A round pizza is $\frac{1}{2}$ of an inch thick and has a diameter of 10 inches. It is cut into 10 congruent pieces. What is the number of cubic inches in the volume of one piece? Express your answer in terms of $\pi$.


A square has sides of length 12, and a circle centered at one of its vertices has a radius of 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.

Solution: \boxed{144 + 75\pi}


Consider a circular park consisting of an outer-ring path for walkers (white), a ring-shaped flower garden (gray) surrounding a central circular fountain (black). The walking path is 6 feet wide, the garden ring is 10 feet wide, and the fountain has a diameter of 20 feet. What is the diameter, in feet, of the circle that is the outer boundary of the walking path?


What is the largest integer that must divide the product of any 5 consecutive integers?


Let \( a, b, c, d, e, f, g, \) and \( h \) be real numbers such that \( abcd = 9 \) and \( efgh = 4 \). Find the minimum value of
\[
(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2.
\]


In how many ways can 8 distinct beads be placed on a necklace? (Note that two arrangements are the same if one can be rotated or reflected to produce the other.)


How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression, where the common difference is less than 30?


Consider two spinners, where one is divided into fifths and the other into thirds. Each section of the first spinner contains numbers: 2, 4, 6, 8, 10, and each section of the second spinner contains numbers: 3, 5, 9. If each spinner is spun once, what is the probability that the product of the results will be a multiple of 4?


If $\sum_{n = 0}^{\infty}\cos^{2n}\theta = 9$, what is the value of $\cos{2\theta}$?


A card is chosen at random from a modified deck of 54 cards (including 2 jokers), and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace?


Calculate $\frac{5 \cdot 7! + 35 \cdot 6!}{8!}$


The fifth term of a geometric sequence of positive numbers is $16$ and the eleventh term is $2$. What is the eighth term of the sequence?


If $\log_4 x^3 + \log_4 x = 5,$ compute $x.$


Let $b_1, b_2, \ldots$ be a sequence such that:

(i) $b_1 = 2$, and

(ii) for any positive integer $n$, $b_{3n}=n^2 \cdot b_n$.

What is the value of $b_{3^{100}}$?

Solution: \boxed{2 \cdot 3^{9900}}


30 people attend a conference. Due to a seating arrangement, each person only shakes hands with the people sitting next to them in a circular table setup. How many unique handshakes occur?


Tim had an initial loan payment due of \$500. A delay in payment subjected the loan to a 2% late fee at every 30-day interval past its due date. If the payment was delayed for a total of 90 days, what is the total amount due with the compounded late fees?


What is the greatest integer less than or equal to $\frac{5^{50} + 4^{50}}{5^{48} + 4^{48}}$


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


A sports coach distributes 40 water bottles among 25 athletes before a game, with each athlete receiving at least one water bottle. Calculate the maximum number of water bottles one athlete could have received, assuming everyone received at least one.


Compute $\arccos (\cos 3).$ All functions are in radians.


A triangle has vertices \( A=(2, 3) \), \( B=(-4, -2) \), and \( C=(7, -1) \). The equation of the bisector of \( \angle B \) can be written in the form \( x - 3y + d = 0 \). Find \( d \).

Solution: \boxed{d = -2}


The distance between two vectors is the magnitude of their difference.  Find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 3 \\ -2 \\ -4 \end{pmatrix} + t \begin{pmatrix} 8 \\ 6 \\ -2 \end{pmatrix}\]is closest to
\[\bold{a} = \begin{pmatrix} 5 \\ 7 \\ 3 \end{pmatrix}.\]


Calculate $\left[\left(20^{10} \div 20^9\right)^3 \cdot 10^6\right] \div 2^{12}$.


Solution: \boxed{17a - 8b + 50c}


Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $3$ when divided by any of $11,$ $4,$ and $3.$


In the function machine shown, the input is 12. What is the output?

[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0)--(width1,height)--(0,height)--cycle); label(a,"$\mbox{In}\mbox{put}$ = 12",(width1/2,height/2));
draw(b,(0,0)--(width2,0)--(width2,height)--(0,height)--cycle); label(b,"Multiply by 2",(width2/2,height/2));
draw(c, (widthC/2,0)--(0,heightC/2)--(-widthC/2,0)--(0,-heightC/2)--cycle);
label(c,"Compare with 22",(0,0));
draw(d,(0,0)--(width3,0)--(width3,height)--(0,height)--cycle); label(d,"Add 8",(width1/2,height/2));
draw(e,(0,0)--(width4,0)--(width4,height)--(0,height)--cycle); label(e,"Add 3",(width1/2,height/2));
draw(f,(0,0)--(widthE,0)--(widthE,0)+lengthE*dir(angleE)--lengthE*dir(angleE)--cycle);
label(f,"$\mbox{Out}\mbox{put}$ = ?",lengthE/2*dir(angleE) + (widthE/2,0));
add(shift(width1/2*left)*a); draw((0,0)--(0,-2),EndArrow(4));
add(shift(5*down + width2/2*left)*b);
add(shift((7+heightC/2)*down)*c); draw((0,-5)--(0,-7),EndArrow(4));
pair leftpt = (-widthC/2,-7-heightC/2), rightpt = (widthC/2,-7-heightC/2);
draw("$\le 22$?",leftpt--(leftpt + 2.5W)); draw((leftpt + 2.5W)--(leftpt + 2.5W+2S),EndArrow(4));
draw("$> 22?$",rightpt--(rightpt + 2.5E),N); draw((rightpt + 2.5E)--(rightpt + 2.5E+2S),EndArrow(4));
rightpt = rightpt + 2.5E+2S;
leftpt = leftpt + 2.5W+2S;
add(shift(leftpt+height*down+.3*width3*left)*d);
add(shift(rightpt+height*down+.7*width4*left)*e);
rightpt = rightpt+.75height*down+.7*width4*left;
leftpt = leftpt+.75height*down+.7*width3*right;
draw(leftpt--rightpt);
pair midpt = (leftpt+rightpt)/2;
draw(midpt--(midpt+2down),EndArrow(4));
add(shift(midpt+.65widthE*left+(2+lengthE*Sin(angleE))*down)*f);[/asy]


Let $Q$ be the plane passing through the origin with normal vector $\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}.$ Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{v},$ $\mathbf{Q} \mathbf{v}$ is the projection of $\mathbf{v}$ onto plane $Q.$

Solution: \boxed{\begin{pmatrix} \frac{5}{9} & \frac{2}{9} & -\frac{4}{9} \\ \frac{2}{9} & \frac{8}{9} & \frac{2}{9} \\ -\frac{4}{9} & \frac{2}{9} & \frac{5}{9} \end{pmatrix}}


A square and four smaller squares are contained within it, each smaller square circumscribing a circle with a radius of 3 cm. What is the area of the larger square and the total area occupied by the four smaller squares?

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


Each row of a seating arrangement seats 7 or 8 people. Fifty-five people are to be seated. How many rows seat exactly 8 people if every seat is occupied?


A rectangular box has a volume of $4320$ cubic inches and a surface area of $1704$ square inches. The sum of the lengths of its $12$ edges totals $208$ inches. If the length, width, and height of the box were to be increased by 4 inches each, what would be the new volume of the box, in cubic inches?

Solution: \boxed{8624 \text{ cubic inches}}


The average age of seven children is 8 years old. Each child is a different age and there is a difference of three years in the ages of any two consecutive children. In years, how old is the oldest child?

Solution: \boxed{17 \text{ years}}


Find the distance between the vertices of the hyperbola \(16x^2 - 64x - 4y^2 + 8y - 11 = 0.\)


Find the sum of the first eight terms in the geometric sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$. Express your answer as a common fraction.


For all values of $x$ for which it is defined, $g(x) = \cot \frac{x}{6} - \cot x$ can be written as
\[g(x) = \frac{\sin mx}{\sin \frac{x}{6} \sin x}.\]
Find the value of $m$.


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

[asy]import TrigMacros;

size(400);

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

draw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-2,10,pi/2,1);
layer();
rm_trig_labels(-2,10, 2);

label("$4$", (0,4), E);
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);
[/asy]


The marble statue of George Washington in a fictional museum has a height of 90 feet. A scale model of the statue has a height of 6 inches. How many feet of the statue does one inch of the model represent?


For how many values of the digit $A$ is it true that $120$ is divisible by $A$ and $451{,}2A8$ is divisible by $8$?


Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 6$ and $ab + ac + bc = 11.$ Find the largest possible value of $a.$

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


The equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ is to be simplified to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\] for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$


Convert the point $(4, -4\sqrt{3}, 4)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$

Solution: \boxed{\left(4\sqrt{5}, \frac{5\pi}{3}, \cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)}


Mr. and Mrs. Lopez have three children. When they get into their family car, two people sit in the front, and three sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?


Line $l_1$ has equation $5x + 4y = 2$ and passes through point $A = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $B$. Line $l_3$ has a positive slope, goes through point $A$, and intersects $l_2$ at point $C$. The area of $\triangle ABC$ is $4$. Determine the slope of line $l_3$.


Spinners $A$ and $B$ are spun. Spinner $A$ has numbers 1 through 5; Spinner $B$ has numbers 1, 3, and 5. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?


Let $x,$ $y,$ $z$ be positive real numbers such that
\[\left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) + \left( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} \right) = 9.\] Determine the maximum value of
\[\left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) \left( \frac{y}{x} + \frac{z}{y} + \frac{x}{z} \right).\]


Let $x,$ $y,$ $z$ be nonnegative real numbers such that $x + y + z = 3.$  Find the maximum value of
\[\frac{xy}{x + y + 1} + \frac{xz}{x + z + 1} + \frac{yz}{y + z + 1}.\]


Determine the sum of the angles in degrees for the sixth roots of the equation \(z^6 = 64i\).


Determine the number of angles between 0 and $2 \pi,$ other than integer multiples of $\frac{\pi}{2},$ such that $\sin \theta,$ $\cos \theta$, and $\tan \theta$ form a geometric progression (G.P.) in some order.


What is the remainder when the binary number $111001101101_2$ is divided by 4? Give your answer in base 10.


If two distinct numbers are selected at random from the first eight prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction.


The average of four different positive whole numbers is $5.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers?


A regular octagon is inscribed in a circle of radius 3 units. In square units, what is the area of the octagon? Express your answer in simplest radical form.


For how many integer values of \( n \) between 1 and 999 inclusive does the decimal representation of \( \frac{n}{1000} \) terminate?


Determine the domain of the function
\[ f(x) = \frac{\sqrt{x^2 - 9x + 20}}{|x - 5| + |x + 2|}. \]

Solution: \boxed{(-\infty, 4] \cup [5, \infty)}


Let $M$ denote the number of $8$-digit positive integers where the digits are in non-decreasing order. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed.)


How many three-digit multiples of seven are there?


A spinner with seven equal sectors labeled from 1 to 7 is used in a game between two players, Alice and Bob. They each spin the spinner once. Alice wins if the absolute difference between their numbers is less than 4. Otherwise, Bob wins. Determine the probability that Alice wins.


The greatest common divisor of two positive integers is $(x+5)$ and their least common multiple is $x(x+5)^2$, where $x$ is a positive integer. If one of the integers is 60, what is the smallest possible value of the other one?

