\[
2ab + 2bc\sqrt{3}.
\]


A parabola with equation $y=x^2+ax+b$ passes through the points $(2,20)$, $(-2,0)$, and $(0,b)$. What are the values of $a$ and $b$?


Find the distance between the points (0,6) and (8,0).


In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?


In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively, where $B$ is at the origin $(0,0)$, $A$ is at $(0,10)$, and $C$ is at $(12,0)$. Point $F$ is on line $BC$ extended such that $E$ is the midpoint of $\overline{BF}$. If the area of $\triangle EFC$ is 18, find the area of $\triangle DBC$.


Compute the product of the roots of the equation \[4x^4 - 3x^3 + 2x^2 - 5x + 6 = 0.\]


The sum of 25 consecutive integers is \( 5^5 \). What is their median?


Quadrilateral $ABCD$ is a square. Circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.


Determine how many different positive values of $x$ will make this statement true: there are exactly $3$ positive two-digit multiples of $x$.


Define the operation \( a \nabla b = 3 + b^a \). What is the value of \( (2 \nabla 3) \nabla 4 \)?


Keisha’s basketball team is redesigning their uniform. The seventh-graders can choose from three colors (black, gold, or red) for the shorts, and the eighth-graders have three different choices (black, white, or green) for the jersey. They won’t confer with each other regarding their choice. Calculate the probability that the shorts will be a different color from the jersey, considering that each color has an equal chance of being selected.


A taxi company charges a fixed starting fee of $20 and then a proportional amount per mile after that. If Ann's total fare was $150 after traveling 60 miles, how much would she be charged if she travels 80 miles?


In a certain region, the rate of taxation is half the amount of the income in thousands: that is, $\frac{x}{2}\%$ tax rate for an income of $x$ thousand dollars. What income, in dollars, will yield the highest take-home pay?


The number $c+\sqrt{d}$ and its radical conjugate have a sum of $0$ and a product of $9$. Find $c+d$.


Angles \(A\) and \(B\) are complementary. If the measure of angle \(A\) is 4 times angle \(B\), what is the measure of angle \(A\) and angle \(B\)?


A toy store manager received a shipment of Robot Rascals just in time for a sale event. The manager decides to display $30\%$ of them on the shelves and keeps the rest in storage. After displaying them, $40\%$ of the toys on the shelves are sold on the first day. If there are 168 Robot Rascals left in storage, how many Robot Rascals were in the initial shipment?


When $500^{150}$ is expanded out, the result is $1$ followed by how many zeros?


Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ intersect at an angle of $60^\circ$. If $DP = 21$ and $EQ = 27$, determine the length of side $DE$.


Find the minimum value of the function
\[ h(x) = x + \frac{1}{x} + \left(\frac{1}{x + \frac{1}{x}}\right)^2 \]
for $x > 0$.


At a conference with $35$ businessmen, eighteen businessmen drank coffee, and fifteen businessmen drank tea. Seven businessmen drank both coffee and tea. Additionally, six businessmen drank juice, with three of them drinking both juice and tea but not coffee. How many businessmen drank neither coffee, tea, nor juice?


What is the sum of the squares of the lengths of the **medians** of a triangle whose side lengths are $13,$ $14,$ and $15$?


In the triangle shown where angles $M$, $N$, and $L$ are the vertices, $MN$ is horizontal, and $ML$ is vertical. Suppose $\sin N = \frac{5}{13}$, and the length of $LM = 10$. Calculate the length of $LN$, the hypotenuse of triangle $MLN$.


Compute
\[\sin^6 0^\circ + \sin^6 1^\circ + \sin^6 2^\circ + \dots + \sin^6 90^\circ.\]


Solve the equation $x(5x - 11) = -2$ and express the solutions in the form $\frac{m+\sqrt{n}}{p}$ and $\frac{m-\sqrt{n}}{p}$, where $m$, $n$, and $p$ have a greatest common divisor of 1. Find $m+n+p$.


A region $S$ in the complex plane is defined by
\[ S = \{x + iy : x^2 + y^2 \leq 4\}. \]
A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac12 + \frac12i\right)z$ is also in $S$?


Simplify $\sqrt[3]{2^{10} \cdot 5^6}$ and express it in the form $c\sqrt[3]{d}$ where $c$ and $d$ are positive integers. What is $c+d$?


Find the projection of the vector $\begin{pmatrix} 6 \\ -3 \end{pmatrix}$ onto the vector $\begin{pmatrix} 3 \\ 1 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 4.5 \\ 1.5 \end{pmatrix}}


$\triangle PQR$ is similar to $\triangle XYZ$. What is the number of centimeters in the length of $\overline{YZ}$? Express your answer as a decimal to the nearest tenth.

[asy]
draw((0,0)--(10,-2)--(8,6)--cycle);
label("10cm",(6,3),NW);
label("7cm",(10.2,2.5),NE);
draw((15,0)--(23,-1.8)--(22,4.5)--cycle);
label("$P$",(10,-2),E);
label("4cm",(21.2,1.3),NE);
label("$Q$",(8,6),N);
label("$R$",(0,0),SW);
label("$X$",(23,-1.8),E);
label("$Y$",(22,4.5),NW);
label("$Z$",(15,0),SW);
[/asy]


Express $5.\overline{341}$ as a common fraction in lowest terms.


How many integers between $\frac{23}{3}$ and $\frac{65}{2}$ are multiples of $5$ or $3$?


The partial fraction decomposition of
\[
\frac{x^2 - 25}{x^3 - 3x^2 - 4x + 12}
\]
is
\[
\frac{A}{x - 1} + \frac{B}{x + 3} + \frac{C}{x - 4}.
\]
Find the product \( ABC \).


All positive integers whose digits add up to 13 are listed in increasing order. What is the sixteenth number in that list?


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 7 cards?


Let \( a, b, c \) be positive real numbers such that
\[
\left( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \right) + \left( \frac{b}{a} + \frac{c}{b} + \frac{a}{c} \right) = 10.
\]
Find the minimum value of
\[
\left( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \right)^2 + \left( \frac{b}{a} + \frac{c}{b} + \frac{a}{c} \right)^2.
\]


The forecast predicts an 80 percent chance of rain for each day of a three-day festival. If it doesn't rain, there is a 50% chance it will be sunny and a 50% chance it will be cloudy. Mina and John want exactly one sunny day during the festival for their outdoor activities. What is the probability that they will get exactly one sunny day?


What is the sum of the digits of the base $8$ representation of $888_{10}$?


Rationalize the denominator of $\frac{5}{\sqrt{180} - 2}$.

Solution: \boxed{\frac{5(3\sqrt{5} + 1)}{88}}


If $x$ and $y$ are positive real numbers such that $6x^3 + 12xy^2 = 3x^2y + 3x^4$, what is the value of $x$?


How many ways are there to put 6 balls in 4 boxes if the balls are not distinguishable and neither are the boxes, with at least one ball in each box?


What is the sum of all the two-digit positive integers whose squares end with the digits 25?


Consider the function $g(x) = x^2 + 3\sqrt{x}$. Evaluate $3g(3) - g(9)$.

Solution: \boxed{-63 + 9\sqrt{3}}


For how many positive integers $x$ is $x^2 + 10x + 25$ between 30 and 60?


Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,200 \pi).$


In triangle $ABC$, $AB = 12$, $AC = 10$, and $BC = 16$. The centroid $G$ of triangle $ABC$ divides each median in the ratio $2:1$. Calculate the length $GP$, where $P$ is the foot of the perpendicular from point $G$ to side $BC$.


A sphere is inscribed in a cube. The edge of the cube measures 8 inches. Calculate both the volume of the inscribed sphere and the free space volume between the inscribed sphere and the cube. Express your answer in terms of $\pi$.

Solution: \boxed{512 - \frac{256}{3} \pi}


Consider a rectangle and four circles, each with a radius of 3 cm. The circles are arranged at the corners of the rectangle in such a way that each circle touches two adjacent sides of the rectangle. What is the area, in square centimeters, of the rectangle?

Solution: \boxed{144 \text{ cm}^2}


What integer is closest to the value of $\sqrt[3]{5^3+7^3}$?


Simplify $\frac{x+3}{4} - \frac{5-2x}{3}$.  Express your answer as a single fraction.

Solution: \boxed{\frac{11x - 11}{12}}


Solution: \boxed{243 \cdot \sqrt[3]{3}}


Let $\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If $AD = 2$ and $BD = 3$, then what is $CD$?


There are two concentric spheres with radii 4 units and 7 units respectively. What is the volume, in cubic units, of the region within the larger sphere and not within the smaller sphere? Express your answer in terms of $\pi$.


Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle such that the triangle is divided into a rectangle and two smaller right triangles. The area of one of the two smaller right triangles is $n$ times the area of the rectangle. If the rectangle has length twice its width, what is the ratio of the area of the other small right triangle to the area of the rectangle? Express your answer as a common fraction in terms of $n$.


Determine how many perfect squares less than 20,000 can be represented as the difference of two consecutive perfect squares.


Let \( m = 2^5 \cdot 3^4 \cdot 4^5 \cdot 6^6 \). Determine the total number of natural-number factors of \( m \).


A segment with endpoints at $A(1, -3)$ and $B(11, 3)$ is extended through $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates for point $C$?

Solution: \boxed{(16, 6)}


The projection of $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ onto a certain vector $\mathbf{u}$ is $\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$. Find the projection of $\begin{pmatrix} 4 \\ -1 \\ 5 \end{pmatrix}$ onto $\mathbf{u}$.

Solution: \boxed{\begin{pmatrix} \frac{14}{3} \\ -\frac{7}{3} \\ \frac{7}{3} \end{pmatrix}}


If $n$ is a positive integer, then let $g(n)$ be the sum of the digits of $\frac{1}{3^{n}}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $g(n) > 15$?


Solution: \boxed{\frac{1}{\sqrt{3}} - \frac{1}{\sqrt{5001}}}


For a positive real number $x > 1,$ consider the Riemann zeta function $\zeta(x)$. Compute
\[\sum_{k=3}^\infty \left\{\frac{\zeta(2k)}{k}\right\}.\]
Note: For a real number $x,$ $\{x\}$ denotes the fractional part of $x.$


The graph of $y=g(x)$, a piecewise linear function, is shown in black below. This graph consists of four line segments, and the distance between grid lines is $1$ unit. Determine the sum of the $x$-coordinates of all points where $g(x) = 2$.

