There are integers $b, c$ for which both roots of the polynomial $x^2 - 2x - 1$ are also roots of the polynomial $x^6 - bx - c$. Determine the product $bc$.


Let $T = 1 - 2 + 3 - 4 + \cdots + 2021 - 2022$. What is the residue of $T$, modulo 2023?


A very large number \( y \) is equal to \( 2^33^44^55^66^77^88^9 \). What is the smallest positive integer that, when multiplied with \( y \), produces a product that is a perfect square?


Let $n$ be the smallest positive integer such that there exist positive real numbers $a$ and $b$ for which
\[(a + bi)^n = -(a - bi)^n.\]
Compute $\frac{b}{a}$.


Given $m$ and $n$ be any two distinct prime numbers, determine the largest prime number which divides all possible numbers of the form $m^2 - n^2$.


How many integers $n \neq 0$ satisfy the inequality $\frac{1}{|n|} > \frac{1}{5}$?


Simplify
\[\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.\]

Solution: \boxed{2 \csc x}


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


Suppose that all four of the numbers \[4 - 3\sqrt{3}, \; -4-3\sqrt{3}, \; 2+\sqrt{5}, \; 2-\sqrt{5}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


A zoo has five pairs of different animals, each consisting of one male and one female. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male lion, how many ways can he feed all the animals?


A club has 30 members. In how many ways can a 5-person executive committee be formed if one of the members must be the president?


The sixth and ninth terms of a geometric sequence of real numbers are $9!$ and $10!$ respectively. What is the first term?

Solution: \boxed{a = \frac{362880}{10^{5/3}}}


What is the smallest positive multiple of $17$ that is $6$ more than a multiple of $73$?


How many integers between 1 and 300 are multiples of both 4 and 6 but not of either 5 or 8?


Find a positive integer that is divisible by 24 and whose cube root is a number between 8.2 and 8.3.


Find $3 \cdot 7^{-1} + 9 \cdot 13^{-1} \pmod{60}$.

Express your answer as an integer from $0$ to $59$, inclusive.


Solution: \boxed{1 + i}


A certain organism begins with 4 cells. Each cell splits and becomes two cells at the end of each day. However, every day, one of the new cells from each original cell turns into a specialized type that no longer divides. If this process continues for 10 days, how many regular cells that are still capable of division will there be at the end of the time period?


Suppose 5 different integers are randomly chosen from between 10 and 59, inclusive. What is the probability that they each have a different tens digit?


There is a unique two-digit positive integer $s$ for which the last two digits of $13\cdot s$ are $52$.


What is the smallest positive integer \( n \) such that \( 5n \equiv 105 \pmod{24} \)?


A right circular cone is sliced into five equal-height sections by planes parallel to its base. What is the ratio of the volume of the second-largest piece to the volume of the largest piece?


Determine the range of the function $y = \log_2 (\sqrt[3]{\sin x})$ for $0^\circ < x < 180^\circ$.

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


In a circle, triangle $ABC$ is inscribed such that $\angle ABC = \angle ACB$. $AD$ and $CD$ are tangents from a point $D$ outside the circle. If $\angle ABC = 3 \angle D$, $\angle ADA$ is an external angle at $D$ formed as $AD$ intersects the circle at another point $A_D$, and $\angle BAC = \alpha$, express $\alpha$ in terms of $\pi$.

Solution: \boxed{\alpha = \frac{\pi}{7}}


An equilateral triangle has a perimeter of 48 inches. A square and a rectangle are such that the square’s perimeter is 64 inches and the rectangle’s perimeter is twice the perimeter of the square. What is the ratio of the length of the side of the triangle to the length of the side of the square? Express your answer as a common fraction.


What is the greatest common factor of 36 and 54?


Two cards are chosen sequentially with replacement from a deck consisting of two standard decks (a total of 104 cards). What is the probability that at least one of the chosen cards is an ace?


A right circular cone is inscribed in a right rectangular prism. The base of the cone is tangent to all four sides of the rectangular base of the prism. If the base of the rectangle has side lengths $a$ and $b$, and the height of both the prism and the cone is $h$, what is the ratio of the volume of the cone to the volume of the prism? Express your answer as a common fraction in terms of $\pi$.


Determine the domain of the function \( h(x) = \frac{x^3 - 3x^2 + 6x - 8}{x^2 - 5x + 6} \).

Solution: \boxed{(-\infty, 2) \cup (2, 3) \cup (3, \infty)}


Calculate the area of the region enclosed by $x^2 + y^2 + 10x + 2 = 5y - 4x + 13$.


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}}


What is the number of square units in the area of trapezoid PQRD with vertices P(0,0), Q(0,-3), R(5,0), and D(5,9)?


Five-sixths of the people in a room are seated in five-sixths of the chairs. The rest of the people are standing. If there are 10 empty chairs, how many people are in the room?


Find the quotient and remainder when $x^5 - 24x^3 + 12x^2 - x + 20$ is divided by $x - 3$.