Solution: \boxed{n = 2000}


There exist real numbers $p$ and $q$ such that
\[\begin{pmatrix} 3 \\ -4 \end{pmatrix} + p \begin{pmatrix} 8 \\ -6 \end{pmatrix} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} + q \begin{pmatrix} -3 \\ 4 \end{pmatrix}.\]Find the ordered pair $(p, q).$

Solution: \boxed{\left( -\frac{1}{14}, -\frac{1}{7} \right)}


How many different counting numbers will each leave a remainder of 6 when divided into 67?


On Jack's 12th birthday in 2010, his mother said, "My age is now three times your age." In what year will Jack's mother be able to say, "My age is now twice your age," on Jack's birthday?


Let $a$ and $b$ be positive real numbers.  Find the minimum value of
\[a^2 + ab + b^2 + \frac{1}{(a+b)^2}.\]


Let $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 4$. If $|z+w| = 5$, what is $ \left | \frac{1}{z} + \frac{1}{w} \right|$?


What is $2\frac{1}{4}$ divided by $\frac{3}{5}$?

Solution: \boxed{3 \frac{3}{4}}


The Great Eighteen Soccer League is divided into three divisions, each containing six teams. Each team plays each of the other teams in its own division three times and every team in the other divisions twice. How many league games are scheduled?


Juan has expanded his stamp collection to include stamps from four countries: Italy and Germany (Europe), and Colombia and Argentina (South America). The prices he paid are: Italy and Germany, $7$ cents each, Colombia $3$ cents each, and Argentina $6$ cents each. The number of stamps from each decade is listed below:
- 1950s: Italy 5, Germany 8, Colombia 7, Argentina 4
- 1960s: Italy 9, Germany 6, Colombia 6, Argentina 8
- 1970s: Italy 10, Germany 12, Colombia 12, Argentina 10
- 1980s: Italy 11, Germany 9, Colombia 15, Argentina 9
Calculate the total cost of his South American stamps issued before the 1980s in dollars and cents.


Given \( w \) and \( z \) are complex numbers such that \( |w+z|=2 \) and \( |w^2+z^2|=15 \), find the smallest possible value of \( |w^3+z^3| \).


The inscribed circle of triangle $DEF$ is tangent to $\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$.


Suppose $m$ is a positive integer such that in base $5$, it can be expressed as $\overline{XYZ}_5$, and in base $9$, it can be expressed as $\overline{ZYX}_9$. Find the largest possible value of $m$ in base $10$.


A ball is dropped from 2000 feet high and bounces back up to half the height it fell. After how many bounces will the ball first reach a maximum height less than 6 inches?


In the diagram below, we have \( AB = 30 \) and \( \angle ADB = 90^\circ \). If \( \sin A = \frac{4}{5} \) and \( \sin C = \frac{1}{4} \), then what is \( DC \)?

[asy]
pair A,B,C,D;
A = (0,0);
B = (8*sqrt(5),16);
D = (8*sqrt(5),0);
C = (8*sqrt(5) + 32*sqrt(2),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]


A fair 10-sided die is rolled once. If I roll $n$, then I win $10-n$ dollars. However, if $n$ is odd, the win is doubled. What is the expected value of my win, in dollars?


Two standard dice are rolled. What is the expected number of 6's obtained? Express your answer as a common fraction.


A function $f$ has the property that $f(2x+3)=x^2-2x+3$ for all real numbers $x$. What is $f(7)$?


Calculate the area of the parallelogram formed by the vectors $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -4 \\ 5 \end{pmatrix}$.


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 $23$ distinct primes.

If $a$ has more distinct prime factors than $b$, then $b$ has at most how many distinct prime factors?


Determine the greatest integer less than or equal to \[\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}.\]


What is the sum of the coordinates of the midpoint of the segment with endpoints $(5, 3, -2)$ and $(-1, 4, 6)$?


In isosceles right triangle $ABC$, where angles $A$ and $B$ are 45 degrees each, the angle bisectors of angles $A$ and $C$ intersect at a point $D$. What is the measure of obtuse angle $ADB$?


Given vectors \(\mathbf{u}\) and \(\mathbf{z}\) such that \(\|\mathbf{u}\| = 5,\) \(\|\mathbf{z}\| = 8,\) and \(\mathbf{u} \cdot \mathbf{z} = 20,\) find \(\|\operatorname{proj}_{\mathbf{z}} \mathbf{u}\|\).


Completely factor the following expression: \[(16y^6 + 36y^4 - 9) - (4y^6 - 6y^4 + 9).\]

Solution: \boxed{6(2y^6 + 7y^4 - 3)}


Find $a + b + ab$ where $a$ and $b$ are the roots of the cubic equation:
\[x^3 - 4x^2 - x + 4 = 0.\]


Determine how many different tens digits are possible in numbers that are divisible by 8.


Solution: \boxed{x = 37.5}


The numbers \(x\) and \(y\) are inversely proportional. When the sum of \(x\) and \(y\) is 30, \(x\) is three times \(y\). What is the value of \(y\) when \(x = -10\)?


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}}}


Let $A$, $B$, and $C$ be non-zero distinct digits less than 5, and consider the equations ${AB_5}+{C_5}={C}0_5$ and ${AB_5}+{BA_5}={CC_5}$ in base 5. Find the three-digit number ${ABC}$.


John has saved up $5235_9$ dollars for a trip to Japan. A round-trip airline ticket costs $1250_8$ dollars. In base ten, how many dollars will he have left for lodging and food?


Given the function $y = f(x)$, the graph of another function $y = h(x)$ is derived by first reflecting $y = f(x)$ in the y-axis and then shifting it six units to the right. Write the expression for $h(x)$ in terms of $f(x)$.

Solution: \boxed{f(6 - x)}


A digit is appended to the left of the number $864$. If the resulting four-digit number is divisible by $5$, how many possibilities are there for the digit that was appended?


Let $p$, $q$, $r$, and $s$ be real numbers such that $|p-q|=1$, $|q-r|=2$, and $|r-s|=5$. What is the sum of all possible values of $|p-s|$?


Caroline can make 15 lassis out of 3 mangoes and 2 bananas. How many lassis can she make out of 15 mangoes and 10 bananas?


In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 3\cos Y$. What is $XY$?


If 25,197,631 hot dogs are packaged in sets of 17, how many will be left over?


Consider the polynomial $g(x) = x^{2009} + 19x^{2008} + 1$ with distinct roots $s_1, s_2, ..., s_{2009}$. A polynomial $Q$ of degree $2009$ has the property that
\[ Q\left(s_j + \dfrac{1}{s_j}\right) = 0 \] for $j = 1, \ldots, 2009$. Determine the value of $\frac{Q(1)}{Q(-1)}$.


In a hyperbola, the center is at $(1, -2)$, one focus is at $(1, 5)$, and one vertex is at $(1, 0)$. The equation of this hyperbola can be written as
\[
\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.
\]
Find $h + k + a + b.$

Solution: \boxed{1 + 3\sqrt{5}}


The graph of the equation \[ 3x^2 + 9y^2 - 6x + 27y = k \] is a non-degenerate ellipse if and only if \( k > a \). What is \( a \)?


What is the units digit of the product of all the odd positive integers between 20 and 120?


Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Let $N$ be the number of such minimally intersecting ordered triples of sets for which each set is a subset of $\{1,2,3,4,5,6,7,8,9\}$. Compute $N$ modulo 1000.


During her run, Lisa jogs at 10 miles per hour when feeling energetic and slows down to 6 miles per hour when tired. If Lisa travels a total of 88 miles in a total time of 10 hours, for how many hours did she feel energetic? Express your answer as a common fraction.


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

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


The image of the point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(10,7)$. Find $m+b$.


The expression $x^2 + 9x + 14$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 4x - 21$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?


Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 50^\circ$, and $\angle ADC = 160^\circ$.  What is the degree measure of $\angle BAD$?


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

Solution: \boxed{\left(\frac{1}{2}, \frac{3}{5}\right)}


How many integers from 1-9 are divisors of the five-digit number 46,320?


The graph of \(y^2 + 2xy + 60|x| = 900\) partitions the plane into several regions. What is the area of the bounded region?


The interior of a right, circular cone is 12 inches tall with a 3-inch radius at the opening. The cone is filled with ice cream, and a full sphere of ice cream rests snugly at the opening of the cone. What is the volume of ice cream in total? Express your answer in terms of $\pi$.


If $x$, $y$, and $z$ are positive integers satisfying $xy + z = yz + x = xz + y = 47$, what is the value of $x + y + z$?


Find the domain of the function $f(x) = \frac{x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1}{x^3 - 9x}$.

Solution: \boxed{(-\infty, -3) \cup (-3, 0) \cup (0, 3) \cup (3, \infty)}


If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 120$ and $\gcd(a,c) = 360$, what is the smallest possible value of $\gcd(b,c)$?


A school has 1234 students that need to be transported for a field trip. Each regular bus can hold a maximum of 45 students, but on this particular day, 3 of the buses available can only hold 30 students each due to maintenance. What is the minimum number of buses needed to transport all the students?


"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0 \leq x < m$. The graphs of $$y \equiv 3x + 1 \pmod{11}$$ and $$y \equiv 7x + 5 \pmod{11}$$ on modulo $11$ graph paper have some points in common. What is the sum of the $x$-coordinates of those points?


Determine the sum of all numbers $x$ for which the function
$$g(x) = \frac{1}{2 + \frac{1}{1 + \frac{1}{x}}}$$
is undefined.


Raj and Sam went to a shop to buy markers for a school project. Raj bought 60 markers, and Sam bought 36. Moreover, each package of markers must also be divisible by 3. What is the largest possible number of markers that could be in each package?


How much greater, in square inches, is the area of a circle of radius 25 inches than a circle of diameter 15 inches? Express your answer in terms of $\pi$.


Simplify: $\frac{2^{n+5} - 4(2^{n+1})}{4(2^{n+4})}$. Express your answer as a common fraction.


What is the greatest common divisor of $8!$ and $11! \cdot 9^2$?


All positive integers whose digits add up to 13 are listed in increasing order: $49, 58, 67, ...$. What is the tenth number in that list?


Six aluminum cans are required to create one new can. In a collection of 500 cans, 20 cans are damaged and can't be recycled. How many total new cans can eventually be made from this collection, considering that newly made cans can also be recycled?


Let
\[g(a,b) = \left\{
\renewcommand{\arraystretch}{3}
\begin{array}{cl}
\dfrac{3ab - a + 4}{3a} & \text{if $a + b \le 5$}, \\
\dfrac{3ab - b - 5}{-3b} & \text{if $a + b > 5$}.
\end{array}
\renewcommand{\arraystretch}{1}
\right.\]
Find $g(3,2) + g(2,4)$.


Determine how many integer values of $b$ are possible if there are exactly four integers $x$ satisfying the inequality
\[x^2 + bx + 1 \le 0.\]


How many lattice points lie on the hyperbola \(x^2 - y^2 = 3000^2\)?


What is the smallest odd number with five different prime factors?


What is the remainder when $1234\cdot 5678$ is divided by $1000$?


It is currently $2\!:\!45\!:\!00 \text{ p.m.}$. What time will it be in $9876$ seconds?