[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);
        yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
    }
    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(-5,5,-5,5);
draw((-4,-4)--(-2,0)--(0,-2)--(2,2)--(4,0),black);
[/asy]


Add $234_8 + 157_8.$ Express your answer in base 8.


What is the base $2$ representation of $123_{10}$?


Cybil and Ronda are friends. The 15 letters from their names are placed on identical cards so that each of 15 cards contains one letter (Cybil's name contributing 6 letters and Ronda giving 9 letters). Without replacement, three cards are selected at random from the 15 cards. What is the probability that at least one letter is from each friend's name? Express your answer as a common fraction.


John now has six children. What is the probability that exactly three of them are girls?


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


How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes if no box can hold more than 4 balls?


The median of a set of consecutive odd integers is 144. If the greatest integer in the set is 153, what is the smallest integer in the set?


Determine how many different whole numbers could be the length of the diagonal (dashed line) in the following quadrilateral:

[asy]
draw((0,0)--(6,6)--(15,2)--(8,-9)--cycle,linewidth(0.7));
draw((0,0)--(15,2),dashed);
label("10",(3,3),NW);
label("12",(10.5,4),NE);
label("18",(11.5,-3.5),SE);
label("15",(4,-3.5),SW);
[/asy]


The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS.


Let $\omega$ be a nonreal root of $z^3 = 1.$ Find the number of ordered pairs $(a,b)$ of integers such that $|a \omega + b| = 2.$


Let $R$ be the set of all possible remainders when a number of the form $3^n$, where $n$ is a nonnegative integer, is divided by 500. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by 500.


In the new version of the game of Frood, dropping $n$ froods still earns $\frac{n(n+1)}{2}$ points. However, eating $n$ froods now earns $20n$ points. What is the least number of froods for which dropping them will earn more points than eating them?


The number of inches in the perimeter of a square equals the number of square inches in the area of its inscribed circle. What is the diameter, in inches, of the circle? Express your answer in terms of pi and in simplest form.


Let $g(x) = x^2 - 4x + 3$. How many distinct real numbers $c$ satisfy $g(g(g(g(c)))) = 5$?


Mike wants to purchase a rectangular mirror (the shaded region) that fits exactly inside a frame. The outer perimeter of the frame measures 100 cm by 140 cm. The width of each side of the frame is 12 cm. What is the area of the mirror?

Solution: \boxed{8816 \text{ cm}^2}


Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) - y) = g(x) + g(g(y) - g(-x)) + 2x \] for all real numbers \( x \) and \( y \).

Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \).


Find the value of $x$ such that $\sqrt{x + 16} = 12$ and then subtract 8 from the squared result.


Of the numbers $\frac{8}{12}, \frac{5}{6}$, and $\frac{9}{12}$, which number is the arithmetic mean of the other two?


Phil rolls 8 fair 6-sided dice. What is the probability that at least two dice show the same number?


How many distinct terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)\] assuming that terms involving the product of $a$ and $e$, and $b$ and $f$ are identical and combine into a single term?


Let $p$ and $q$ satisfy $pq = 15$ and $p + q = 8$. What are the values of $p^2 + q^2$ and $p^3 + q^3$?


How many integers, $x$, satisfy the inequality $|7x - 5| \le 9$?

Solution: \boxed{3 \text{ integers}}


Determine the sum of the two solutions to the equation $81 - 27x - x^2 = 0$.


If 20 bahs are equal to 30 rahs, and 12 rahs are equal in value to 20 yahs, how many bahs are equal in value to 1200 yahs?


Julien and his Canadian friend Lucas are at a café in Lyon that accepts both euros and Canadian dollars. They want to buy a pie, but neither of them has enough money. If the pie costs 12 euros and Lucas has a Canadian ten-dollar bill, how many euros does Julien need to contribute to the cost of the pie if 1 euro = 1.5 CAD?

Solution: \boxed{5.33 \text{ euros}}


My club has 30 members. However, 3 of the members are new and cannot yet be part of the executive committee. In how many ways can I choose a 5-person executive committee from the remaining eligible members?


In a regular tetrahedron embedded in 3-space, the centers of the four faces are the vertices of another smaller tetrahedron. The edge length of the original tetrahedron is 3 units. Calculate the ratio of the volume of the smaller tetrahedron to that of the original tetrahedron. Provide the solution as simplified fractions.


How many four-digit numbers whose digits add up to $10$ are divisible by $11$?


How many positive integer multiples of $3003$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 99$?


The line $y = -\frac{1}{2}x + 8$ crosses the x-axis at $P$ and the y-axis at $Q$. Point $T(r, s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?


The mean (average) of 8, 15, and 21 is equal to the mean (average) of 18 and $x$. What is the value of $x$?


The sum of the areas of two squares is 85, while the difference in their areas is 41. Find the sum of their perimeters.

Solution: \boxed{4(\sqrt{63} + \sqrt{22})}


Calculate the sum of the coefficients of the polynomial $4(2x^{8} + 5x^5 - 6) + 9(x^6 - 8x^3 + 4)$ when fully simplified.


Solution: \boxed{-46 - 9i}


Megan has lost Fatima's phone number. Megan knows that the first three digits are either 296, 299, or 298. The remaining four digits are 0, 1, 6, 7, and 8, but she isn't sure of the order of these digits. Two of these digits are used twice. If Megan randomly dials a seven-digit number that meets these conditions, what is the probability that she dials Fatima's correct number? Express your answer as a common fraction.


For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate?


The table below represents the distance in miles between selected international cities. If two different cities from the table are chosen at random, what is the probability that the distance between them is less than $8000$ miles? Express your answer as a common fraction.

\begin{tabular}{|c|c|c|c|c|}
\hline
& Berlin & Sydney & Tokyo & Miami \\ \hline
Berlin & & 9900 & 5800 & 5000 \\ \hline
Sydney & 9900 & & 4800 & 9200 \\ \hline
Tokyo & 5800 & 4800 & & 7400 \\ \hline
Miami & 5000 & 9200 & 7400 & \\ \hline
\end{tabular}


The polynomial $g(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $g(3i) = g(3+i) = 0$. What is $a+b+c+d$?


Solution: \boxed{13 - 2i}


For how many integer values of $n$ between 1 and 565 inclusive does the decimal representation of $\frac{n}{570}$ terminate?


Find the area of the region bounded by the graph of \( r = 2\sec \theta \), the graph of \( r = 2\csc \theta \), the \( x \)-axis, and the \( y \)-axis.


Circle C is in the interior of circle D. The diameter of circle D is 20 cm. What is the diameter of circle C for which the ratio of the shaded area to the area of circle C is 7:1?

Solution: \boxed{5\sqrt{2}\text{ cm}}


If the system of equations
\[
4x - 2y = c,
\]
\[
6y - 12x = d.
\]
has a solution, find \( \frac{c}{d} \), assuming \( d \neq 0 \).


Let $g(x)$ be the function defined on $-1\le x\le 1$ by the formula $$g(x) = 1 - \sqrt{1 - (x-1)^2}.$$ This is a graph of $y=g(x)$. If a graph of $x=g(y)$ is overlaid on the graph above, then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth?


Molly flips a fair coin eight times, each time surprisingly landing on heads. After the coin flips, she rolls a fair six-sided die. Calculate the probability that:
a) Molly will flip tails on her next coin flip, and
b) the die roll results in either a 1 or a 2.


Find the maximum value of $2^x - 16^x$ over all real numbers $x$.


A solid has a square base of side length \( s \). The top face is also a square, parallel to the base with length \( 1.5s \), and the height of the solid from the base to the top face is \( h \). Given that \( s = 8 \) and \( h = 10 \), calculate the volume of the solid.


Determine the smallest positive integer value of $e$ for the polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
with integer coefficients, given that its roots are $-3$, $7$, $8$, and $-\frac{1}{4}$.


Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{8}{9}.\]


Find the number of functions \(f : \mathbb{R} \to \mathbb{R}\) such that
\[f(x + y) f(x - y) = (f(x) - f(y))^2 - 4x^2 f(y)\]
for all real numbers \(x\) and \(y\).


Find a positive integer that is divisible by 18 and has a square root between 24 and 24.5.


Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/2000$. Find $n$.


If 5 taps are equivalent to 4 tops, and 3 tops are equivalent to 12 teps, how many taps are equivalent to 48 teps?

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


Find the number of complex numbers \( z \) such that \( |z| = 1 \) and
\[
\left| \frac{z}{\overline{z}} - \frac{\overline{z}}{z} \right| = 2.
\]


A collection of six positive integers has a mean of 5.5, a unique mode of 4, and a median of 6. If a 10 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth.


Calculate the value of the expression $[(a \times b - c) - (a + b \times c)] - [(a \times c - b) - (a - b \times c)]$, where $a=3$, $b=7$, and $c=2$.


What is the greatest number of Saturdays that can occur in the first $45$ days of a year?


A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $252$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$.


Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(x) + f(y) - f(x)f(y) \ge 2\]
for all real numbers $x,$ $y.$


Suppose that $(u_n)$ is a sequence of real numbers satisfying
\[u_{n+2} = 3u_{n+1} + 2u_n\]
and that $u_3 = 10$ and $u_6 = 256$. What is $u_5$?


The third and fourth terms of a geometric sequence are 24 and 36, respectively. What is the first term of the sequence?


In a redesign of his company's logo, Wei decided to use a larger square and more circles. Each circle is still tangent to two sides of the square and its adjacent circles, but now there are nine circles arranged in a 3x3 grid instead of a 2x2 grid. If each side of the new square measures 36 inches, calculate the total shaded area in square inches.

Solution: \boxed{1296 - 324\pi}


Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p  + q + r + s = 10$ and
\[ pq + pr + ps + qr + qs + rs = 20. \]
Find the largest possible value of $s$.

Solution: \boxed{\frac{5 + \sqrt{105}}{2}}


A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $81\,972$ is divisible by $9$ because the sum of its digits, $8 + 1 + 9 + 7 + 2 = 27$, is divisible by $9.$ If $994\,561\,2A5$ is divisible by $9,$ where $A$ represents one digit, what is the sum of the possible values of $A?$


Let $R$ be the set of all possible remainders when a number of the form $3^n$, where $n$ is a nonnegative integer, is divided by 500. Let $S$ be the sum of all elements in $R$. Determine the remainder when $S$ is divided by 500.


A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 8 and an altitude of 16, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.


A five-gallon container in the shape of a perfect cylinder has its height tripled and the diameter doubled. How many gallons does the new container hold?


Find the domain of the function $g(x) = \cot(\arcsin(x^3))$.