In triangle $XYZ$, $XY = 5$, $YZ = 12$, $XZ = 13$, and $YM$ is the angle bisector from vertex $Y$. If $YM = k \sqrt{2}$, then find $k$.


How many three-digit perfect cubes are divisible by 16?


Find the largest constant $m$, so that for any positive real numbers $a, b, c, d, e$, the following inequality is true:
\[
\sqrt{\frac{a}{b+c+d+e}} + \sqrt{\frac{b}{a+c+d+e}} + \sqrt{\frac{c}{a+b+d+e}} + \sqrt{\frac{d}{a+b+c+e}} + \sqrt{\frac{e}{a+b+c+d}} > m.
\]


How many perfect squares less than 20,000 can be represented as the difference of squares of two integers that differ by 2?


A spinner has four equal sections labeled 1, 2, 3, and 4. If you spin the spinner three times and form a three-digit number from the three outcomes, where the first outcome is the hundreds digit, the second outcome is the tens digit, and the third outcome is the units digit, what is the probability that the resulting three-digit number is divisible by 4? Express your answer as a common fraction.


A company has the following salary structure according to the table below. Determine the median salary of the 63 employees in this company.

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$140,000$\\\hline
Vice-President&4&$\$92,000$\\\hline
Director&12&$\$75,000$\\\hline
Associate Director&8&$\$55,000$\\\hline
Administrative Specialist&38&$\$24,000$\\\hline
\end{tabular}


A triangle has three different integer side lengths and a perimeter of 24 units. What is the maximum length of any one side?


How many positive four-digit integers are divisible by $5$ and end in $0$?


Let \( a_1, a_2, \ldots \) be a sequence determined by the rule \( a_n = \frac{a_{n-1}}{2} \) if \( a_{n-1} \) is even and \( a_n = 3a_{n-1} + 1 \) if \( a_{n-1} \) is odd. For how many positive integers \( a_1 \le 3000 \) is it true that \( a_1 \) is less than each of \( a_2 \), \( a_3 \), and \( a_4 \)?


If $\frac{5}{2}(r + s + t + u) = 25$, what is the average of $r$, $s$, $t$, and $u$?


Consider the following data from a new season graph, showing the number of home runs hit in April by the top hitters in the baseball league:
- 5 players hit 6 home runs each.
- 6 players hit 8 home runs each.
- 4 players hit 10 home runs each.

Calculate the mean number of home runs hit by these players.


How many positive integers $n$ satisfy \[(n + 8)(n - 4)(n-12)(n+5)<0\]?


The dilation, centered at \( 1 + 2i \), with a scale factor of 4, takes \( 3 + i \) to which complex number? Additionally, find the conjugate of the resulting complex number.

Solution: \boxed{9 + 2i}


If I choose five cards from a standard $52$-card deck, with replacement, what is the probability that the first four cards drawn represent each of the four suits?


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


Add $857_{12} + 296_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.


Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300.$ What is the least positive difference between a number selected from sequence $A$ and a number selected from sequence $B?$

$\bullet$ Sequence $A:$ $3,$ $6,$ $12,$ $24,$ $48,$ $\ldots$

$\bullet$ Sequence $B:$ $15,$ $45,$ $75,$ $105,$ $135,$ $\ldots$


A flag consists of three horizontal stripes, each potentially colored red, white, blue, green, or yellow. No two adjacent stripes should share the same color, and no color should appear more than once on the flag. How many distinct flags can be designed under these conditions?


Let \( A = (0, 0) \), \( B = (3, 4) \), and \( C = (p, q) \) be three points on the parabola \( y = -x^2 + 5x \), where \( 0 \le p \le 3 \). Find the largest possible area of triangle \( ABC \).


Westvale Middle School has a dedicated four-square league with twelve players, including Michael and Sarah. Each recess, the twelve players are divided into two games, each consisting of six players in no particular order. Throughout the semester, each possible set of six players plays together exactly once. Determine how many times Michael played in the same game as Sarah.


Consider the polynomials \[f(x) = 2 - 15x + 4x^2 - 5x^3 + 7x^4\] and \[g(x) = 4 - 3x - 8x^3 + 12x^4.\] Determine the value of $c$ such that the polynomial $f(x) + c g(x)$ has degree 3.


Sandy's daughter has a playhouse in the backyard, which includes one shaded exterior wall and two identical sloped roof faces. The exterior wall is 10 feet wide and 7 feet high, and each of the roof sections is 10 feet wide and 6 feet high at the peak. The special siding used to cover these areas is sold only in 10-foot by 15-foot sections, costing $35 each. If Sandy can cut the siding as needed, how much will she spend on the siding?


Suppose $x$ and $y$ are inversely proportional. If $x=40$ when $y=9$, find the value of $x$ when $y=20$. Additionally, find the value of $z$ directly proportional to $y$, with $z=45$ when $y=10$.


Let $\mathbf{A}$ be a $3 \times 3$ matrix with real entries such that $\mathbf{A}^4 = \mathbf{0}$. Find the number of different possible matrices that $\mathbf{A}^3$ can be. If you think the answer is infinite, then enter "infinite".