Solution: \boxed{5\!:\!29\!:\!36 \text{ p.m.}}


$PQRS$ shown below is a cube with volume 8. Find the volume of pyramid $PQST$.

[asy]
import three;
triple P,Q,R,S,T,U,V,W;
P = (0,0,0);
Q = (2,0,0);
R = (2,2,0);
S= (0,2,0);
T = (0,0,2);
U = Q+T;
V = R + T;
W = S + T;
draw(Q--R--S);
draw(Q--P--S,dashed);
draw(T--U--V--W--T);
draw(P--T,dashed);
draw(Q--U);
draw(R--V);
draw(S--W);
label("$P$",P,S);
label("$Q$",Q,W);
label("$R$",R,S);
label("$S$",S,E);
label("$T$",T,N);
label("$U$",U,W);
label("$V$",V,SW);
label("$W$",W,E);
[/asy]


Seven balls are numbered 1 through 7 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be odd and greater than 15? Express your answer as a common fraction.


A box contains eight cards. Four of the cards are green on both sides, two cards are green on one side and yellow on the other, and two of the cards are yellow 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 yellow, what is the probability that the other side is yellow?


How many positive three-digit integers are multiples of both 6 and 8?


How many perfect squares are three-digit and divisible by $4$?


Find the sum of the coefficients in the polynomial \( 2(4x^8 - 5x^3 + 6) + 8(x^6 + 3x^4 - 4) \) when it is fully simplified.


How many abundant numbers are there less than 50?


The number of diagonals of a regular polygon is subtracted from the number of sides of the polygon and the result is 2. What is the number of sides of this polygon?


In rectangle $ABCD$, $AB = 9$ cm, $BC = 12$ cm, and $DE = DF$. The area of triangle $DEF$ is one-third the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.


An airplane has three sections: First Class (24 seats), Business Class (30% of the total number of seats), and Economy (2/3 of the total number of seats). How many seats does the plane have?


What is the period of \( y = \tan x + \csc x \)?


A bag contains 15 red marbles, 10 blue marbles, and 5 green marbles. Four marbles are selected at random and without replacement. What is the probability that two marbles are red, one is blue, and one is green? Express your answer as a common fraction.


Solve the equation $x^2 - 6x + 15 = 25$ for $x$. If $c$ and $d$ are the solutions with $c \geq d$, find the value of $3c + 2d$.

Solution: \boxed{15 + \sqrt{19}}


Five concentric circles are drawn with radii of 1, 3, 5, 7, and 9. The innermost circle is painted black, the next ring is white, the following ring is black, the next one is white, and the outermost ring is black again. Calculate the ratio of the total black area to the total white area, and express your answer as a common fraction.


How many integers \( n \neq -1, 0, 1 \) satisfy the inequality \(\frac{1}{|n|} \geq \frac{1}{12}\)?


A line contains the points $(-2, 7)$, $(7, k)$, and $(21, 4)$. What is the value of $k$?

Solution: \boxed{k = \frac{134}{23}}


Find the sum of the coefficients in the polynomial $-3(x^8 - x^5 + 2x^3 - 6) + 5(x^4 + 3x^2) - 4(x^6 - 5)$.


How many two-digit positive integers are congruent to 2 (mod 4)?


Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ f(x) f(y) - f(xy) = x + y + 1 \] for all real numbers \( x \) and \( y \).

Let \( n \) be the number of possible values of \( f(0) \), and let \( s \) be the sum of all possible values of \( f(0) \).  Find \( n \times s \).


A basketball league has twelve teams. During the regular season, each of the twelve teams plays exactly four games against each of the other teams. Additionally, each team must participate in two knockout matches. Calculate the total number of games played during the regular season and knockout matches combined.

Solution: \boxed{276 \text{ total games}}


Let \(O\) be the origin, and let \((a,b,c)\) be a fixed point. A plane passes through \((a,b,c)\) and intersects the \(x\)-axis at \((d,0,0)\), the \(y\)-axis at \((0,e,0)\), and the \(z\)-axis at \((0,0,f)\), where \(d\), \(e\), and \(f\) are non-zero constants, all distinct from \(O\). Let \((p,q,r)\) be the center of the sphere passing through \(O\), \(A\), \(B\), and \(C\). Find
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]


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


What is the smallest positive four-digit number divisible by 6 which has three even and one odd digit?


Suppose the point $(2,3)$ is on the graph of $y=\frac{f(x)}{3}$. Then determine one point that must be on the graph of $y=\frac{f^{-1}(x)}{3}$ and find the sum of that point's coordinates.


Seven balls are numbered 1 through 7 and placed in a bowl. Josh will randomly choose a ball from the bowl, observe its number, then return it to the bowl. He will then randomly choose another ball and observe its number. What is the probability that the sum of the two numbers will be odd and greater than 9? Express your answer as a common fraction.


A bag contains 4 green, 3 blue, and 5 yellow chips. If all 12 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that all green chips are drawn consecutively, all blue chips are drawn consecutively, and all yellow chips are drawn consecutively, and in the specific order of either green-blue-yellow or yellow-green-blue?


Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers.


The graph of the rational function $\frac{1}{p(x)}$ is shown. If $p(x)$ is a quadratic and $p(-3) = 9$, find $p(x).$

[asy]
size(8cm);
import graph;

Label f;
f.p=fontsize(6);

real f(real x) {return 1/(2*(x+2)*(x-2));}

int gridsize = 5;
draw((-gridsize,0)--(gridsize,0), black+1bp, Arrows(8));
draw((0,-gridsize)--(0, gridsize), black+1bp, Arrows(8));
label("$x$", (gridsize, 0), E);
label("$y$", (0, gridsize), N);
label("$0$", (0,0),SE, p=fontsize(8pt));
for (int i=-gridsize+1; i<0; ++i){
    label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt));
    label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));}
for (int i=1; i<=gridsize-1; ++i){
    label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt));
    label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));}

draw(graph(f,-5,-2.05));
draw(graph(f,-1.95,1.95));
draw(graph(f,2.05,5));
draw((-2,-5)--(-2,5), dashed);
draw((2,-5)--(2,5), dashed);
[/asy]

Solution: \boxed{\frac{9}{5}x^2 - \frac{36}{5}}


What is the probability that John gets fewer heads than tails if he flips 12 coins?


Suppose that $x$ is inversely proportional to $y$. If $x = 5$ when $y = 15$, find $x$ when $y = -30$.


The graphs of four functions, labelled (3) through (6), are shown. Note that the domain of function (5) is $$\{-6,-5,-4,-3,-2,0,1,3\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;

picture pic1,pic2,pic3,pic4;

draw(pic1,(-8,0)--(8,0),Arrows(4));
draw(pic1,(0,-8)--(0,8),Arrows(4));
draw(pic2,(-8,0)--(8,0),Arrows(4));
draw(pic2,(0,-8)--(0,8),Arrows(4));
draw(pic3,(-8,0)--(8,0),Arrows(4));
draw(pic3,(0,-8)--(0,8),Arrows(4));
draw(pic4,(-8,0)--(8,0),Arrows(4));
draw(pic4,(0,-8)--(0,8),Arrows(4));

real f(real x) {return x^3-2;}
real h(real x) {return -x-2;}
real k(real x) {return 5/x;}

real x;

draw(pic1,graph(f,-2,2),Arrows(4));
draw(pic3,graph(h,-8,2),Arrows(4));
draw(pic4,graph(k,-8,-0.125*5),Arrows(4));
draw(pic4,graph(k,0.125*5,8),Arrows(4));

dot(pic2,(-6,3)); dot(pic2,(-5,-5)); dot(pic2,(-4,1)); dot(pic2,(-3,0));
dot(pic2,(-2,2)); dot(pic2,(0,-4)); dot(pic2,(1,-3)); dot(pic2,(3,-1));

label(pic1,"(3)",(0,-9));
label(pic2,"(5)",(0,-9));
label(pic3,"(6)",(0,-9));
label(pic4,"(4)",(0,-9));

add(pic1);
add(shift(20)*pic2);
add(shift(0,-20)*pic3);
add(shift(20,-20)*pic4);
[/asy]


Let set $\mathcal{A}$ be a 100-element subset of $\{1,2,3,\ldots,120\}$, and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$


How many three-digit numbers starting with an even digit and ending with an odd digit have the property that their digits, read left to right, are in strictly increasing order?


Find a monic polynomial of degree $4$, in $x$, with rational coefficients such that $\sqrt{3} + \sqrt{5}$ is a root of the polynomial.

Solution: \boxed{x^4 - 16x^2 + 4}


Find
\[
\cos \left( 4 \arccos \frac{2}{5} \right).
\]


In a basketball league, the number of games won by five teams is displayed in a graph, but the team names are missing. The clues below provide information about the teams:

1. The Sharks won more games than the Falcons.
2. The Knights won more games than the Wolves, but fewer than the Dragons.
3. The Wolves won more than 22 games.

Given the bar graph data: 24 wins, 27 wins, 33 wins, 36 wins, 38 wins. How many games did the Knights win?


$x$ is a real number with the property that $x+\frac{1}{x} = 5$. Let $T_m = x^m + \frac{1}{x^m}$. Determine the value of $T_6$.


In a bag, there are only red, blue, and yellow marbles. The ratio of red marbles to blue marbles to yellow marbles is $2:3:4$. There are 24 blue marbles in the bag. How many marbles are there in total?


Every week, Judy goes to the supermarket and purchases: $6$ carrots at $\$1$ each (buy-one-get-one-free offer this week), $3$ bottles of milk at $\$3$ each, $2$ pineapples at $\$4$ each (half price this week), $3$ bags of flour at $\$5$ each, and one $\$8$ container of ice cream. Judy also has a coupon for $\$6$ off any order of $\$30$ or more. How much does Judy spend on this shopping trip?


How many positive integers, not exceeding 150, are multiples of 2 or 3 but not 5?


Suppose I have 5 shirts, 5 ties, and 4 pairs of pants. If an outfit requires a shirt, pants, and may include a tie and/or a belt (with two choices of belts or opting not to wear a belt), how many different outfits can I make?


If $k$ and $\ell$ are positive 5-digit integers such that $\gcd(k,\ell)=5$, what is the smallest possible value for $\mathop{\text{lcm}}[k,\ell]$?


What is the area enclosed by the graph of \( |x| + |3y| = 9 \)?


Triangle $PAB$ is formed by three tangents to circle $O$ and $\angle APB = 50^\circ$. Find $\angle AOB$.


In the Zorgian language, there are 4 words: "zor", "glib", "mek", and "troz". In a sentence, "zor" cannot come directly before "glib", and "mek" cannot come directly before "troz"; all other sentences are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Zorgian?


The Happy Baker baked a round cake with a boundary described by the equation \(x^2 + y^2 + 1 = 2x + 6y\). She needs to determine the size of her cake to properly decorate it. What is the radius of the cake?


What is the sum of the digits of the base-3 expression for $345_{10}$?


In $\triangle ABC$, the ratio $AC:CB$ is $2:5$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $Q$ ($A$ is between $Q$ and $B$). Determine the ratio $QA:AB$.


Calculate $\sqrt{\sqrt[4]{0.00000081}}$. Express your answer as a decimal to the nearest tenth.