Solution: \boxed{[-1, 0) \cup (0, 1]}


A hemisphere-shaped bowl with radius 2 feet is filled with chocolate. However, due to spillage, only 90% of the chocolate is salvaged and evenly distributed between 64 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in feet?


What is the area, in square units, of a regular octagon inscribed in a circle whose area is $400\pi$ square units? Express your answer in simplest radical form.

Solution: \boxed{800 \sqrt{2}}


Let \(p\) and \(q\) satisfy \(pq = 8\) and \(p+q = 7\). What is the value of \(p^2 + q^2\) and \(p^3 + q^3\)?

Solution: \boxed{p^2+q^2 = 33 \quad \text{and} \quad p^3+q^3 = 175}


What is the remainder when \(2021 \cdot 2022 \cdot 2023 \cdot 2024 \cdot 2025\) is divided by 17?


How many positive integers smaller than $500,000$ are powers of $2$, but are not powers of $4$?


Suppose there exist distinct complex numbers $r$, $s$, $t$, and $u$ such that for every complex number $z$, we have
\[
  (z - r)(z - s)(z - t)(z - u) = (z - cr)(z - cs)(z - ct)(z - cu).
\]
Compute the number of distinct possible values of $c$.


The function $f : \mathbb{R} \to \mathbb{R}$ satisfies
\[f(x) + 3f(1 - x) = 3x^2 + x\]
for all $x$. Determine $f(5)$.


The students in Mr. Thompson's literature class are assigned to read a $900$-page book. Four friends, Alice, Bob, Chandra, and Daniel are in the class. Alice reads a page in $15$ seconds, Bob reads a page in $45$ seconds, Chandra reads a page in $30$ seconds, and Daniel reads a page in $60$ seconds. Chandra and Daniel decide to team read the book. Chandra will start from page $1$ to a certain page, and Daniel will read from the next page to page $900$. Calculate the last page that Chandra should read so that she and Daniel spend the same amount of time reading.


A right circular cylinder, whose height is twice its radius, is inscribed in a right circular cone. The cone has a diameter of 14 and an altitude of 16, with the axes of the cylinder and cone aligned. Determine the radius of the cylinder and express your answer as a common fraction.


Find the angle in radians between the vectors $\begin{pmatrix} 3 \\ -2 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}$.

Solution: \boxed{\cos^{-1}\left(\frac{-2}{\sqrt{238}}\right)}


Solution: \boxed{-2 - \sqrt{3}}


Let $f(x)$ be a polynomial of degree 3009 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{3009}.$  There are exactly 1505 distinct values among
\[|r_1|, |r_2|, \dots, |r_{3009}|.\]What is the minimum number of real roots that $f(x)$ can have?


A right circular cylinder is inscribed in a right circular cone. The cone has diameter 12 and altitude 18, and the axes of the cylinder and cone coincide. The diameter of the cylinder is equal to its height. Find the radius of the cylinder. Express your answer as a common fraction.


Each letter represents a non-zero digit. What is the value of $t?$ \begin{align*}
c + o &= u \\
u + n &= t + 1 \\
t + c &= s \\
o + n + s &= 15
\end{align*}


Let $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 4:1.$ Suppose further that $\overrightarrow{A} = (1,2)$ and $\overrightarrow{B} = (4,3)$. Then, express $\overrightarrow{P}$ as a combination of $\overrightarrow{A}$ and $\overrightarrow{B}$, and determine the ordered pair $(t, u)$ such that
\[\overrightarrow{P} = t\overrightarrow{A} + u\overrightarrow{B}.\]

Solution: \boxed{\left(\frac{1}{5}, \frac{4}{5}\right)}


A math symposium will hold a series of lectures with seven different lecturers. Dr. Smith's lecture depends on Dr. Jones's lecture, so Dr. Smith must be scheduled at some time after Dr. Jones. Additionally, Dr. White's lecture depends on both Dr. Jones's and Dr. Smith's lectures, so Dr. White must speak after both of them. In how many ways can the seven lecturers be scheduled?


If $m$ and $n$ are even integers, how many terms in the expansion of $(m+n)^8$ are even?


The graph of the rational function $\frac{1}{q(x)}$ displays vertical asymptotes at $x = -2$ and $x = 3$. Given that $q(1) = -10$, determine the expression for $q(x)$.

Solution: \boxed{\frac{5}{3}x^2 - \frac{5}{3}x - 10}


Define a new sequence of complex numbers by \( z_1 = 1 \) and
\[ z_{n + 1} = z_n^2 \cdot (1+i) \]
for all \( n \ge 1 \). In the complex plane, find the distance of \( z_5 \) from the origin.


Suppose that $A$ and $B$ are digits in base $d > 7$ such that $\overline{AB}_d + \overline{BA}_d = 202_d$. Find $A_d - B_d$ in base $d$.


How many different positive integers can be represented as a difference of two distinct members of the set $\{2, 3, 7, 8, 9, 12\}$?


Determine the number of digits in the value of $2^{15} \times 3^2 \times 5^{12}$.


To perform maintenance at the center $C$ of a circular garden, a maintenance team uses a 20-foot plank from point $A$ to $B$ and then a 12-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$. Determine the area of the circular garden. Express your answer in terms of $\pi$.

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


Three cards are dealt sequentially from a standard deck of 52 cards without replacement. What is the probability that the first card is an Ace, the second card is a 10, and the third card is a Jack?


The product of three consecutive integers, including a negative number, is -336. What is their sum?


Find the remainder when $7^{7^{7^7}}$ is divided by 1000.


How many three-digit positive integers \( y \) satisfy \( 4528y + 563 \equiv 1407 \pmod{29} \)?


If $0.6 = \frac{y}{468 + x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.


Let $\mathbf{p},$ $\mathbf{q},$ and $\mathbf{r}$ be unit vectors such that
\[\mathbf{p} + \mathbf{q} + 2 \mathbf{r} = \mathbf{0}.\]Find the angle between $\mathbf{p}$ and $\mathbf{q},$ in degrees.


A prism has 18 edges. How many faces does it have?


A ball was floating in a river when the river froze. The ball was removed (without breaking the ice), leaving a hole $30$ cm across at the top and $10$ cm deep. What was the radius of the ball (in centimeters)?


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 that a real number $x$ satisfies \[\sqrt{64-x^2} - \sqrt{36-x^2} = 4.\] What is the value of \[\sqrt{64-x^2} + \sqrt{36-x^2}?\]


What is the greatest common factor of 36 and 54?


The base-10 numbers 123 and 58 are multiplied. The product is then written in base-8. What is the units digit of the base-8 representation?


Evaluate $\left|\frac{3}{4} - 3i\right|$, then add $\frac{5}{12}$ to the result.

Solution: \boxed{\frac{9\sqrt{17} + 5}{12}}


The fourth and sixth terms of a geometric sequence of real numbers are $6!$ and $7!$ respectively. What is the first term?

Solution: \boxed{\frac{720 \sqrt{7}}{49}}


What is the number of degrees in the larger angle formed by the hour and minute hands of a clock at 3:30? Express your answer as a decimal to the nearest tenth.


Captain Zarnin of Planet Hvan has expanded his battle station and now has five job openings: Assistant Engineer, Weapons Maintenance, Field Technician, Radio Specialist, and Security Analyst. Zarnin has received 30 resumes and found himself less than thrilled about one-third of them, and does not hire them. The rest, he thinks, could fill any of the open posts. In how many ways can Zarnin staff his battle station?


Brand X soda now offers, "We will give you 30$\%$ more soda than Brand Y for a total price that is 15$\%$ less than Brand Y's price!" What is the new ratio of the unit price of Brand X soda to the unit price of Brand Y soda? Express your answer as a common fraction.


Consider a "36-inch" television screen where the ratio of the horizontal length to the height is still $4:3$. What is the horizontal length (in inches) of this television screen? Additionally, calculate the screen area.

Solution: \boxed{622.08 \text{ square inches}}


Determine the number of non-similar regular 1500-pointed stars, where a regular $n$-pointed star follows similar geometric and combinatorial properties as described for the 1000-pointed star.


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


Jonas selected an angle $x$ strictly between 30 and 120 degrees and wrote the values of $\sin x$, $\cos x$, $\tan x$, and $\sec x$ on four different cards. He then gave those cards to four students: Matilda, Philippa, Gordon, and Cynthia, one card to each. After sharing their values with each other, only Matilda was certain which trigonometric function produced the value on her card. Compute the sum of all possible values that Jonas wrote on Matilda's card.


Let $P$ be a point in coordinate space where all coordinates are positive integers and their sum is a prime number. The line between the origin and $P$ is drawn. The angle between this line and the $x$-, $y$-, and $z$-axis are $\alpha, \beta,$ and $\gamma,$ respectively. Given $\cos \alpha = \frac{1}{4}$ and $\cos \beta = \frac{2}{5},$ determine $\cos \gamma.$


The planning committee at a corporation has 12 members, of which 5 are managers. A five-person management subcommittee must be formed that includes at least two managers. How many distinct management subcommittees are possible?


Two cards are drawn sequentially from a standard 52-card deck. What is the probability that the first card is a spade and the second card is an ace?


Consider a triangle with side lengths $\ln 20$, $\ln 90$, and $\ln m$, where $m$ is a positive integer. Determine the number of possible values for $m$ to form a triangle with positive area.


How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt[3]{500}$?


What is the smallest possible area, in square units, of a right triangle with two sides measuring $7$ units and $8$ units?


What is the value of $n$ in the equation $n + (n + 1) + (n + 2) + (n + 3) = 20$?


Thirty gremlins and fifteen imps are at the Annual Mischief Convention. Due to a spell, each imp shakes hands with only half of the gremlins. All gremlins are friendly and shake hands with each other as well as with any imp that approaches them. Each pair of creatures shakes hands at most once. How many handshakes occur at this convention?


Suppose that $a$ and $b$ are positive integers such that $a-b=8$ and $\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 16$. Find the smallest possible value of $b$.


If \( g(x) = \frac{x+2}{x-5} \), then for what value of \( x \) is \( g^{-1}(x) \) undefined?


For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 20\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 50x + c.\] What is $f(1)$?


Calculate the number of real solutions $(x,y,z,w)$ to the system of equations:
\begin{align*}
x &= z+w+zwz, \\
y &= w+x+wzx, \\
z &= x+y+xyx, \\
w &= y+z+zyz.
\end{align*}


Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at an angle of $45^\circ$. If $AD = 12$ and $BE = 16$, then calculate the area of $\triangle ABC$.


If $\frac{1}{8}$ of $2^{36}$ is $4^x$, then what is the value of \( x \) ?


Point $(x, y)$ is randomly selected from the rectangular region defined by vertices $(0, 0), (3013, 0), (3013, 3014),$ and $(0, 3014)$. What is the probability that $x > 8y$?


Triangle $XYZ$ is isosceles with angle $X$ congruent to angle $Z$. The measure of angle $Z$ is five times the measure of angle $Y$. What is the number of degrees in the measure of angle $X$?


Given two functions \( r(x) = |x| - 3 \) and \( s(x) = -|x| \), evaluate \( s(r(x)) \) at \( x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \) and find the sum of these eleven values.


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)--(7,7)--(15,2)--(9,-10)--cycle,linewidth(0.7));
draw((0,0)--(15,2),dashed);
label("11",(3.5,3.5),NW);
label("13",(11,4.5),NE);
label("21",(12, -4),SE);
label("15",(4.5,-4),SW);
[/asy]


A square is inscribed in the ellipse
\[\frac{x^2}{4} + \frac{y^2}{8} = 1,\]
such that its sides are parallel to the coordinate axes. Find the area of this square.


The points $(0,5)$ and $(5,0)$ are adjacent vertices of a square. What is the area of the square?


I have a drawer containing 7 forks, 7 spoons, 7 knives, and 7 teaspoons. If I reach in and randomly remove four pieces of silverware, what is the probability that I get one fork, one spoon, one knife, and one teaspoon?


What is the remainder when \( 5x^3 - 18x^2 + 31x - 40 \) is divided by \( 5x - 10 \)?


In a triangle, two of the side lengths are 5 and 12, and the angle between them is $150^\circ.$ Find the length of the third side.

Solution: \boxed{\sqrt{169 + 60\sqrt{3}}}


An organization starts with six leaders and some number of regular members, totaling twenty people. Each year, all leaders are replaced, and each regular member recruits three new members. After this expansion, six new leaders are elected from outside the organization. How many people will there be in the organization after four years?


How many degrees are there in the measure of angle \(Q\)?

[asy]
size (6cm,6cm);
pair A,B,C,D,E,F;

A=(0,1);
B=(3,0);
C=(6,0.5);
D=(7,3.5);
E=(4.5,6);
F=(1.5,5);

draw (A--B--C--D--E--F--A,linewidth(1));

label("$Q$",A,SW);
label("$138^\circ$",B,S);
label("$85^\circ$",C,SE);
label("$130^\circ$",D,NE);
label("$120^\circ$",E,N);
label("$95^\circ$",F,NW);

draw(anglemark(B,A,F),blue);

[/asy]


Find $m$ if $m\cdot m! + 2m! = 5040$, where $m! = m\cdot (m-1)\cdot (m-2)\cdots 2\cdot 1$.


Katie’s savings account balance over five consecutive months is depicted in the following graph. Calculate her average monthly balance during this period. The graphical data is as follows:
- January: $120
- February: $240
- March: $180
- April: $180
- May: $240


Determine the average number of rounds played by each golfer from a new set of data presented in the following chart:
```
Rounds of Golf Played:  4, 5, 6, 7, 8
Number of Golfers:      3, 6, 3, 2, 7
```
Express your answer to the nearest whole number.


In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel with lengths of 10 and 30 units respectively, and the altitude from $A$ to $CD$ is 18 units. The diagonals $AC$ and $BD$ intersect at point $G$, which divides $AC$ in the ratio 1:2 starting from $A$. Points $E$ and $F$ are the midpoints of $AD$ and $BC$ respectively. Calculate the area of quadrilateral $EFCD$ in square units.


How many distinct ordered pairs of positive integers $(m, n)$ are there such that the sum of the reciprocals of $m$ and $n$ equals $\frac{1}{6}$?


What is the greatest common divisor of $7!$ and $\frac{9!}{4!}?$ Express your answer as an integer.


How many integers between 500 and 2500 are divisible by all of the numbers 10, 15, and 18, and are also divisible by 7?


**Two sides of a right triangle have the lengths 6 and 8. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth.**


Let $\omega$ be a nonreal root of $x^4 = 1.$ Compute
\[(1 - \omega + \omega^3)^4 + (1 + \omega - \omega^3)^4.\]


The complex numbers $\alpha, \beta,$ and $\gamma$ are zeros of a polynomial $Q(z) = z^3 + sz + t$ such that $|\alpha|^2 + |\beta|^2 + |\gamma|^2 = 360$. The points corresponding to $\alpha, \beta,$ and $\gamma$ in the complex plane form an equilateral triangle. Calculate the square of the side length of this triangle.


Let \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) be unit vectors such that the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\theta\), and the angle between \(\mathbf{c}\) and \(\mathbf{a} \times \mathbf{b}\) is \(90^\circ - \theta\). If \(\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{1}{2},\) find the smallest possible value of \(\theta,\) in degrees.


A Mersenne number is defined as \(2^n - 1\) where \(n\) can be any integer that is a power of 2. Find the largest Mersenne number that is prime and less than 500.


How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^3\cdot5^7}$?


Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number entered on five slips. Five slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all five slips bear the same number. Let $q$ be the probability that three of the slips bear a number $a$ and the other two bear a number $b\ne a$. What is the value of $q/p$?


Jane is currently 30 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, both Dick's and Jane's ages will be two-digit numbers, and Dick's age will be obtained by interchanging the digits of Jane's age. Let $d$ be Dick's current age. How many ordered pairs of positive integers $(d, n)$ are possible?


A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?


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

[asy]
import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.5,xmax=3.5,ymin=-4.25,ymax=6.25;

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 f1(real x){return sgn(3x-3)*abs(3x-3)^(1/3)+3*sgn(3x-3)*abs(3x-3)^(1/5)+abs(3x-3)^(1/7);}
draw(graph(f1,xmin,xmax),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=q(x)$",(xmax+0.25,f1(xmax)),E);
[/asy]

The value of $q(1)$ is an integer. What is it?


For certain real values of $a, b, c,$ and $d,$ the equation $x^4 + ax^3 + bx^2 + cx + d = 0$ has four non-real roots. The product of two of these roots is $8 - 3i$ and the sum of the other two roots is $-2 + 6i$ where $i^2 = -1.$ Find $b.$

Solution: \boxed{b = 56}


Suppose we have a right triangle $DEF$ with the right angle at $E$ such that $DF = \sqrt{85}$ and $DE = 7.$ A circle is drawn with its center on $DE$ such that the circle is tangent to $DF$ and $EF.$ If $Q$ is the point where the circle and side $DF$ meet, what is $FQ$?


Find the largest solution to the equation \[\lfloor x \rfloor = 10 + 50\{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$


Twelve points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the twelve points as vertices?


What is the largest perfect square factor of 4410 and what is the least common multiple (LCM) of this factor with 18?


Two mathematicians were both born in the last 300 years. Each lives (or will live) to be 80 years old. Each mathematician is equally likely to be born at any point during those 300 years. What is the probability that they were contemporaries for any length of time?


Consider a regular tetrahedron with side length 1. A tetrahedron has 4 vertices. Calculate the sum of the areas of all triangles whose vertices are also vertices of this tetrahedron. Express your answer in the form \( a + b\sqrt{c} \), where \( a \), \( b \), and \( c \) are integers. Find \( a + b + c \).


What is the number of square centimeters in the area of a circle with a diameter of 8 meters? Express your answer in terms of \( \pi \).


Triangle $DEF$ has vertices with coordinates $D(1, 4),$ $E(6, 9),$ and $F(-5, 7)$. The triangle is reflected about line $M$. The image points are $D'(1, -6),$ $E'(6, -11),$ and $F'(-5, -9)$. What is the equation of line $M$?

Solution: \boxed{y = -1}


A $6 \times 9$ rectangle is used to form two different cylinders by rolling the rectangle along its length and its width. Determine the ratio of the volume of the larger cylinder to the volume of the smaller cylinder. Express your answer as a common fraction.


Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $256\sqrt{2}$ square feet and $576\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?


Solution: \boxed{\csc 20}


Determine all real numbers \( b \) such that the inequality \( |x^2 + 2bx + 2b| \leq 1 \) has exactly one solution in \( x \).


Let $a,$ $b,$ $c$ be positive real numbers. Determine the set of all possible values of
\[ g(a,b,c) = \frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}. \]

Solution: \boxed{(1, 2)}


A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?


There are four points that are $7$ units from the line $y = 20$ and $10$ units from the point $(10, 20)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 100) < 0.\]


How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?


Let $P$ be a point in coordinate space, where all the coordinates of $P$ are positive. The line between the origin and $P$ is drawn. The angle between this line and the $x$-, $y$-, and $z$-axis are $\alpha,$ $\beta,$ and $\gamma,$ respectively. If $\cos \alpha = \frac{2}{5}$ and $\cos \beta = \frac{3}{5}$, and it is known that $z = 2y$, find $\cos \gamma.$


It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the \( x \) and \( y \) axes and so that the medians to the midpoints of the legs lie on the lines \( y = 4x + 1 \) and \( y = nx + 2 \). Determine the number of different constants \( n \) for which such a triangle exists.


Find the number of positive integers $n \le 1500$ such that $21n$ is a perfect square.


Let \( z \) be a nonreal complex number such that \( |z| = 2 \). Find the real part of \( \frac{2}{1 - z} \).


How many positive divisors does the number $36$ have, and what is their sum?


Let $a$ and $b$ be real numbers. One of the roots of $x^3 + ax + b = 0$ is $2 + i \sqrt{5}$. Find $a + b$.


A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \end{pmatrix} + t \begin{pmatrix} 4 \\ 5 \end{pmatrix}.\]
A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -7 \\ 14 \end{pmatrix} + u \begin{pmatrix} 2 \\ 3 \end{pmatrix}.\]
If $\theta$ is the acute angle formed by the two lines, then find $\cos \theta.$


In a right triangle JKL, the hypotenuse KL measures 13 units, and side JK measures 5 units. Determine $\tan L$ and $\sin L$.


Using only the digits 5, 6, 7, 8, how many positive nine-digit integers can be made that are palindromes?


At the end of a professional bowling league, the bottom two of the top seven bowlers (#6 and #7) have a playoff match, and the loser receives $7$th position. The winner of this match faces #5 in another game. The loser is awarded the $6$th position, and the winner continues to face #4, #3, #2, and #1 consecutively in separate matches. Similar to the original structure, each game's loser receives the prize one rank better than the previous game's position, leading up to the match with #1 where the winner gets the 1st prize and the loser gets the 2nd prize. Calculate the number of different orders in which bowlers #1 through #7 could be awarded prizes.