Given the points $(2, 3)$, $(10, 9)$, and $(6, m)$, where $m$ is an integer, determine the sum of all possible values of $m$ for which the area of the triangle formed by these points is a maximum.


Determine the value of $h$ such that the equation
\[\frac{x - 3}{hx + 2} = x\] has exactly one solution.


Let \(\mathbf{u},\) \(\mathbf{v},\) and \(\mathbf{w}\) be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{4} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}.\) Find \(\sin \phi.\)


A library has eight identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the library and at least two are checked out?


In a triangle, two angles measure 60 and 30 degrees. If the side opposite the 30-degree angle measures $9$ units, find the sum of the lengths of the two remaining sides. Express your answer as an exact value, simplifying where necessary.

Solution: \boxed{18 + 9\sqrt{3}}


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

Find the number of possible values of $g(3)$ and the sum of these values. Let $m$ be the number of possible values and $t$ be the sum of these values. Determine the product $m \times t$.


A square is divided into two congruent rectangles horizontally and then one of those rectangles is further divided into two congruent rectangles vertically. If the perimeter of the square is 160 units, what is the perimeter of one of the smaller rectangles created by the vertical division?

Solution: \boxed{80 \text{ units}}


Points $A$, $B$, $C$, $D$, and $E$ lie on a line in that order. If $AB=3$ units, $BC=7$ units, $CD=4$ units, and $AE=20$ units, what is the ratio of $AC$ to $DE$?


What is the greatest four-digit number which is a multiple of 17?


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


What must be the value of the coefficient \(c\) in \(P(x) = x^3 + 3x^2 + cx + 15\) so that \(x - 3\) is a factor of \(P\)?


In a school trip, there are 7 students and a teacher who want to take rides in the park. The park allows for 4 riders at a time on each ride. After selecting a riding group of 4, they are to choose between riding either the Ferris wheel or the roller coaster. How many different combinations of riding groups and ride choices can they make?


John, also known for his peculiar choices, is crafting a rectangular storage container with square bases. He prefers the height of the container to be twice the side length of the square base. Determine the minimum height he should use if he wants his container to have a total surface area of at least 150 square units while using the least amount of material possible.


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

$\bullet$ The only integer in the list that occurs more than once is $10,$

$\bullet$ its median is $11,$ and

$\bullet$ its average (mean) is $12.$

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


Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length?


Evaluate the expression $\frac{1}{(5^{2})^{4}} \cdot (5)^{15}$.


Expand the product \((x + 3)(x - 4)(x + 1)\).

Solution: \boxed{x^3 - 13x - 12}


Adam has 20 coins consisting of rare and common types. He knows that 5 of his rare coins are worth $15$ in total. Each common coin is worth $1$ less than each rare coin. If he has an equal number of rare and common coins, how much is his entire collection worth?


Let $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the side lengths, and let $R$ denote the circumradius. Find $OH^2$ if $R = 9$ and $a^2 + b^2 + c^2 = 83$.


Let \( P \) be the parabola with equation \( y = x^2 + 5 \) and let \( Q = (10, 10) \). There are real numbers \( r \) and \( s \) such that the line through \( Q \) with slope \( m \) does not intersect \( P \) if and only if \( r < m < s \). What is \( r + s \)?


For each positive integer \( n \), let \( n! \) denote the product \( 1 \cdot 2 \cdot 3 \cdot \ldots \cdot (n-1) \cdot n \).

What is the remainder when \( 10! \) is divided by \( 13 \)?


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


What is the remainder when \(8x^4 - 10x^3 + 16x^2 - 18x + 5\) is divided by \(4x - 8\)?


Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 942^\circ.$


Let $c$ and $d$ be real numbers. One of the roots of $x^3 + cx + d = 0$ is $2 + i \sqrt{2}$. Find $c + d$.


Let $g(n)$ be a function such that for a given integer $n$, returns the smallest integer $m$, where $m$ is the smallest possible integer such that $m!$ is divisible by $n$. Given that $n$ is a multiple of 35, what is the smallest value of $n$ such that $g(n) > 15$?


In $\triangle ABC$, the ratio $AC:CB$ is $2:5$. The bisector of the angle at $C$ intersects $AB$ at point $P$. Find the ratio $PA:PB$.


A circular park consists of an outer walking path, a ring-shaped garden, and a central circular pond. The walking path is 10 feet wide, the garden ring is 8 feet wide, and the pond has a diameter of 16 feet. What is the area of the walking path?


A pyramid with a volume of 60 cubic feet has a rectangular base. If the length of the base is tripled, the width is doubled, and the height is increased by $20\%$, what is the new volume of the pyramid, in cubic feet?

Solution: \boxed{432 \text{ cubic feet}}


Consider the polynomial equation \[ x^4 + ax^3 + bx^2 + cx + d = 0, \] where \( a, b, c, \) and \( d \) are rational numbers. This polynomial has roots \( 2-\sqrt{5} \) and \( 1 \). Find the value of \( c \) if the polynomial also has a root that is three times an integer.


Find the number of triples $(x, y, z)$ of real numbers that satisfy
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z + 1), \\
y &= 2020 - 2021 \operatorname{sign}(x + z + 1), \\
z &= 2020 - 2021 \operatorname{sign}(x + y + 1).
\end{align*}
Note: For a real number $a,$
\[\operatorname{sign}(a) = \left\{
\begin{array}{cl}
1 & \text{if } a > 0, \\
0 & \text{if } a = 0, \\
-1 & \text{if } a < 0.
\end{array}
\right.\]


$\triangle PQR$ is similar to $\triangle STU$. The length of $\overline{PQ}$ is 7 cm, and the length of $\overline{QR}$ is 10 cm. If the length of $\overline{ST}$ is 4.9 cm, what is the length of $\overline{TU}$ in centimeters? Express your answer as a decimal to the nearest tenth.


What is the sum of the digits of the base-2 expression for $365_{10}$?


Find the sum of the coefficients in the polynomial \( 3(x^8 - 2x^5 + 4x^3 - 7) - 5(2x^4 - 3x^2 + 8) + 6(x^6 - 3) \).


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}}}


A point $P$ is randomly selected from the square region with vertices at $(\pm 2, \pm 2)$. What is the probability that $P$ is within two units of the vertex at $(2,2)$? Express your answer as a common fraction in terms of $\pi$.


In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid, one in each quadrant. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value of $A+B$?


A certain amount of helium gas is held in a 3.00 liter container under a pressure of 4 kPa. If the same amount of helium is transferred to a new container with a volume of 6.00 liters while keeping the temperature constant, what will the final pressure of the helium be in kPa?


Find the exact value, in radians, of $\arctan \frac{1}{3} + \arctan \frac{3}{8}.$

Solution: \boxed{\arctan \frac{17}{21}}


Tamika selects two different numbers at random from the set $\{10, 11, 12\}$ and adds them. Carlos takes two different numbers at random from the set $\{4, 6, 7\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.


Javier is excited to visit six attractions at Disneyland. Two of these attractions are a parade and a fireworks show, which must be seen in that exact order, but can occur at any point in his day. How many different ways can he order his visit to all six attractions, considering the constraints on the parade and fireworks show?


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


In how many ways can 8 distinct beads be placed on a bracelet? (Again, two arrangements are considered the same if one can be rotated or reflected to create the other.)


Write down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in this list?


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

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


Given that $\sqrt{5}+3$ is a root of the equation \[x^3 + ax^2 + bx + 16 = 0\]and that $a$ and $b$ are rational numbers, compute $b.$


A point is selected at random from the portion of the number line from 0 to 10. What is the probability that the point is closer to 7 than to either 0 or 10? Express your answer as a fraction.


Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = 7,$ $\|\mathbf{c}\| = 2,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = 2\mathbf{c}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b},$ then find $\sin \theta.$


Two concentric circles have radii 24 and 33 units, respectively. They define a ring-shaped region between them. A new circle is to be drawn such that its area matches that of the ring-shaped region. Determine the radius of this new circle, and express your answer in simplest radical form.


A ball is dropped from a height of 15 feet and bounces back up to half the distance it just fell each time. After how many bounces will the ball first reach a maximum height less than 0.5 feet?


The Imaginary Island uses an alphabet of ten letters: A, B, D, F, G, K, L, M, N, R. Suppose their license plates consist of five letters. How many license plates are possible that begin with either A or B, end with R, cannot contain the letter F, and have no letters that repeat?


Two friends meet each day for a walk in the park, arriving independently at random times between 4 p.m. and 5 p.m., and each stays for exactly $n$ minutes. The probability that either one arrives while the other is already there is $50\%$. Express $n$ as $d - e\sqrt{f}$, where $d, e,$ and $f$ are positive integers, and $f$ is not divisible by the square of any prime. Find $d + e + f$.


When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is 3. When $p(x)$ is divided by $x + 3,$ the remainder is -4. Find the remainder when $p(x)$ is divided by $(x + 1)(x + 3).$

Solution: \boxed{\frac{7}{2}x + \frac{13}{2}}


When the vectors $\begin{pmatrix} -6 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Find $\mathbf{p}.$

Solution: \boxed{\begin{pmatrix} -\frac{12}{17} \\ \frac{54}{17} \end{pmatrix}}


Let \(\bold{v} = \begin{pmatrix} 7 \\ -4 \\ 3 \end{pmatrix}\), \(\bold{w} = \begin{pmatrix} 13 \\ -1 \\ 2 \end{pmatrix}\), and \(\bold{u} = \begin{pmatrix} 1 \\ 0 \\ 6 \end{pmatrix}\). Find the volume of the parallelepiped formed by vectors \(\bold{v}\), \(\bold{w}\), and \(\bold{u}\).


George wants to buy pencils for his class. He has $\$9.30$ and each pencil costs $\$1.05$, tax included. If he buys more than 8 pencils, he gets a 10% discount on the total cost. How many pencils can George buy at most?


Solution: \boxed{\frac{1}{41 + 17\sqrt{5} - 2\sqrt{2}}}


A frozen yogurt shop offers five flavors of yogurt and seven different toppings. How many combinations of one flavor, two different toppings, and an option to double one of the toppings or not double at all, are available?


In Ms. Green's 30-student chemistry class, no two students have the same initials, and each student's first name and last name begin with the same letter. If the letter "Y" is considered a consonant, what is the probability of randomly picking a student whose initials are vowels? Express your answer as a common fraction.


Starting with the number 150, Shaffiq repeatedly divides his number by two and then takes the greatest integer less than or equal to that number. How many times must he do this before he reaches the number 1?


What is the base ten equivalent of $54321_7$?


Ms. Johnson's ninth-grade class wants to participate in the annual three-person-team basketball tournament. Before the main sixteen team tournament, there is a preliminary round where four teams compete to fill the last two spots in the sixteen team bracket. The losing team of each game is eliminated from the tournament. How many games will be played in total to determine the winner, starting from the preliminary round?


When three standard dice are tossed, and the numbers $a, b, c$ are obtained, find the probability that $$(a-1)(a-6)(b-1)(b-6)(c-1)(c-6) \neq 0$$


A circle with center $C$ is tangent to the positive $x$-axis and $y$-axis and externally tangent to a circle centered at $(5,0)$ with a radius of $2$. What is the sum of all possible radii of the circle with center $C$?


Maria, Juan, Carlos, and Manu take turns flipping a coin in the given order. The first one to flip tails wins. What is the probability that Manu will win? Express your answer as a common fraction.


Suppose $x$ is a multiple of 120. Find the greatest common divisor of $g(x) = (3x + 4)(5x + 3)(11x + 6)(x + 11)$ and $x$.


In the right-angled triangle $LMN$, suppose $\sin N = \frac{3}{5}$. If the length of $LM$ is 15, calculate the length of $LN$.


The base of a triangular piece of paper $ABC$ is $15\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base. The area of the triangle that projects below the base is $25\%$ that of the area of the triangle $ABC$. What is the length of $DE$, in cm?

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


In triangle $XYZ$, where $XZ = 13$ units and $XY=12$ units, and angle $XYZ$ is a right angle at $Y$. There are two squares, $WXZY$ and $YZUV$, formed along the legs $XZ$ and $YZ$, respectively. What is the sum of the areas of the two squares?

Solution: \boxed{194 \text{ square units}}


All positive integers whose digits add up to 13 are listed in increasing order: $49, 58, 67, ...$. What is the eleventh number in that list?


Calculate the product of $0.\overline{256}$ and $12$, and express your result as a fraction in its simplest form.


The repeating decimal for $\frac{4}{13}$ is $0.cdc\ldots$ What is the value of the sum $c + d$?


A polynomial $f(x)$ of degree $15$ is divided by $d(x)$ resulting in a quotient $q(x)$ of degree $9$ and a remainder $5x^4 + 2x^2 - 3x + 7$. Determine the degree of $d(x)$.


How many more digits does the base-4 representation of $2023_{10}$ have than the base-9 representation of $2023_{10}$?


In triangle \( \triangle ABC \), \( O \) is the center of the circumcircle with \( \angle BOC = 130^\circ \) and \( \angle AOB = 150^\circ \). Determine the measure of \( \angle ABC \).


What is the base ten equivalent of $23456_{7}$?


The solutions to the equation $z^5 = -32i$ can be expressed in the form
\begin{align*}
z_1 &= s_1 (\cos \alpha_1 + i \sin \alpha_1), \\
z_2 &= s_2 (\cos \alpha_2 + i \sin \alpha_2), \\
z_3 &= s_3 (\cos \alpha_3 + i \sin \alpha_3), \\
z_4 &= s_4 (\cos \alpha_4 + i \sin \alpha_4), \\
z_5 &= s_5 (\cos \alpha_5 + i \sin \alpha_5),
\end{align*}
where $s_k > 0$ and $0^\circ \le \alpha_k < 360^\circ.$ Find $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_5,$ in degrees.


Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is either a spade or an Ace and the second card is an Ace?


What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 15$, $y = 10$ and the $y$-axis? Express your answer as a decimal to the nearest tenth.


A dog discovers $521_{7}$ different paths it can take during its daily walk. How many paths are there in decimal (base 10)?


In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the internal angle at $C$ intersects $AB$ at point $D$. Find the ratio $AD:DB$.


The score on a certain 150-point test varies directly with the square of the time a student spends preparing for the test. If a student receives 90 points on a test for which she spent 2 hours preparing, what score would she receive on the next test if she spent 3 hours preparing?


Let \( x \) and \( y \) be positive real numbers. Find the minimum value of:
\[
\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 1024 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 1024 \right).
\]


Solve for $x$: $x = \frac{\sqrt{8^2 + 15^2}}{\sqrt{49 + 64}}$. Express as a common fraction.


A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.


Calculate how many diagonals can be drawn in a regular dodecagon (12-sided polygon), where a diagonal connects every third vertex.


Let $\star(x)$ denote the sum of the digits of a positive integer $x$. Define $\mathcal{S}$ as the set of positive integers such that for all elements $n$ in $\mathcal{S}$, $\star(n) = 15$ and $0 \leq n < 10^7$. Compute the value of $\star(m)$, where $m$ is the number of elements in $\mathcal{S}$.


What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 18] = 90$?


Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. There are four such connected triangles in total. The hypotenuse of the largest triangle is 10 centimeters. What is the length of the longer leg of the smallest triangle? Express your answer as a common fraction.


Let $u$ and $v$ be the two real roots of the quadratic equation $x^2 - 2x\sqrt{3} + 1 = 0$. Determine $u^6 + v^6$.


Simplify $\left( \frac{1}{3m} \right)^{-3} \cdot (3m)^4$.


Triangles $PQR$ and $XYZ$ are similar. Triangle $XYZ$ is also similar to triangle $ABC$. Given $PQ = 8$, $QR = 16$, $BC = 24$ units, and $YZ = 12$ units, find the length of segment $AB$.


In triangle $ABC$, angle $ACB$ is 60 degrees, and angle $CBA$ is 50 degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of the diameter which goes through $A$. Find the angle $DAE$, in degrees.


Find the smallest positive integer $N$ such that $N$ and $N^2$ end in the same sequence of three digits $abc$ when written in base $10$, where $a$ is not zero.


If $f(x)$ is a monic quartic polynomial such that $f(-2)=-4$, $f(1)=-1$, $f(3)=-9$, and $f(5)=-25$, find $f(0)$.


The Eiffel Tower in Paris is 1,083 feet tall. A tourist purchases a souvenir model of the Eiffel Tower that is 8 inches tall. How many feet of the actual Eiffel Tower does one inch of the model represent?

Solution: \boxed{135.375 \text{ feet}}


A 12-foot by 15-foot floor is tiled with square tiles of size 2 feet by 2 feet. Each tile includes a pattern of four white quarter circles of radius 1 foot centered at each corner of the tile. The remaining portion of the tile is shaded. Calculate the total shaded area of the floor.

Solution: \boxed{180 - 45\pi \text{ square feet}}


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots, as shown below?
[asy]size(3cm,3cm); for (int i=0; i<5; ++i) { for (int j=0; j<5; ++j) { filldraw(Circle((i, j), .05), black, black); } } [/asy] (Two rectangles are different if they do not share all four vertices.)


Point \(O\) is the center of an ellipse with major axis \(\overline{AB}\) and minor axis \(\overline{CD}\). Point \(F\) is one focus of the ellipse. If \(OF = 8\) and the diameter of the inscribed circle of triangle \(OCF\) is 4, compute the product \((AB)(CD)\).


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


Let \((x, y)\) be a solution to the system of equations
\[
\begin{aligned}
\lfloor x \rfloor - \{y\} &= 1.7, \\
\{x\} + \lfloor y \rfloor &= 4.4.
\end{aligned}
\]
Compute \(|x - y|\).


How many total days were there in the years from 2010 through 2015?


Find the remainder when the sum $1! + 2! + 3! + 4! + 5! + \cdots + 10!$ is divided by $7$.


Simplify the following expression:
$$ \frac{12!}{10! - 2\cdot 9!} $$


Find the positive integer $n$ such that
\[\sin \left( \frac{\pi}{3n} \right) + \cos \left (\frac{\pi}{3n} \right) = \frac{\sqrt{2n}}{3}.\]


What is the degree measure of the smaller angle formed by the hands of a clock at 2:30?


How many positive factors of 360 are perfect squares?


In triangle $PQR$, $PQ = PR$, and $\overline{PS}$ is an altitude. Point $T$ is on the extension of $\overline{QR}$ such that $PT = 12$. The values of $\tan \angle RPT$, $\tan \angle SPT$, and $\tan \angle QPT$ form a geometric progression, and the values of $\cot \angle SPT$, $\cot \angle RPT$, $\cot \angle SPQ$ form an arithmetic progression. Determine the area of triangle $PQR$.


In the diagram, $BP$ and $BQ$ quadrisection $\angle ABC$ with $BP$ being the first cut, and $BQ$ being the last. $BM$ bisects the angle immediately preceding $BQ$. Find the ratio of the measure of $\angle MBQ$ to the measure of $\angle ABQ$.


In triangle $XYZ$, the sides are tangent to a circle with center $O$. $\angle XYZ = 75^\circ$ and $\angle YXO = 40^\circ$. Find $\angle YZX$.


A square pyramid with a base edge of 24 inches and an altitude of 18 inches has a smaller similar pyramid cut from its apex. The altitude of the smaller pyramid is one-third that of the original pyramid. What fraction of the original pyramid's volume is the volume of the remaining frustum?


$ABCDEFGH$ is a cube with each side of length $a$. Point $G$ is the top far corner, and point $A$ is the bottom near corner. Point $E$ is directly above point $A$. Find $\cos \angle GAE$.


The function \( g(x) \) satisfies:
\[ 4g(x) - 3g\left(\frac{1}{x}\right) = x^2 + 1 \] for all \( x \neq 0 \). Find \( g(3) \).


Let $x$ and $y$ be positive real numbers such that $x + y = 2.$ Find the set of all possible values of $\frac{1}{x} + \frac{1}{y}.$

Solution: \boxed{[2, \infty)}


Two different numbers are randomly selected from the set $T = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\}$. The probability that their sum is 13 would be greater if the number $m$ had first been removed from set $T$. What is the value of $m$?


Let $S$ be the set of all real values of $x$ with $0 < x < \pi$ such that $\sin x$, $\cos x$, and $\cot x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\cot^2 x$ over all $x$ in $S$.


How many four-digit numbers satisfy the property that the second digit is the average of the first and the third digits?


Determine the smallest positive integer $n$ such that $7^n \equiv n^7 \pmod 5$.


If $c > 0$ and $d > 0$ are integers, what is the smallest possible positive value of $\left(\frac{c+d}{c-d}\right)^2 + \left(\frac{c-d}{c+d}\right)^2$?


Find the minimum value of:
\[(8 - x)(6 - x)(8 + x)(6 + x).\]


Andrew and John are both Beatles fans. Their respective collections share 12 of the same albums. Andrew has 23 albums in his collection. In addition to these albums, Andrew also has 5 pieces of exclusive Beatles memorabilia which John does not have. John has 8 albums that are not in Andrew's collection. How many items are there in either Andrew's or John's collection or memorabilia, but not both?


In triangle $ABC$, angle $ACB$ is $40$ degrees, and angle $CBA$ is $80$ degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of the diameter which goes through $A$. Find the angle $DAE$, in degrees.


One of the asymptotes of a hyperbola has equation $y = -2x + 5$. The foci of the hyperbola have the same $x-$coordinate, which is $2$. Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".

Solution: \boxed{y = 2x - 3}


Consider the lines:
\[
y = 3x + 4, \quad 2y = 2x + 6, \quad 3y = 9x - 1, \quad y = -\frac{1}{3}x + 5, \quad 4y = -12x + 3.
\]
Let's say a pair of lines is $\emph{good}$ if the two lines are either parallel or perpendicular to each other. Among all the pairs of lines shown, how many pairs are good?


Determine the value of $y$ that satisfies $\sqrt[3]{y\sqrt[4]{y^5}}=4$.


Find the maximum value of the expression:
\[
\sin \theta_1 \cos \theta_2 + \sin \theta_2 \cos \theta_3 + \sin \theta_3 \cos \theta_4 + \sin \theta_4 \cos \theta_5 + \sin \theta_5 \cos \theta_1,
\]
over all real numbers $\theta_1, \theta_2, \theta_3, \theta_4, \theta_5$.