Four primes $p$, $q$, $r$, $s$ form an increasing arithmetic sequence with $p > 5$ and common difference 4. What is the ones digit of $p$?


What is the average of all the integer values of $M$ such that $\frac{M}{70}$ is strictly between $\frac{2}{5}$ and $\frac{3}{10}$?


The volume of a rectangular solid is \( 512 \text{ cm}^3 \), its total surface area is \( 384 \text{ cm}^2 \), and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.

Solution: \boxed{96 \text{ cm}}


Let \(a\), \(b\), and \(c\) be positive real numbers such that
\[
\left( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \right) + \left( \frac{b}{a} + \frac{c}{b} + \frac{a}{c} \right) = 9.
\]
Find the value of
\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]


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?


How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, and F? (Note that A, E, and F are vowels, any word is valid, not just English language words, and letters may be used more than once.)


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 7 cards?


Let vectors \(\mathbf{a} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}\). There exist scalars \(p\), \(q\), and \(r\) such that
\[
\begin{pmatrix} 3 \\ 7 \\ 0 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r (\mathbf{a} \times \mathbf{b}).
\]
Find \(r\).


What is the last digit of the decimal expansion of $\frac{1}{2^{15}}$?


What is the greatest number of digits in the product of a 5-digit whole number and a 4-digit whole number?


The school's boys basketball team has 16 players, including a set of twins, Bob and Bill, and a set of triplets, Chris, Craig, and Carl. In how many ways can we choose 7 starters if all three triplets must be in the starting lineup and both twins must either both be in the lineup or both not be in the lineup?


Consider the functions:
$$
\begin{array}{ccc}
f(x) & = & 4x^2 + 3x + 7 \\
g(x) & = & x^2 - kx + 5
\end{array}
$$
Find the value of $k$ if $f(5) - g(5) = 0$.


At the start of a cycling trip, you cycle at 15 miles per hour helped by a tailwind. Later, as you turn back against the wind, your effective speed drops to 10 miles per hour due to wind resistance. If you travel a total of 150 miles in 12 hours, for how many hours did you have the help of the tailwind? Express your answer as a common fraction.


Solution: \boxed{(4y - 6)^2}


The sum of 81 consecutive integers is $9^5$. What is their median?


In how many ways can 7 distinct beads be placed on a bracelet? (Note that two arrangements are the same if one can be rotated or reflected to produce the other.)


Solve the inequality
\[|x - 3| + |x + 4| < 10.\]

Solution: \boxed{(-5.5, 4.5)}


Let $x$ and $y$ be real numbers such that
\[ xy - \frac{x}{y^2} - \frac{y}{x^2} + \frac{x^2}{y^3} = 4. \]
Find the product of all possible values of $(x - 2)(y - 2)$.


Solution: \boxed{\frac{\sqrt{10 + 2\sqrt{5}}}{4}}


Find the coefficient of the $x^2$ term in the expansion of the product $(2x^3 + 4x^2 - 3x + 1)(3x^2 - 2x - 5)$.


The hypotenuse of a right triangle measures 12 inches, and one angle is $30^\circ$. What is the number of square inches in the area of the triangle?


Solution: \boxed{-4y^2 - 17y - 8}


How many positive perfect square integers are factors of the product $\left(2^{12}\right)\left(3^{15}\right)\left(5^{18}\right)$?


The students in Mr. Smith's literature class are assigned to read a 900-page book. Three students, Elena, David, and Fiona, are part of this class. Elena reads a page in 50 seconds, David reads a page in 75 seconds, and Fiona reads a page in 40 seconds. David and Fiona decide on a 'team reading' approach where Fiona starts from page 1 and David starts from where Fiona stops through page 900. Determine the last page Fiona should read such that both spend an equal amount of time reading.


Let $G$ be the centroid of triangle $XYZ$. If $GX^2 + GY^2 + GZ^2 = 90$, find $XY^2 + XZ^2 + YZ^2$.


Calculate the distance between the points $(5, -3)$ and $(-7, 4)$. Additionally, determine if the points $(5, -3)$, $(-7, 4)$, and $(5, 4)$ form a right triangle.


The base of a triangular sheet of paper \( ABC \) is \( 15 \text{ cm} \) long. The paper is folded such that the crease \( DE \) is parallel to the base \( AB \). The area of the triangle that projects below the base after folding is \( 25\% \) of the area of triangle \( ABC \). Determine the length of \( DE \), in cm.

Solution: \boxed{7.5 \text{ cm}}


Solution: \boxed{-2x^2 - 2x + \frac{11}{4}}


Julio is packing candies for a carnival giveaway. He packs them in groups of 5, and always has 4 candies left over. When packed in groups of 6, there are 3 left over. If the candies are packed in groups of 8, there are 5 candies left over. Additionally, when he counts all the candies together, their number is less than 150. Determine the least number of candies Julio could have.


How many three-digit numbers are multiples of neither 3 nor 11?


Find the product of all real values of $k$ for which $\frac{1}{3x} = \frac{k-2x}{5}$ has exactly one real solution.


Given that $\sec x - \tan x = \frac{3}{4},$ find all possible values of $\cos x$.


Solve for $x$: $4 \cdot 4^x + \sqrt{16 \cdot 16^x} = 32$.


Find the sixth term of the geometric sequence with the first term $5$ and the second term $1.25$.


When a biased coin is flipped five times, the probability of getting heads exactly twice is equal to that of getting heads exactly three times. Find the probability, expressed as a fraction $\frac{p}{q}$ in lowest terms, that the coin shows heads exactly four times out of the five flips.


A four-inch cube ($4\times4\times4$) of silver weighs 12 pounds and is worth $800$. What would be the worth of a six-inch cube of the same material? Round your answer to the nearest dollar.


How many even perfect square factors does $2^6 \cdot 5^3 \cdot 7^8$ have?


What is the degree of the polynomial obtained when expanding the expression $\left(2x^5+3x^3+x-14\right) \cdot \left(3x^{11}-9x^8+9x^4+30\right) - (x^3+5)^8$?


To solve $51^2$, Jacob calculates $50^2$ and adds 101. What number does Jacob subtract from $50^2$ to compute $49^2$?


Find the difference between $4321_7$ and $1234_7$ in base $7$.


Let triangle $ABC$ be such that $AX = XYZ = ZY = YB = BC$, and the measure of angle $ABC$ is $150^\circ$. What is the measure of angle $BAC$?


A classroom contains 48 chairs, and they need to be arranged in a rectangular array with at least two chairs in every row and column. How many distinct arrays can be formed?


Regular hexagon $PQRSTU$ has vertices $P$ and $R$ at $(2,2)$ and $(10,3)$, respectively. Calculate the area of the hexagon.


How many ways are there to put 7 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?


Let \( g(n) \) be a function that, given an integer \( n \), returns an integer \( k \), where \( k \) is the smallest possible integer such that \( k! \) is divisible by \( n \). Given that \( n \) is a multiple of 18, what is the smallest value of \( n \) such that \( g(n) > 18 \)?

Solution: \boxed{n = 342}


**Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $256\sqrt{2}$ square feet and $576\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?**


Calculate the infinite product $3^{\frac{1}{2}} \cdot 9^{\frac{1}{4}} \cdot 27^{\frac{1}{8}} \cdot 81^{\frac{1}{16}} \dotsm.$


Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/500$. Find $n$.


The numbers $a,$ $b,$ $c,$ $d$ are equal to 2, 3, 5, 7, in some order. Find the largest possible value of
\[2ab + 2bc + 2cd + 2da.\]


Susan flips an unfair coin 8 times. The coin has a $\frac{1}{3}$ probability of landing heads and a $\frac{2}{3}$ probability of landing tails. What is the probability that she flips exactly 3 heads?


A conference hall is setting up seating arrangements in rows. Each row can accommodate $13$ chairs, and currently, there are $156$ chairs in total. The hall expects exactly $100$ attendees and wants complete rows with the smallest number of empty seats. How many chairs should be removed?


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


When the binary number $101110110101_2$ is divided by 8, what is the remainder (give your answer in base 10)?


What is the coefficient of \(x^3\) when the polynomial \(3x^4 - 2x^3 + 4x^2 - 3x + 2\) is multiplied by \(x^2 - 4x + 3\)?


Compute $1010^2 - 990^2 - 1005^2 + 995^2$ without using a calculator.


How many $x$-intercepts does the graph of the parabola $x = -3y^2 + 4y + 2$ have?


Point $(x, y)$ is randomly picked from the rectangular region with vertices at $(0, 0), (3030, 0), (3030, 3031), and (0, 3031)$. What is the probability that $x > 3y$? Express your answer as a common fraction.


In the diagram, \( G \) and \( H \) are the midpoints of \( \overline{PQ} \) and \( \overline{QR} \) respectively, where \( P(0,8) \), \( Q(0,0) \), and \( R(10,0) \). Find the sum of the \( x \) and \( y \) coordinates of \( I \), the point of intersection of \( \overline{PH} \) and \( \overline{GQ} \).


In Mr. Jackson's 26-student computer science class, each student has distinct initials, and each student's first name and last name begin with the same letter. If the letter "Y" is no longer considered a vowel, what is the probability of randomly picking a student whose initials are vowels (excluding "Y") from the class? Express your answer as a common fraction.


What is the greatest common multiple of 10 and 15 that is less than 120?


A flag is made of three horizontal strips of fabric, each of a solid color, either red, white, blue, green, or yellow. If no two adjacent strips can be the same color, and no color can repeat anywhere in the flag, how many distinct flags are possible?


The polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
has coefficients that are all integers, and has roots $-3$, $6$, $10$, and $-\frac{1}{2}$. If $e$ is a positive integer, then find its smallest possible value.


Given the polynomial \(g(x)\) satisfies \(g(x + 1) - g(x) = 12x + 2\). Find the leading coefficient of \(g(x)\).


Solution: \boxed{\csc 20}


Suppose that $n, n+1, n+2, n+3, n+4, n+5, n+6$ are seven consecutive integers.

Determine a simplified expression for the sum of these seven consecutive integers.

Solution: \boxed{7n + 21}


In 1970, there were 300,000 cases of a certain disease in a country. By 2000, the disease cases reduced to 1,000. Estimate the number of disease cases that would have been reported in 1990 if the number of disease cases reported from 1970 to 2000 decreased linearly.


Find the dimensions and perimeter of a non-square rectangle where the number of square units in its area is numerically equal to twice the number of units in its perimeter, and the dimensions are integers.