Express $0.\overline{137} + 0.\overline{024}$ as a common fraction.


How many positive four-digit integers of the form $\_\_35$ are divisible by 15?


What is the sum of all integers in the set $\{1,2,3,4,5,6,7,8,9,10\}$ that are primitive roots $\pmod{11}$?


How many two-digit positive integers are congruent to 1 (mod 4)?


Circle \( D \) has a radius of 8 cm. What is the area of the largest possible inscribed equilateral triangle in circle \( D \)?

Solution: \boxed{48 \sqrt{3} \text{ square centimeters}}


A square has sides of length 12, and a circle centered at the center of the square has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.


Define a new operation $a \clubsuit b = \frac{2a^2}{b^2}$. Calculate the value of $(4 \clubsuit (2 \clubsuit 8)) \clubsuit 2$.


Find the number of real solutions to the equation
\[\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = x.\]


A sphere is inscribed in a right circular cylinder. Given that the height of the cylinder is 10 inches and the radius of the base of the cylinder is 5 inches, calculate the volume of the inscribed sphere. Express your answer in terms of \(\pi\).

Solution: \boxed{\frac{500}{3} \pi}


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? (Two rectangles are different if they do not share all four vertices.)


Alice and Bob are playing the Smallest Positive Integer Game again. This time, Alice says, "My number is 60." Bob replies, "What kind of smallest number is that? Every prime factor of your number is also a prime factor of my number."

What is the smallest possible number that Bob could have?


Let \( p, q, r \) be the roots of the polynomial \( 4x^3 - 6x^2 + 17x - 10 = 0 \). Find the sum \( pq + pr + qr \).


What is the smallest positive number that is prime and $12$ less than a perfect square whose root is a prime number?


Find the remainder when $102938475610$ is divided by $12$.


A binary operation $\diamondsuit$ is defined such that $x \diamondsuit (y \diamondsuit z) = (x \diamondsuit y) \cdot z$ and $x \diamondsuit x = 1$ for all nonzero real numbers $x, y,$ and $z$. Solve the equation $1008 \diamondsuit (3 \diamondsuit x) = 50$.


A line with slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(6,0)$ and the $y$-axis at $D$. The lines intersect at $E(3,3)$. Find the area of the shaded quadrilateral $OBEC$.


I randomly select an integer $p$ from 1 to 15 inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 5p - 3q = -1$? Express your answer as a common fraction.

Solution: \boxed{\frac{5}{15} = \frac{1}{3}}


Find the quotient when $x^6 + 5$ is divided by $x - 1.$

Solution: \boxed{x^5 + x^4 + x^3 + x^2 + x + 1}


The product of three different positive integers is equal to \( 11^3 \). What is the sum of the three integers?


Let $a$, $b$, and $c$ be distinct integers where $b$ is the average of $a$ and $c$, i.e., $b = \frac{a+c}{2}$. Given that
\[\frac{a}{b-c} + \frac{b}{c-a} + \frac{c}{a-b} = 0,\]
find the value of
\[\frac{a}{(b-c)^3} + \frac{b}{(c-a)^3} + \frac{c}{(a-b)^3}.\]


Two diagonals of a regular hendecagon (an 11-sided polygon) are chosen. What is the probability that their intersection lies inside the hendecagon?


Real numbers $c$ and $d$ are chosen with $1<c<d$ such that no triangle with positive area has side lengths $1, c,$ and $d$ or $\tfrac{1}{d}, \tfrac{1}{c},$ and $1$. What is the smallest possible value of $d$?

Solution: \boxed{\frac{3 + \sqrt{5}}{2}}


Three tour guides are leading eight tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?


What is the area, in square units, of a triangle whose vertices are at $(4, -1)$, $(2, 7)$, and $(4, 4)$?


In $\triangle ABC$, where $AC = BC = 10$ and $AB = 6$, point $D$ is on line $AB$ such that $B$ is between $A$ and $D$. The distance from $C$ to $D$, $CD$, is $12$. What is the length of $BD$?

Solution: \boxed{\sqrt{53} - 3}


Triangle \( \triangle PQR \) has side-lengths \( PQ = 15, QR = 30, \) and \( PR = 20. \) The line through the incenter of \( \triangle PQR \) parallel to \( \overline{QR} \) intersects \( \overline{PQ} \) at \( X \) and \( \overline{PR} \) at \( Y. \) What is the perimeter of \( \triangle PXY? \)


Two sides of a right triangle have 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.


Find the radius of the circle with equation $25x^2 - 50x + 25y^2 + 100y + 125 = 0.$


Two real numbers \(x\) and \(y\) satisfy \(x + y = 4\) and \(x^3 - y^3 = 64\). Compute \(xy\).


Evaluate the product $\lfloor -3.5 \rfloor \cdot \lceil 3.5 \rceil \cdot \lfloor -2.5 \rfloor \cdot \lceil 2.5 \rceil \cdot \lfloor -1.5 \rfloor \cdot \lceil 1.5 \rceil$.


What is the remainder when the sum of the numbers from $11065$ to $11085$ inclusive, each incremented by $2$, is divided by $10$?


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


The average of the numbers $1, 2, 3,\dots, 148, 149,$ and $x$ is $50x$. What is $x$?


Let $P(x) = b_0 + b_1x + b_2x^2 + \dots + b_mx^m$ be a polynomial with integer coefficients, where $0 \le b_i < 5$ for all $0 \le i \le m$.

Given that $P(\sqrt{5})=23+19\sqrt{5}$, compute $P(3)$.


Humanity discovers 15 habitable planets, of which 7 are Earth-like and 8 are Mars-like. Earth-like planets require 3 units of colonization effort, whereas Mars-like planets need only 1 unit. If a total of 18 units of colonization effort is available, how many different combinations of planets can be colonized, given that all planets are unique?


The integer $m$ is the smallest positive multiple of 18 such that every digit of $m$ is either 9 or 0. Compute $\frac{m}{18}$.


Palindromes are numbers that read the same forwards and backwards, like 5775. What is the least possible four-digit palindrome that is divisible by 5?


Given five members of a club -- Alice, Bob, Carol, Dave, and Eve -- in how many ways can three be chosen to be the three officers (president, secretary, and treasurer) of the club, assuming no person holds more than one job and Dave cannot be the treasurer?


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


A solid rectangular block is created by stacking together $N$ congruent 1-cm cubes face to face. When the block is positioned to expose three faces, exactly $143$ of the 1-cm cubes are completely covered from view. Determine the smallest possible value of $N.$


On an 8x8 chessboard, squares are labeled according to the formula $\frac{1}{i+j-1}$, where $i$ is the row number and $j$ is the column number, both starting from 1. Eight squares are chosen such that exactly one square is chosen in each row and each column. Find the minimum sum of the labels of the eight chosen squares.


Evaluate the sum
\[\text{cis } 60^\circ + \text{cis } 70^\circ + \text{cis } 80^\circ + \dots + \text{cis } 140^\circ\]
and express it in the form \( r \, \text{cis } \theta \), where \( r > 0 \) and \(0^\circ \leq \theta < 360^\circ\). Find \( \theta \) in degrees.


Compute \[ \left\lfloor \dfrac {2020^3}{2018 \cdot 2019} - \dfrac {2018^3}{2019 \cdot 2020} \right\rfloor,\]where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


According to the table below, how many dollars are in the median value of the 67 salaries paid to this company's employees?

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$140{,}000$\\\hline
Senior Manager&4&$\$95{,}000$\\\hline
Manager&15&$\$80{,}000$\\\hline
Assistant Manager&7&$\$55{,}000$\\\hline
Clerk&40&$\$25{,}000$\\\hline
\end{tabular}


A bag contains 7 red, 8 green, 9 yellow, 10 blue, and 4 purple jelly beans. A jelly bean is selected at random. What is the probability that it is either blue or purple?


Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, different from points $A$ and $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve, which misses two points. Calculate the area of the region bounded by this curve to the nearest integer.


How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|} \geq \frac{1}{10}$?


In a class, 50 students took a quiz. What is the largest number of students who can be awarded extra credit if they score above the class mean?


For how many positive integral values of \( a \) is it true that \( x = 3 \) is the only positive integer solution of the system of inequalities:
$$
\begin{cases}
2x + 1 > 3x - 2 \\
4x - a > -11
\end{cases}
$$


Find the smallest positive integer $b$ such that $x^4 + b^2$ is not prime for any integer $x$, where $b$ is even.


A convex polyhedron $Q$ has $30$ vertices, $70$ edges, and $42$ faces, of which $26$ are triangular, $12$ are quadrilaterals, and $4$ are pentagons. Determine the number of space diagonals in polyhedron $Q$.


The sequence $a_1, a_2, \ldots$ is geometric with $a_1 = a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_4 a_1 + \log_4 a_2 + \cdots + \log_4 a_8 = 840,$ find the number of possible ordered pairs $(a,r)$.


In $\triangle ABC,$ $AB=AC=24$ and $BC=20.$ 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 $ADEF$?


Find the largest constant $n$, such that for any positive real numbers $a,$ $b,$ $c,$ $d,$ and $e,$
\[\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > n.\]


What is the greatest integer $n$ such that $n^2 - 13n + 30 < 0$?


A store owner originally sells jackets for $\$25$ each. He decides to offer a $40\%$ discount for an end-of-season sale. After the season, the remaining jackets are discounted again by $25\%$ of the new sale price. Finally, to clear out inventory, an additional $10\%$ discount is applied to the last reduced price. What is the final sale price of a jacket?


In the game of Zog, dropping $n$ zogs gives a score of the sum of the first $n$ even positive integers. Eating $n$ zogs earns $8n$ points. What is the least number of zogs for which dropping them will earn more points than eating them?


Compute the product of \(0.\overline{456}\) and \(8\), and express your result as a fraction in its simplest form.


What is the greatest common divisor of $442872$ and $312750$?


Find the number of cubic centimeters in the volume of a cylinder formed by rotating a square with side length 20 centimeters about its vertical line of symmetry, and also calculate the total surface area of the cylinder. Express your answer in terms of $\pi$.


A large bronze statue of George Washington is 90 feet tall. A scale model of this statue placed in a museum is 6 inches tall. How many feet of the statue does one inch of the model represent?


Consider a sequence that starts at 1,024,000 and each successive term is obtained by dividing the previous term by 3. Determine the last integer of this sequence.


In a newly designed circular park, there is an outer walking path, an inner garden ring, and a central circular pond. The walking path is 10 feet wide everywhere, the garden ring is 12 feet wide everywhere, and the pond has a diameter of 16 feet. Determine the diameter, in feet, of the circle that forms the outer boundary of the walking path.


How many integers between 800 and 1000 have the sum of their digits equal to 22 and are divisible by 5?


Suppose $\cos S = 0.5$ in a right triangle where $SP = 10$. What is $SR$?

[asy]
pair S,P,R;
S = (0,0);
P = (10,0);
R = (0,10*tan(acos(0.5)));
draw(S--P--R--S);
draw(rightanglemark(S,P,R,18));
label("$S$",S,SW);
label("$P$",P,SE);
label("$R$",R,N);
label("$10$",P/2,S);
[/asy]


How many three-digit numbers are multiples of neither 8 nor 6?