If $5x + 11 \equiv 3 \pmod{19}$, determine the integer between $0$ and $18$, inclusive, to which $3x + 7$ is congruent $\pmod{19}$.


Determine all angles \(\theta,\) \(0 \leq \theta \leq 2\pi,\) such that for all real numbers \(x,\) \(0 \leq x \leq 2,\)
\[ x^2 \sin \theta - x(2 - x) + (2 - x)^2 \cos \theta > 0. \]

Solution: \boxed{\left( \frac{\pi}{12}, \frac{5\pi}{12} \right)}


If $\frac{9}{x^2} = \frac{x}{25}$, what is the value of $x$?


How many positive factors of 144 are also multiples of 18?


What is the smallest integer $y$ such that $\frac{y}{4} + \frac{3}{7} > \frac{2}{3}$?


Find the radius of the circle given by the equation $x^2 + 10x + y^2 - 4y + 29 = 0$.


What is the perimeter, in cm, of quadrilateral $EFGH$ if $\overline{EF} \perp \overline{FG}$, $\overline{GH} \perp \overline{FG}$, $EF=15$ cm, $GH=5$ cm, and $FG=8$ cm?

Solution: \boxed{28 + 2\sqrt{41} \text{ cm}}


The polynomial \( P(x) = 3x^3 + dx^2 + ex + f \) has the property that the mean of its zeros, twice the product of its zeros, and the sum of its coefficients are all equal. The $y$-intercept of the graph of \( y = P(x) \) is 9. What is \( e \)?


Find all real values of $x$ that satisfy \(\frac{x^3 -3x^2 + 2x}{x^2 - 3x + 2} \le 0\).

Solution: \boxed{(-\infty, 0]}


What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper?


What is the smallest positive four-digit integer equivalent to 3 mod 8?


Simplify the expression $\dfrac{20}{21} \cdot \dfrac{35}{54} \cdot \dfrac{63}{50}$.


When the binary number $100101110011_2$ is divided by 8, what is the remainder (give your answer in base 10)?


The greatest common divisor of two positive integers is $(x+6)$ and their least common multiple is $x(x+6)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one?


What is the value of $\dfrac{5 \times 6 + 4}{8}?$


The first term of this sequence is 3, and each successive term is the sum of all the previous terms. What is the value of the first term that exceeds 10000?


If $a$, $b$ are nonzero real numbers such that $a^2 + b^2 = 18ab$, find the value of $\left|\frac{a+b}{a-b}\right|$.


Find the number of even digits in the base-7 representation of $528_{10}$.


Simplify the expression $1 - (2 - (3 - (4 - (5 - z))))$.


Determine the coefficient of the \(x^2\) term in the expansion of \((3x^2 + 4x + 5)(6x^2 + 7x + 8)\).


For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate?


A torus is formed by revolving a circle in the $xz$-plane centered at $(4, 0, 2)$ with radius $2$ around the $z$-axis. This torus rests on a flat table represented by the $xy$-plane. Determine the radius of the largest spherical ball that can be placed atop the center of the torus while still touching the horizontal plane.


The first term of a given sequence is 2, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 10,000?


Let \( x \) and \( y \) be positive real numbers such that \( x^3 + y^3 = 3(x + y) \). Simplify
\[ \frac{x}{y} + \frac{y}{x} - \frac{3}{xy}. \]


The number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:

1. The Falcons won more games than the Hawks.
2. The Warriors won more games than the Knights but fewer than the Royals.
3. The Knights won more than 25 games.

How many games did the Warriors win? The wins recorded are 20, 26, 30, 35, 40, and 45.


The formula for the total surface area of a cylinder is \(SA = 2\pi r^2 + 2\pi rh\), where \(r\) is the radius and \(h\) is the height. If a right cylinder has a radius of 3 feet and a total surface area of \(27\pi\) square feet, what is the height of this cylinder?


The number $0.84375$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?


What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by any of $19,$ $7,$ and $3?$


Solve for $x$: $\log_2 x + \log_8 x = 5$.


In $\triangle ABC,$ $AB = AC = 13$ and $BC = 10.$ 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 $ADEF$?


For how many positive integers $n\geq 2$ is $10101_n$ a prime number?


Fully simplify the following expression: $[(2+3+6+7)\div3] + [(3\cdot6+9)\div4]$.


Let $r_1, r_2, r_3$ be the roots of the polynomial \( x^3 - 3x^2 + 8 + 5 = 0 \). Find the monic polynomial, in terms of $x$, whose roots are $3r_1, 3r_2,$ and $3r_3$.

Solution: \boxed{x^3 - 9x^2 + 351}


Consider the sequence starting with the number 8820, created by first repeatedly dividing by 2 until no longer divisible by 2, and then subsequently dividing by 5 as long as divisibility by 5 continues. How many integer terms are in this sequence?


Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ so that $P$ is divisible by $5^k$.


How many even three-digit numbers are there such that the sum of the tens and units digits is 11, and the hundreds digit is odd?


A positive two-digit number is odd and is a multiple of 9. The product of its digits is a perfect square. What is this two-digit number?


Chewbacca has 15 pieces of cherry gum, 25 pieces of grape gum, and 10 pieces of apple gum. Some pieces are loose, and others are in complete packs, with each pack containing exactly $y$ pieces of gum. If Chewbacca loses one pack of cherry gum and finds one pack of apple gum, then the ratio of cherry gum he has to the number of grape gum will be the same as if he finds 2 extra packs of grape gum. Find $y$.


Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 100$. Find $\frac n{100}$.


Barbara collects two types of ancient coins, type A and type B. She has 18 coins in total. She finds out that 8 of these coins, which are all of type A, are worth 24 dollars in total. Additionally, she has confirmed that 6 of the type B coins total up to 21 dollars. If Barbara has 12 coins of type A, how much is her entire collection worth?


Solution: \boxed{\frac{1 + \sqrt{5}}{4}}


How many different routes are there from point $A$ to point $B$ if you can only move to the right or down along the drawn segments in a 3x3 grid?

[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,15)--(0,15)--cycle);
draw((5,0)--(5,15));
draw((10,0)--(10,15));
draw((0,5)--(15,5));
draw((0,10)--(15,10));
dot((0,15));
dot((15,0));
label("$A$",(0,15),NW);
label("$B$",(15,0),SE);
[/asy]


In a regular tetrahedron embedded in 3-dimensional space, the centers of the four faces are the vertices of a smaller tetrahedron. If the vertices of the larger tetrahedron are located on the surface of a sphere of radius \(r\), find the ratio of the volume of the smaller tetrahedron to that of the larger tetrahedron. Express your answer as a simplified fraction.


The partial fraction decomposition of
\[ \frac{x^2 - 4}{x^3 + x^2 - 11x - 13} \]
is
\[ \frac{A}{x - 1} + \frac{B}{x + 3} + \frac{C}{x - 4}. \]
Find the product \( ABC \).


Rectangle $ABCD$ is 10 cm by 5 cm. $M$ is the midpoint of $\overline{BC}$, and $N$ is the midpoint of $\overline{CD}$. What is the number of square centimeters in the area of region $AMCN$?

Solution: \boxed{25 \text{ cm}^2}


A classroom has 9 desks arranged in a row. Two students, Alex and Jessie, choose their desks at random. What is the probability that they do not sit next to each other?


What is the modulo $7$ remainder of the sum $1+2+3+\ldots+198+199+200?$


Determine the ratio $\frac{s}{r}$, where $r$ is the total number of rectangles and $s$ is the number of squares formed by the grid of a $7\times7$ checkerboard. Express $\frac{s}{r}$ in its simplest form and find the sum of the numerator and denominator.


A Mersenne prime is defined to be a prime number of the form $2^n - 1$, where $n$ must itself be a prime. What is the smallest Mersenne prime greater than 30?


A parabola with the equation $y = ax^2 + bx + c$ is reflected about the x-axis and then both the original and its reflection are translated vertically, but in different directions by 3 units. If the new equations of the parabolas are $y=f(x)$ for the original and $y=g(x)$ for its reflection, what is the equation of $y = (f-g)(x)$?

Solution: \boxed{2ax^2 + 2bx + 2c + 6}


Eleven stones are arranged in a straight line. They are counted left to right as $1,2,3, \ldots, 11$, and then from right to left, so that the stone previously counted as 10 is now 12. The pattern continues to the left until the stone previously counted as 1 is counted as 21. The pattern then reverses so that the stone originally counted as 2 is counted as 22, and so on. This counting continues. Which of the original stones is counted as 123?


A 12-foot by 15-foot floor is tiled with square tiles of size 2 feet by 2 feet. Each tile has a pattern consisting of one white circle of radius 1 foot located at the center of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

Solution: \boxed{180 - 45\pi}


Let the first term of a geometric sequence be $\frac{2}{3}$, and let the second term be $10$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by five million?


How many distinct arrangements of the letters in the word "balloon" are there, considering the repeated 'l' and 'o'?


Let $a$ and $b$ be the roots of the equation $x^2 - mx + 3 = 0$. Suppose that $a^2 + \frac{1}{b^2}$ and $b^2 + \frac{1}{a^2}$ are the roots of the equation $x^2 - sx + t = 0$. Determine the value of $t$.


In the seven-digit integer $4B6,\!792X$, what is the largest digit $B$ so that the seven-digit integer will be divisible by 4?


Let $z$ be a complex number such that
\[z^2 + |z|^2 = 5 + 4i.\]
Find $|z|^2.$


Among all the roots of the polynomial equation
\[ z^6 - z^4 + z^2 - 1 = 0, \]
find the maximum real part of a root.


What is the greatest possible value of $a+b$ such that $a^2 + b^2 = 130$ and $ab = 45$?


Evaluate the expression $\dfrac{\sqrt[4]{7}}{\sqrt[7]{7}}$ and find $7$ raised to what power?


Find the area and the perimeter of a triangle with side lengths 10, 12, and 12.


Jenny has 10 stamp books that each contain 50 pages. Each page in her books contains 8 stamps. Jenny decides to reorganize her stamp books such that each page contains 12 stamps. This will give her more space to collect stamps without having to buy new books. Under her new system, Jenny fills up 6 complete books, still with 50 pages per book. Her seventh book now contains 37 pages filled with 12 stamps per page and 1 last page with the remaining stamps. How many stamps are on that last page?


Andrew's grandfather's age is fifteen times Andrew's age. If Andrew's grandfather was 55 years old when Andrew was born, how old is Andrew now?


A choir has 90 members who plan to perform in a formation during an event. They need to stand in rows with the same number of members per row. The number of members per row should be between 4 and 15. Determine the possible number of different row-lengths the choir can be arranged in.


Find the difference between $1234_5$ and $432_5$ in base $5$.


What is the sum of the digits of the base $8$ representation of $888_{10}$?


Let \(x,\) \(y,\) and \(z\) be positive real numbers and let \(k\) be a positive constant. Find the minimum value of
\[
\frac{k \cdot 4z}{2x+y} + \frac{k \cdot 4x}{y+2z} + \frac{k \cdot y}{x+z}.
\]


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 5 and 20 respectively. A height from $A$ and $B$ to $CD$ is 12 units. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$?


Let \[f(x) = \left\{
\begin{array}{cl}
x^2 - 6 & \text{ if }x < 10, \\
x - 15 & \text{ if } x \geq 10.
\end{array}
\right.\] What is $f(f(f(20)))$?


Calculate the sum $3_8 + 4_8 + 5_8 + \cdots + 100_8$ and express your answer in base $8$.


What is the least three-digit whole number, the product of whose digits is 8?


On a $12$-hour clock, an elapsed time of ten hours looks the same as an elapsed time of $100$ hours. Because of this, we can say that ten hours is "clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $10$ hours and is "clock equivalent'' to its square number of hours?


In the diagram, $PQR$ is a straight line, and point $S$ is connected to $Q$ and $R$. Additionally, $T$ is another point not on line $PQR$, creating triangle $QST$. Given $\angle PQS = 120^\circ$, $\angle QRS = 50^\circ$, and $\angle QST = 30^\circ$, what is the value of $x$, where $x$ is the measure of $\angle QSR$?

[asy]
draw((-2,0)--(8,0),linewidth(0.7));
draw((8,0)--(5,-5.5)--(0,0),linewidth(0.7));
draw((5,-5.5)--(2,2),linewidth(0.7)); // Triangle QST
label("$P$",(-2,0),W); label("$Q$",(0,0),N); label("$R$",(8,0),E);
label("$S$",(5,-5.5),S); label("$T$",(2,2),N);

label("$120^\circ$",(0,0),NW); label("$50^\circ$",(7,0),S);
label("$30^\circ$",(2,1),NE);

label("$x^\circ$",(5,-4.5));

[/asy]


The sum of the first 1010 terms of a geometric sequence is 100. The sum of the first 2020 terms is 190. Find the sum of the first 3030 terms.


Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of
\[ A\cdot M\cdot C + A\cdot M + M\cdot C + C\cdot A?\]


Given that $$(x+y+z)(xy+xz+yz)=24$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=9$$ for real numbers $x$, $y$, and $z$, what is the value of $xyz$?


Determine the base $b$ such that the solution of the equation $7^{x+7} = 9^x$ can be expressed in the form $x = \log_b 7^7$.


Compute the product of $0.\overline{456}$ and $7$, and write your result as a fraction in simplified form.


The operation $*$ for non-zero integers is defined by $a * b = \frac{1}{a} + \frac{1}{b}$. If $a+b = 10$ and $a \times b = 24$, calculate $a*b$. Moreover, confirm whether the product of $a$ and $b$ is greater than the sum of $a$ and $b$.


What is the largest $2$-digit prime factor of the integer $n = {250\choose 125}$?


Let
\[ g(x) = \left\{
\begin{array}{cl}
-x + 2 & \text{if } x \le 1, \\
3x - 7 & \text{if } x > 1.
\end{array}
\right. \]
How many solutions does the equation $g(g(x)) = 5$ have?


In rhombus \(EFGH\), angle \(E\) measures \(135^\circ\). What is the number of degrees in the measure of angle \(G\)?


When the graph of $y = 3x^2 + 2x - 5$ is shifted six units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$.


Determine $c$ such that $\lfloor c\rfloor$ satisfies \[3x^2 + 8x - 35 = 0\] and $\{ c \} = c - \lfloor c \rfloor$ satisfies \[4x^2 - 12x + 5 = 0.\]


What is the probability that when we roll six fair 6-sided dice, they won't all show the same number?


While strolling in the park, Jia walked 80 feet toward a picnic area in 40 minutes. At this rate, how many minutes will it take her to walk the remaining 100 yards to the picnic area?


How many odd numbers between $1000$ and $9999$ have distinct digits?


Find the radius of the circle with equation $x^2 - 8x + y^2 - 10y + 29 = 0$.


Find the quadratic polynomial, with real coefficients, which has \(1 + 2i\) as a root, and where the coefficient of \(x\) is \(10\).

Solution: \boxed{-5x^2 + 10x - 25}


What is the constant term of the expansion of $\left(5x + \frac{2}{5x}\right)^8$?


For two real values of $m$, the equation $3x^2 + mx + 16 = 0$ has exactly one solution in $x$. What is the positive value of $m$?


For real numbers $s$, the points of intersection of the lines $x + 3y = 9s + 4$ and $x - 2y = 3s - 3$ are plotted. All the plotted points lie on a line. Find the slope of this line.


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $90$, and $BC=12$. Find the volume of the tetrahedron.


Add $15_6 + 25_6.$ Express your answer in base $6.$


Find $XY$ in the triangle below.

[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (2,0);
R = Q+rotate(60)*(2,0);
draw(P--Q--R--P, linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$", P, S);
label("$Y$", Q, S);
label("$Z$", R, NE);
label("$24$",(Q+R)/2, NE);
label("$60^\circ$", (1.3, 0), N);
[/asy]


Suppose real numbers $u$ and $v$ exist such that
\[
\begin{pmatrix} 3 \\ -1 \end{pmatrix} + u \begin{pmatrix} 9 \\ -6 \end{pmatrix} = \begin{pmatrix} 4 \\ 0 \end{pmatrix} + v \begin{pmatrix} -3 \\ 4 \end{pmatrix} + \begin{pmatrix} 1 \\ 1 \end{pmatrix}.
\]
Find the ordered pair $(u,v)$.

Solution: \boxed{\left(\frac{7}{9}, -\frac{5}{3}\right)}


Let $t(x)$ have domain $\{-2, -1, 0, 1\}$ and range $\{-1, 1, 3, 5\}$. Let $u(x)$ have domain $\{0, 1, 2, 3\}$ and is defined by $u(x) = x + 2$. Determine the sum of all possible values of $u(t(x))$.


A parabola has vertex \( V = (0,0) \) and focus \( F = (0,1) \). Let \( P \) be a point in the first quadrant, lying on the parabola, so that \( PF = 51 \). Find \( P \).

Solution: \boxed{(10\sqrt{2}, 50)}


A point \((x,y)\) is randomly selected from inside the rectangle with vertices \((0,0)\), \((6,0)\), \((6,2)\), and \((0,2)\). What is the probability that \(x < 2y\)?


The arithmetic mean, geometric mean, and harmonic mean of $a$, $b$, $c$ are $7$, $6$, $5$ respectively. What is the value of $a^2+b^2+c^2$?


Find the positive value of $x$ which satisfies
\[\log_3 (x + 3) + \log_{\sqrt{3}} (x^2 + 5) + \log_{\frac{1}{3}} (x + 3) = 3.\]

Solution: \boxed{\sqrt{3\sqrt{3} - 5}}


When the height of a cylinder is tripled and its radius is increased by $150\%$, by what factor $X$ is the cylinder's volume multiplied?


What is the 59th digit after the decimal point in the decimal representation of $\frac{1}{17}$?


Let $n$ be a positive integer. If $b \equiv (5^{3n} + 6)^{-1} \pmod{11}$, what is the remainder when $b$ is divided by $11$?


150 people were surveyed and asked whether they believe the color teal is more "blue" or "green". 90 people responded that teal is "kinda blue," 45 stated it is both "kinda blue" and "kinda green", and 25 people considered teal as neither "kinda blue" nor "kinda green".

How many of those 150 people believe that teal is "kinda green"?


At the beginning of my bike ride, I feel energetic and can travel at 25 miles per hour. After a while, I become tired and travel only 15 miles per hour. If I also take a 1-hour rest without covering any distance, and the total distance I travel is 150 miles in a total time of 9 hours, how many hours did I feel energetic? Express your answer as a common fraction.


What is the smallest positive value of $x$ such that $x + 1234$ results in a palindrome?


In a new shooting match setup, nine clay targets are arranged in three hanging columns with three targets each. A marksman must break all the targets following these rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
Determine the number of different orders in which the nine targets can be broken.


A sample of helium gas is at a pressure of 8 kPa in a 3-liter container. If the gas is then transferred to a 6-liter container while maintaining the same temperature, what will the new pressure of the helium be in kPa?

Solution: \boxed{4 \, \text{kPa}}


If $\tan x + \tan y = 10$ and $\cot x + \cot y = 12$, what is $\tan(2x+2y)$?


Let \( x, y, z \) be distinct, nonzero real numbers such that
\[ x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}. \]
Find \( |xyz| \).


Find an ordered triple $(x, y, z)$ that satisfies the following system of equations:
\begin{align*}
3x + 2y - z &= 1, \\
4x - 5y + 3z &= 11.
\end{align*}

Solution: \boxed{(1, 1, 4)}


How many three-digit perfect cubes are divisible by $27$?


In Weather Town, the forecast predicts a 75% chance of rain each day during the upcoming five-day festival. On days it doesn’t rain, the weather will be sunny. Jasmine and Lee hope for exactly two sunny days during this time, as they plan indoor activities otherwise. What is the probability they get the weather they want? Present your answer as a fraction.


Jessica is taking a 20-question, multiple-choice Biology Quiz. Each question has four answer choices. Jessica randomly guesses the answers for the last five questions. What is the probability that she will get at least two of these last five questions correct?


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

Solution: \boxed{x^4 - 3x^2 + 9}


A triangle has side lengths of 9, 12, and 15 units. Calculate both the area of the triangle and the height from the right angle vertex to the hypotenuse.


Below is the graph of $y = a \cos (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$  Find $b.$

[asy]import TrigMacros;

size(400);

real f(real x)
{
	return 3*cos(2*x + pi/2) + 4;
}

draw(graph(f,-2*pi,2*pi,n=700,join=operator ..),blue);
trig_axes(-2*pi,2*pi,-2,10,pi/2,1);
layer();
rm_trig_labels(-5,5, 2);

label("$5$", (0,5), E);
label("$6$", (0,6), E);
label("$7$", (0,7), E);
label("$8$", (0,8), E);
label("$9$", (0,9), E);
label("$10$", (0,10), E);
[/asy]


Solution: \boxed{27x^3 + 2x^5}


Points $A$, $B$, $C$, and $D$ are located on $\overline{AB}$ such that $AB = 4AD = 8BC$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $A$ and $C$?