Let \( a \) and \( b \) be angles such that
\[ \cos(a + b) + \sin(a - b) = \cos a + \cos b. \]
Find the maximum value of \(\sin a\).


What is the units digit of the product of all the odd positive integers between 20 and 200?


Determine the number of digits in the value of $3^{10} \times 5^{6}$.


Find all real numbers \(y\) such that
\[
2 \le \frac{y}{3y-4} < 5.
\]
(Give your answer in interval notation.)

Solution: \boxed{\left(\frac{10}{7}, \frac{8}{5}\right]}


A tree is supported by a rope which extends from the top of the tree to a point on the ground 4 meters from its base. When Mia walks 3 meters from the base of the tree toward the point where the rope is attached to the ground, her head just touches the rope. Mia is 1.6 meters tall. How many meters tall is the tree?


Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$?


You have 6 red shirts, 4 green shirts, 7 pairs of pants, 10 green hats, and 9 red hats, all distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, ensuring the shirt and hat are not the same color?


How many 3-digit numbers have the property that the units digit is exactly twice the tens digit?


In parallelogram $PQRS$, the measure of angle $PQR$ is twice the measure of angle $QRS$. What is the measure of angle $PSR$?


Suppose that \( p \) is a prime number and the equation \( 2017_p + 402_p + 114_p + 230_p + 7_p = 301_p + 472_p + 503_p \) holds. How many possible values of \( p \) are there?


Let \[f(x) = \left\{
\begin{array}{cl}
3x + 9 & \text{if } x < -3, \\
-x^2 + 2x + 8 & \text{if } x \ge -3.
\end{array}
\right.\] Find the sum of all values of $x$ such that $f(x) = 6.


In 2010, the sum of the year's digits is $3$.  What is the first year after 2010 in which the sum of the digits is 5?


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


Let \( \mathbf{B} = \begin{pmatrix} x & 3 \\ -4 & y \end{pmatrix} \) for some real numbers \( x \) and \( y \). If
\[ \mathbf{B} + \mathbf{B}^{-1} = \mathbf{0}, \] then find \( \det \mathbf{B} \).


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 2:3$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 20, determine the area of $\triangle ABC$.


Solution: \boxed{-1 + 2^{258}}


How many degrees are in the measure of the angle formed by the hour and minute hands of a clock when the time is 3:15 PM?


What are the sum of the tens digit and the units digit in the decimal representation of $8^{351}$?


In a right-angled triangle $ABC$ with $\angle C = 90^\circ$, the heights from $A$ and $B$ to side $BC$ and $AC$ respectively are $5$ and $15$. What is the smallest integer value for the third altitude from $C$ onto $AB$?


A rectangle is dissected into nine non-overlapping squares. The widths (side lengths) of the squares are such that when arranged correctly, they perfectly fill the rectangle without any gaps or overlaps. The side lengths of the squares follow these relationships:
1. The sum of the side lengths of the first and second smallest squares equals the side length of the third smallest square.
2. The sum of the side lengths of the first and third smallest squares equals the side length of the fourth smallest square.
3. The sum of the side lengths of the third and fourth smallest squares equals the side length of the fifth smallest square.
4. The sum of the side lengths of the fourth and fifth smallest squares equals the side length of the sixth smallest square.
5. The sum of the side lengths of the second, third, and fifth smallest squares equals the side length of the seventh smallest square.
6. The sum of the side lengths of the second and seventh smallest squares equals the side length of the eighth smallest square.
7. The sum of the side lengths of the first, fourth, and sixth smallest squares equals the side length of the ninth smallest square.
8. The sum of the side lengths of the sixth and ninth smallest squares equals the sum of the seventh and eighth smallest squares.

Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.


There are three pairs of real numbers \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) that satisfy both \( x^3 - 4xy^2 = 3003 \) and \( y^3 - 4x^2y = 3002 \). Compute \( \left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right) \).

Solution: \boxed{\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right) = \frac{3}{3002}}


Call a set of integers "widely spacy" if it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are widely spacy?


What is the number of square units in the area of a hexagon formed by the points (0,0), (1,4), (3,4), (4,0), (3,-4), and (1,-4)?


Let $p,$ $q,$ and $r$ be distinct complex numbers such that
\[\frac{p}{1 - q} = \frac{q}{1 - r} = \frac{r}{1 - p} = m.\] Determine all possible values of $m$ and find their sum.


How many four-digit positive integers are there with a thousands digit $5$ that are also divisible by $5$?


Let $n$ be the integer such that $0 \le n < 29$ and $5n \equiv 1 \pmod{29}$. What is $\left(3^n\right)^2 - 3 \pmod{29}$?

Express your answer as an integer from $0$ to $28$, inclusive.


Find the sum of the first seven terms in the geometric sequence $\frac15,\frac1{25},\frac1{125},\dots$. Express your answer as a common fraction.


At a national contest, 25 Mathletes each get their own room in a hotel with room numbers ranging from 1 to 25. If the Mathletes staying in rooms 15 and 20 do not arrive, what is the median room number of the other 23 Mathletes?


In triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. If $\angle ABC = 54^\circ$ and $\angle ACB = 16^\circ$, find the measure of $\angle BHC$, in degrees.


Line segment $\overline{AB}$ is extended past $B$ to $Q$ such that $AQ:QB = 5:2.$
\[
\overrightarrow{Q} = s \overrightarrow{A} + v \overrightarrow{B}
\]
for some constants $s$ and $v.$ Determine the ordered pair $(s, v)$.

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


Five painters, each working at the same rate, can complete a painting job in $1.8$ work-days. If only two painters are available and one of them works twice as efficiently as the others, how many work-days will it take for them to finish the job?


Compute the determinant of the following matrix:
\[
\begin{vmatrix}
1 & -3 & 3 & 2 \\
0 & 5 & -1 & 0 \\
4 & -2 & 1 & 0 \\
0 & 0 & 0 & 6
\end{vmatrix}.
\]


What is the measure of the smaller angle between the hour and minute hands of a clock when the time is 6:30 p.m.?


Calculate \( 5^{-1} + 25^{-1} \pmod{23} \). Express the result as an integer from 0 to 22 inclusive.


Find the smallest possible munificence of a monic quadratic polynomial in the form $f(x) = x^2 + px - 1$ over the interval $-1 \le x \le 1$.


Let $C'$ be the circle with equation $x^2+6y-4=-y^2+12x-12$. Find the value of $a'+b'+r'$, where $(a',b')$ is the center of $C'$ and $r'$ its radius.

Solution: \boxed{3 + \sqrt{37}}


It costs 5 cents to copy 3 pages. How many pages can you copy for $\$15$?


Solve the compound inequality
\[3 - \frac{1}{3x + 4} < 5 \quad \text{and} \quad 2x + 1 > 0.\]

Solution: \boxed{\left(-\frac{1}{2}, \infty \right)}


A school has between 300 and 400 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into four distinct sections of students. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


Solve for $y$ in terms of $b$, where $b \neq 0$, given the determinant:
\[
\begin{vmatrix}
y + b & -y & y \\
-y & y + b & -y \\
y & -y & y + b
\end{vmatrix} = 0.
\]

Solution: \boxed{y = -\frac{b}{3}}


What is the range of the function \( g(x) = \frac{x}{(x+1)^2} \)?

Solution: \boxed{(-\infty, \frac{1}{4}]}


Simplify $21 \cdot \frac{8}{15} \cdot \frac{1}{14}$.


How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes, given that each box must contain at least one ball?


What is the smallest positive integer \(x\) that, when multiplied by \(600\), produces a product that is a multiple of \(1152\)?


Alice has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal rounded to the nearest thousandth.


What is the remainder when the sum of the first 150 counting numbers is divided by 5550?


Calculate $\binom{18}{10}$ given that $\binom{16}{7} = 11440$ and you need to find $\binom{16}{9}$ first, knowing that $\binom{16}{16-9} = \binom{16}{7}$.


Let $S$ be the sum of all positive integers $n$ such that $n^2 + 12n - 1947$ is a perfect square. Find the remainder when $S$ is divided by $1000$.


The third and fourth terms of a decreasing geometric sequence are 18 and 12, respectively. What is the first term of the sequence?


Find the greatest common divisor of $50404$ and $40303$.


Determine the number of relatively prime dates in the month with the second fewest relatively prime dates.


Solution: \boxed{2^{48} + 3^{40}}


Parallelogram $EFGH$ with $E(3,4)$, $F(5,8)$, $G(7,4)$, and $H(5,0)$ is reflected across the $x$-axis to $E'F'G'H'$ and then $E'F'G'H'$ is reflected across the line $y=x-1$. This is done such that $H'$ is the image of $H$, and $H''$ is the image of $H'$. What is the ordered pair of $H''$ in the coordinate plane?

Solution: \boxed{(1, 4)}


Pirate Pete shares his treasure with Pirate Paul in an interesting way. Pete first says, "One for me, one for you," giving himself one coin and starting Paul's pile with one coin. Then Pete says, "Two for me, and two for you," giving himself two more coins but making Paul's pile two coins in total. This pattern continues until Pete gives himself $x$ more coins but makes Paul's pile $x$ coins in total. At this point all of the coins have been distributed, and Pirate Pete has exactly five times as many coins as Pirate Paul. How many gold coins do they have in total?


A spinner with seven congruent sectors numbered from 1 to 7 is used. If Jane and her brother each spin the spinner once, and Jane wins if the absolute difference of their numbers is less than 4, what is the probability that Jane wins? Express your answer as a common fraction.


Solve the following system of equations for $x$ and $y$:
\begin{align*}
4x - 7y &= -20,\\
9x + 3y &= -21.
\end{align*}
Express your answer as an ordered pair $(x, y)$.

Solution: \boxed{\left(-\frac{69}{25}, \frac{32}{25}\right)}


Let's consider a similar problem with different values for voltage \( V \) and impedance \( Z \). Suppose \( V = 2 + i \) and \( Z = 2 - 4i \). Find the current \( I \).


A line is parameterized by a parameter $t$. The vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$, and the vector on the line at $t = 4$ is $\begin{pmatrix} 8 \\ -7 \end{pmatrix}$. Find the vector on the line at $t = -3$.

Solution: \boxed{\begin{pmatrix} -6 \\ 21 \end{pmatrix}}


Find the matrix that corresponds to rotating about the origin by an angle of $135^\circ$ clockwise.

Solution: \boxed{\begin{pmatrix} -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{pmatrix}}


A cube has side length $8$ meters. Its vertices are alternately colored red and blue. What is the volume of the tetrahedron whose vertices are the blue-colored ones?

Solution: \boxed{\frac{512}{3} \text{ cubic meters}}


Suppose that the roots of $x^3+2x^2+5x-8=0$ are $a$, $b$, and $c$, and that the roots of $x^3+px^2+qx+u=0$ are $a+b$, $b+c$, and $c+a$. Find $u$.


A rectangular box has a volume of $5000$ cubic inches and a surface area of $1950$ square inches. The sum of the lengths of its $12$ edges is $240$ inches. What would be the volume of the box, in cubic inches, if its length, width, and height were each increased by two inches?