A cube made of silver measuring 4 inches on each side weighs 5 pounds and is valued at $300. Calculate the value of a silver cube that measures 6 inches on each side, rounding your answer to the nearest dollar.


Sarah walks 8 blocks west and then 15 blocks south. If each block is one-fifth of a mile, how many miles did she walk?


A box contains eight cards. Four of the cards are blue on both sides, two cards are blue on one side and green on the other, and two of the cards are green on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is green, what is the probability that the other side is also green?


Determine how many numbers less than 50 are abundant numbers.


The marble statue of George Washington in a fictional museum has a height of 80 feet. A scale model of the statue has a height of 5 inches. How many feet of the statue does one inch of the model represent?


A regular dodecagon \(Q_1 Q_2 \ldots Q_{12}\) is drawn in the coordinate plane with \(Q_1\) at \((2,0)\) and \(Q_7\) at \((4,0)\). If \(Q_n\) is the point \((x_n, y_n)\), compute the numerical value of the product:
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \ldots (x_{12} + y_{12} i).
\]


In $\Delta ABC$, $\overline{FG} \parallel \overline{AB}, CF=5$ cm, $FA=15$ cm, and $CG=8$ cm. What is the number of centimeters in the length of $\overline{CB}$?

[asy]
pair A,B,C,D,E;
A = (-3,-6); B = (6,-6); C = (0,0); D = A/4; E = B/4;
draw(E--D--C--B--A--D);
label("A",A,W); label("B",B,dir(0)); label("C",C,N); label("D",D,W);
label("E",E,dir(0));
[/asy]

Solution: \boxed{32\text{ cm}}


A line is parameterized by a parameter $t$. The vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$, at $t = 4$ is $\begin{pmatrix} 8 \\ -5 \end{pmatrix}$, and at $t = 5$ is $\begin{pmatrix} 10 \\ -9 \end{pmatrix}$. Find the vector on the line at $t = 0.$

Solution: \boxed{\begin{pmatrix} 0 \\ \frac{17}{3} \end{pmatrix}}


Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $b,c,a$ is a geometric progression. What is the smallest possible value of $a$?


The function \( f : \mathbb{R} \to \mathbb{R} \) satisfies
\[
f(x) + 3f(1 - x) = 4x^2
\]
for all \( x \). Find \( f(5) \).


There are 6 blue chips, 4 yellow chips, and 2 red chips in a bag. One chip is drawn from the bag. That chip is then placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors?


Determine the median and the mean of the set \(\{m, m+2, m+4, m+11, m+18\}\) where the median of the set is \(16\).


Solve the equation by completing the square: $$100x^2+60x-90 = 0.$$ Rewrite the equation in the form \((dx + e)^2 = f\), where \(d\), \(e\), and \(f\) are integers and \(d > 0\). What is the value of \(d + e + f\)?


What is the probability that Hannah gets fewer than 4 heads if she flips 12 coins?


Luna now writes down all the five-digit numbers that contain each of the digits 1, 2, 3, 6, and 9 exactly once. What is the smallest number in Luna's list that is divisible by 6?


Determine how many integers \( j \) exist such that \( 1 \le j \le 5000 \) and \( f(j) = 1 + j + 2\sqrt{j} \), where \( f(j) \) denotes the sum of all positive divisors of the integer \( j \).


Given that $9!=362880$, what is the value of $9!\div4!$?


An ellipse has its foci at $(1, 1)$ and $(1, 5)$. Given that it passes through the point $(-4, 3)$, 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 $a+k.$


Let \[f(x) =
\begin{cases}
5x+20 &\text{if }x<3, \\
3x-9&\text{if }x\ge3.
\end{cases}
\]If $f(x)=1$, find the sum of all possible values of $x$.


If $x$, $y$, and $z$ are positive numbers satisfying $xy=40$, $xz=80$, and $yz=120$, what is the value of $x+y+z$?


How many different primes are in the prime factorization of $85\cdot87\cdot89\cdot91$?


Rationalize the denominator of $\frac{7}{3+\sqrt{5}}$. Write the answer in the form $\frac{A\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. If the greatest common divisor of $A$, $C$, and $D$ is 1, determine $A + B + C + D$.


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


When the binary number $101011100101_2$ is divided by 8, what is the remainder (give your answer in base 10)?


A frustum of a right circular cone has an altitude of $30$ centimeters, and the areas of its lower and upper bases are $400\pi$ sq cm and $100\pi$ sq cm, respectively. Compute the altitude of the smaller cone that was removed from the original larger cone to form this frustum.

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


Alice and Bob each think of a polynomial. Each of their polynomials is monic, has degree 3, and has the same positive constant term. The product of their polynomials is \[z^6 + 2z^5 + 5z^4 + 8z^3 + 10z^2 + 8z + 9.\] What is the constant term of Bob's polynomial?


The Trumpington marching band has $24m$ members. When they line up in rows of 30, there are 6 members left over. If $m$ is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching band?


Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG=5$. If $EFGH$ has two sides with distinct integer lengths, then what is the area of $EFGH$? Express your answer in simplest radical form.


Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\clubsuit$ and the second card is a Queen?


Eric is trying to remember the six-digit code to his locker. He knows that he only used digits 1 through 6 (possibly repeated), that every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible codes does Eric need to try?


Solution: \boxed{\frac{\log_2 7}{6}}


If two sides of a right triangle are 8 units and 15 units, what is the least possible length, in units, of the third side? Express your answer in simplest radical form.


Billy wants to determine the average number of cookies in a package of cookies. He purchases 9 packages of cookies, opens them, and counts the number of cookies in each one. He finds that the packages have 9, 11, 13, 15, 15, 17, 19, 21, and an unusually low 5 cookies. Based on these observations, what is the average number of cookies in a package?


Calculate the sum of the series $2 + 4 + 8 + 16 + 32 + \cdots + 512 + 1000$.


What is the coefficient of $a^3b^3$ in $(a+b)^6\left(c+\dfrac{1}{c}\right)^8$?


Find the minimum value of the function
\[ f(x) = x^2 + \frac{1}{x^2} + \frac{1}{x^2 + \frac{1}{x^2}} \]
for $x > 0$.


Find the remainder when \( 8x^4 - 18x^3 + 5x^2 - 3x - 30 \) is divided by \( 2x - 4 \).


The quadratic $x^2 - 20x + 49$ can be written in the form $(x+b)^2 + c$, where $b$ and $c$ are constants. What is the value of $b+c$?


Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| = 1$ and $\arg(c) = \arg(a) + \arg(b)$. Suppose that
\[ a z^2 + b z + c = 0. \]
Find the largest possible value of $|z|$.

Solution: \boxed{\frac{1 + \sqrt{5}}{2}}


The line $y = \frac{1}{3}x + 3$ is parameterized by:
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -9 \\ s \end{pmatrix} + t \begin{pmatrix} l \\ -6 \end{pmatrix}.\]
Find the ordered pair $(s,l)$.

Solution: \boxed{(0, -18)}


Solution: \boxed{2^{\frac{6 \log_2 3}{3 \log_2 3 - 1}}}


Two lines have the same non-zero $y$-intercept. The first line has a slope of 12 and an $x$-intercept of $(u, 0)$. The second line has a slope of 8 and an $x$-intercept of $(v, 0)$. What is the ratio of $u$ to $v$? Express your answer as a common fraction.


What is the 7th term of an arithmetic sequence of 25 terms with first and last terms of 3 and 78, respectively?


Find the greatest common divisor of $7429$ and $12345$.


Find the number of complex numbers $z$ satisfying $|z| = 1$ and
\[
\left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 2.
\]


Two vertices of an obtuse triangle are $(3,6)$ and $(0,0)$. The area of the triangle is 36 square units. The third vertex is located somewhere on the line where $y = 0$ and $x < 0$. Find the coordinates of the third vertex.

Solution: \boxed{(-12, 0)}


What is the arithmetic mean of the integers from -3 through 6, inclusive? Express your answer as a decimal to the nearest tenth.


Simplify the following expression:
$$7y - 3x + 8 + 2y^2 - x + 12$$

Solution: \boxed{2y^2 + 7y - 4x + 20}


What is the largest integer value of $y$ such that $\frac{y^2 + 3y + 10}{y-4}$ is an integer?


Evaluate the determinant given by:
\[
\begin{vmatrix}
2 & a & b \\
2 & a + b & b + c \\
2 & a & a + c
\end{vmatrix}.
\]

Solution: \boxed{2(ab + bc - b^2)}


A store offers a discount of 40% on all items. A few days later, an additional 10% discount is applied to the already reduced prices. The store advertises that the items are now 55% off the original price. What is the difference between the actual discount and the store's claimed discount?


In Mr. Green's 30-student physics class, no two students have the same initials, and each student's first name and last name begin with the same letter. Including "Y" as a vowel, what is the probability that a randomly picked student's initials are vowels, assuming all 26 letters of the alphabet are used as initials? Express your answer as a common fraction.


How many values of $x$ with $0^\circ \le x < 360^\circ$ satisfy both $\sin x = -0.65$ and $\cos x < 0$?


How many lattice points lie on the hyperbola $x^2 - y^2 = 3^4 \cdot 17^2$?


Given the graph of \( y = g(x) \) for \( -4 \leq x \leq 4 \), determine the range of \( g(x) - 2x \). The graph of \( y = g(x) \) is as follows, with \( g(x) \) defined piecewise linearly through points \( (-4, -2) \), \( (-2, -1) \), \( (0, 0) \), \( (2, 3) \), and \( (4, 4) \).

Solution: \boxed{[-4, 6]}


The asymptotes of a hyperbola are \(y = x + 2\) and \(y = 4 - x.\) Also, the hyperbola passes through \( (4,4) \). Find the distance between the foci of the hyperbola.


Find the volume of the region in space defined by
\[|x + 2y + z| + |x + 2y - z| \le 12\]
and \(x, y, z \ge 0.\)


Simplify
\[\frac{3 + 3\sin x - 3\cos x}{3 + 3\sin x + 3\cos x}.\]

Solution: \boxed{\tan \frac{x}{2}}


Suppose \( a \) and \( b \) are positive integers such that \( \gcd(a, b) \) is divisible by exactly \( 5 \) distinct primes and \( \mathop{\text{lcm}}[a, b] \) is divisible by exactly \( 20 \) distinct primes.

If \( a \) has fewer distinct prime factors than \( b \), then what is the maximum number of distinct prime factors that \( a \) can have?