Solution: \boxed{7198 \text{ cubic inches}}


The graphs of the function \( g(x) = 2x + c \) and its inverse \( g^{-1}(x) \) intersect at the point \((-4, d)\). Given that \( c \) and \( d \) are integers, determine the value of \( d \).

Solution: \boxed{d = -4}


A cube is painted so that one side is yellow, two sides are purple, and three sides are orange. How many different such cubes can be painted, considering two cubes to be identical if one can be rotated to match the other?


Find the matrix $\mathbf{M}$ that rotates the elements of a 2x2 matrix clockwise by 90 degrees. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & a \\ d & b \end{pmatrix}.\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


Consider a $7\times7$ chessboard, which has its lines dividing the board into smaller rectangles and squares. Calculate the number of rectangles and squares formed by these lines. Then find the ratio of the number of squares to rectangles, simplified to its lowest terms, and determine the sum of the numerator and denominator of this fraction.


A square has a side length of 12 cm. At each corner of the square, a three-quarter circle is drawn with a radius equal to one-quarter of the side length. Calculate the area in square centimeters of the unshaded region inside the square. Express your answer in terms of $\pi$.

Solution: \boxed{144 - 27\pi}


Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$ such that it is tangent to circle $C$ at point $R$. Meanwhile, a secant from $Q$ intersects $C$ at points $D$ and $E$, such that $QD < QE$. If $QD = 4$ and $QR = ED - QD$, then what is $QE$?


In triangle $XYZ$, we have $\angle X = 90^\circ$, $YZ = 10$, and $\tan Z = 2\cos Y$. What is $XY$?

Solution: \boxed{XY = 5\sqrt{3}}


Find the sum $a+b$ where $\log_2{800}$ is between the consecutive integers $a$ and $b$.


Cora and Dana play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where
\[ g(n) = \left\{
\begin{array}{cl}
8 & \text{ if } n \text{ is a multiple of 3 and 2}, \\
4 & \text{ if } n \text{ is a multiple of 3 but not of 2}, \\
3 & \text{ if } n \text{ is only a multiple of 2}, \\
1 & \text{ if } n \text{ is not a multiple of 2 or 3}.
\end{array}
\right.\]
Cora rolls the die six times and gets the numbers 5, 4, 3, 6, 2, and 1. Dana rolls and gets the numbers 6, 3, 4, 3, 5, and 3. What is the product of Cora's total points and Dana's total points?


Let $c$ and $d$ be the roots of $x^2 - 6x + 8 = 0.$ Compute
\[c^3 + c^4 d^2 + c^2 d^4 + d^3.\]


Eight sticks with lengths 1, 2, 4, 8, 16, 32, 64, and 128 inches are placed in a box. Three of the sticks are randomly selected. What is the probability that a triangle can be formed by joining the endpoints of the sticks? Express your answer as a common fraction.

Solution: \boxed{\frac{0}{56} = 0}


If 76,394,821 pencils are packaged in sets of 7, how many pencils will be left over?


Let $g$ be defined by  \[g(x) = \left\{
\begin{array}{cl}
3-x & \text{ if } x \leq 2, \\
3x-x^2 & \text{ if } x>2.
\end{array}
\right.\] Calculate $g^{-1}(-4)+g^{-1}(1)+g^{-1}(5)$.


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


A theatre is setting up seats in rows for a play. Each row contains 15 seats, and currently, there are 225 seats in total. The theatre anticipates that 160 people will attend. To minimize the number of empty seats while still having complete rows, how many seats need to be removed?


The graph of $h(x)$ is to be derived from the graph of $y = f(x)$, described below. Determine $h(x)$ in terms of $f(x)$. Suppose $f(x)$ is given as:
- $f(x) = -2 - x$ for $x$ in $[-3, 0]$
- $f(x) = \sqrt{4 - (x - 2)^2} - 2$ for $x$ in $[0, 2]$
- $f(x) = 2(x - 2)$ for $x$ in $[2, 3]$

The graph of $h(x)$ is the graph of $f(x)$ reflected over the x-axis and then shifted left by 6 units. What is $h(x)$ in terms of $f(x)$?

Solution: \boxed{h(x) = -f(x + 6)}


Given that $\lceil x \rceil - \lfloor x \rfloor = 1$, what is $\lfloor x \rfloor + x - \lceil x \rceil$?

Solution: \boxed{x - 1}


Determine the maximum integer value of the expression:
\[
\frac{3x^2 + 9x + 13}{3x^2 + 9x + 5}.
\]


The expression \(x^2 - 16x + 64\) can be written in the form \((x - a)(x - b)\), where \(a\) and \(b\) are both nonnegative integers and \(a > b\). What is the value of \(3b - a\)?


Solution: \boxed{3x^6(18 - 77x^7)}


For each permutation $b_1,b_2,b_3,\cdots,b_{8}$ of the integers $1,2,3,\cdots,8$, form the sum
\[|b_1-b_2| + |b_3-b_4| + |b_5-b_6| + |b_7-b_8|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


Find the function $f(x)$ such that
\[f(x) - 2 f \left( \frac{1}{x} \right) = (x + 2)^x.\]
Given that additional information $f(3)$ and $f(1/3)$ satisfies the relations, solve for $f(3)$.

Solution: \boxed{-\frac{125 + 2\left(\frac{7}{3}\right)^{1/3}}{3}}


What is the sum of all of the divisors of $540$ which are not divisible by $2$?


Let \( r(x) \) have a domain of \(\{ -2, -1, 0, 1 \}\) and a range of \(\{ 1, 3, 5, 7 \}\). Let \( s(x) \) have a domain of \(\{ 0, 1, 2, 3, 4, 5 \}\) and be defined by \( s(x) = 2x + 1 \). What is the sum of all possible values of \( s(r(x)) \)?


Five plastic bottles can be recycled to make a new bottle. Starting with 3125 plastic bottles, how many new bottles can eventually be made? (Remember that the first new bottles that are made can then be recycled into even newer bottles!) Do not include the original 3125 bottles in your count.


In the diagram, $ABCD$ and $EFGD$ are squares each of area 25. If $H$ is the midpoint of both $BC$ and $EF$, find the total area of polygon $ABHFGD$.

[asy]
unitsize(3 cm);

pair A, B, C, D, E, F, G, H;

F = (0,0);
G = (1,0);
D = (1,1);
E = (0,1);
H = (E + F)/2;
A = reflect(D,H)*(G);
B = reflect(D,H)*(F);
C = reflect(D,H)*(E);

draw(A--B--C--D--cycle);
draw(D--E--F--G--cycle);

label("$A$", A, N);
label("$B$", B, W);
label("$C$", C, S);
label("$D$", D, NE);
label("$E$", E, NW);
label("$F$", F, SW);
label("$G$", G, SE);
label("$H$", H, SW);
[/asy]


In the local ultimate frisbee league, teams have 8 members, and each of the 4 teams takes turns hosting tournaments. At each tournament, each team selects two members of that team to be on the tournament committee, except the host team, which selects four members. How many possible 10 member tournament committees are there?


Consider the sequence starting at 9720, where each subsequent term is obtained by dividing the preceding term by 2. How many integers are in this sequence?


On a similar one-way single-lane highway, cars still travel at a uniform speed, but now the safety rule has been updated: the distance from the back of one car to the front of the next is exactly one car length for each 20 kilometers per hour of the speed. Given that each car is 4 meters long and that cars maintain any constant speed, find the maximum number of cars (denoted as $N$) that can pass by a counting device placed beside the road in one hour. Calculate the quotient of $N$ divided by 10.


An environmental agency needs to hire new employees so that:
- 120 new employees will be able to monitor water pollution,
- 150 new employees will be able to monitor air pollution,
- 100 new employees will be able to monitor soil pollution,
- 50 employees will be able to monitor both water and air pollution,
- 30 employees will be able to monitor both air and soil pollution,
- 20 employees will be able to monitor both water and soil pollution,
- 10 employees will be able to monitor all three types of pollution.
What is the minimum number of employees that need to be hired?


Regular square $ABCD$ and regular triangle $ABE$ are drawn on opposite sides of line segment $AB$ such that they are coplanar. What is the degree measure of exterior angle $CBE$?


What is $1001101_2 + 101101_2$? Write your answer in base $10$.


The least common multiple of \( x \), \( 15 \) and \( 21 \) is \( 105 \). \( x \) must also be a power of a prime number. What is the greatest possible value of \( x \)?


Suppose in a right triangle $PQR$, with $\angle PQR = 90^\circ$, we have $\cos Q = \frac{3}{5}$. If $PR = 5$, find the length of $PQ$.


Consider a cubic polynomial
\[ x^3 + bx^2 + cx + d = 0 \]
where all coefficients are integers. Let $m$ be the exact number of integer roots of the polynomial, counting multiplicity. Determine all possible values of $m$.

Solution: \boxed{0, 1, 2, 3}


Determine the smallest integer $B$ such that there exists a sequence of consecutive integers, including $B$, that adds up to 2023.


The least common multiple of $x$, $15$, $18$, and $21$ is $630$. What is the greatest possible value of $x$?


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 $n$.


Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_4b_6 = 17$, what is $b_3b_7$?


Determine the value of $a$ such that the equation
\[\frac{x - 3}{ax - 2} = x\]
has exactly one solution.


Our water polo club has expanded to 20 members. I want to form a starting team consisting of 9 players. One member must be the goalie and another must be the captain. The goalie is chosen first and the captain is chosen among the remaining 8 players of the starting lineup. How many ways can I select this starting team?


A polynomial of degree $15$ is divided by $d(x)$ to give a quotient of degree $8$ and a remainder of $2x^2 + 5x - 8$. What is $\deg d$?


If $\left\lfloor n^2/3 \right\rfloor - \lfloor n/2 \rfloor^2 = 3$, find all integer values of $n$.


Calculate the sum of the coefficients in the polynomial $4(2x^8 + 5x^6 - 3x + 9) + 5(x^7 - 3x^3 + 2x^2 - 4)$ when it is fully simplified.


Keisha's basketball team is deciding on a new uniform style. This time, the seventh-graders will choose the color of the shorts from three options: black, gold, or blue. The eighth-graders will choose the color of the jersey from three options: black, white, or gold. Each group will choose independently and each option is equally likely. What is the probability that the color of the shorts will differ from the color of the jersey? Express your answer as a common fraction.


Susan rolls five fair 12-sided dice, each numbered from 1 to 12. What is the probability that exactly three of these dice show a prime number?


How many distinct sequences of six letters can be made from the letters in EQUALITY if each sequence must begin with Y and end with T, and no letter can appear in a sequence more than once?


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


Given a line parameterized by a parameter \( t \), where the vector at \( t = 0 \) is \( \begin{pmatrix} 2 \\ 3 \end{pmatrix} \), and the vector at \( t = 5 \) is \( \begin{pmatrix} 12 \\ -37 \end{pmatrix} \), find the vector on the line at \( t = -3 \).

Solution: \boxed{\begin{pmatrix} -4 \\ 27 \end{pmatrix}}


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


A modified deck of 54 cards includes 13 ranks in each of 4 suits and 2 jokers that are suit and rank-less. What is the probability that one of the top two cards is the King of $\diamondsuit$?


When the least common multiple of two positive integers is divided by their greatest common divisor, the result is 45. If one integer is 75, what is the smallest possible value of the other integer?