A circle passes through the point $(1,2),$ and is tangent to the parabola $y = x^2$ at $(1,1).$ Additionally, the circle is tangent to the line $y = 2x + 3$ at some point. Find the center of the circle.

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


The Johnson family has 5 sons and 4 daughters. In how many ways can they be seated in a row of 9 chairs such that at least 2 boys are next to each other?


An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$.


Suppose $f(x)$ is a function defined for all real $x$, and the function $f$ is invertible.
Determine how many points of intersection exist between the graphs of $y = f(x^3)$ and $y = f(x^6)$.


What is the value of $\dfrac{(5 \times 4) + 6}{10}?$


A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction.

Solution: \boxed{4 - 2\sqrt{2}}


Let $F_n$ be the Fibonacci sequence, defined by $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute the sum
\[
\sum_{n=0}^\infty \frac{F_n}{5^n}.
\]


An ellipse has foci at $F_1 = (0,3)$ and $F_2 = (4,0)$. The ellipse intersects the $x$-axis at the origin and one other point. What is the other point of intersection?

Solution: \boxed{\left(\frac{56}{11}, 0\right)}


In $\Delta ABC$, $AC = BC$, $m\angle DCB = 50^{\circ}$, and $CD \parallel AB$. What is the number of degrees in $m\angle ECD$?

[asy] pair A,B,C,D,E; B = dir(-50); A = dir(-130); D = (.5,0); E = .4 * dir(50);
draw(C--B--A--E,EndArrow); draw(C--D,EndArrow);
label("$A$",A,W); label("$C$",C,NW);label("$B$",B,E);label("$D$",D,E);label("$E$",E,E);
[/asy]


Express the quotient $3102_5 \div 23_5$ in base $5$.


What is the coefficient of \(x^3y^5\) in the expansion of \(\left(\frac{2}{3}x - \frac{4}{5}y\right)^8\)?


Determine the number of different paths from point $A$ to point $B$ on a 3x3 grid, where movement is allowed only to the right or downward.

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


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'' where the digits $A$, $B$, $C$, and $D$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible?


Two pairs of real numbers $(x,y)$ satisfy the equation $x+y = 2xy = 6$. Given that the solutions $x$ are in the form $x = \frac{p \pm q\sqrt{r}}{s}$ where $p$, $q$, $r$, and $s$ are positive integers and the expression is completely simplified, what is the value of $p + q + r + s$?


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


What is the modular inverse of $7$, modulo $800$?

Express your answer as an integer from $0$ to $799$, inclusive.


Calculate $1010101_2 + 111000_2$ and write your answer in base $10$.


A scalene triangle has side lengths which are distinct primes greater than 3, and the length of its perimeter is also prime. What is its smallest possible perimeter?


In triangle $XYZ$, angle bisectors $\overline{XK}$ and $\overline{YL}$ intersect at $Q$. If $XY = 8$, $XZ = 6$, and $YZ = 4$, find $\frac{YQ}{QL}$.


A soccer team is deciding on a new uniform. The under-12 players will choose the color of the socks (red or blue) and the under-14 players will choose the color of the shirt (red, blue, or green). Each group selects independently and each color choice is equally likely. What is the probability that the socks will be a different color than the shirt? Express your answer as a common fraction.


Determine the value of $a + b$ if the sequence $3, ~8, ~13, \ldots, ~a, ~b, ~38$ is an arithmetic sequence.


A flagpole is supported by a wire which extends from the top of the flagpole to a point on the ground 4 meters from its base. When Nick walks 3 meters from the base of the flagpole towards the point where the wire is attached to the ground, his head just touches the wire. Nick is 1.7 meters tall. How many meters tall is the flagpole?


Evaluate $\log_3 81\sqrt{81}$. Express your answer as an improper fraction.


Each vertex of this parallelogram has integer coordinates. The parallelogram has vertices at $(2, 3)$, $(5, 7)$, $(11, 7)$, and $(8, 3)$. What is the value of the sum of its perimeter $p$ and area $a$?


Find a unit vector that is orthogonal to both $\begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} \frac{7}{\sqrt{89}} \\ \frac{-6}{\sqrt{89}} \\ \frac{-2}{\sqrt{89}} \end{pmatrix}}


The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both red? Express your answer as a common fraction.


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


A large circular pie with a diameter of $18\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Determine the length squared, $l^2$, of the longest line segment that can be drawn in one of these sector pieces.


Consider the function
\[ g(x) = \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}}. \]
Find the sum of all real numbers \(x\) that are not in the domain of \(g(x)\).


Ronnie has 40 pieces of strawberry gum and 60 pieces of banana gum. Some of the pieces are in complete packs, while others are loose. Each complete pack contains exactly \( y \) pieces of gum. If Ronnie loses one pack of strawberry gum, then the ratio of the number of pieces of strawberry gum he has to the number of pieces of banana gum remains the same as if he instead finds 3 packs of banana gum. Find \( y \).


Each of $b_1,$ $b_2,$ $\dots,$ $b_{50}$ is equal to $1$ or $-1.$ Find the minimum positive value of
\[\sum_{1 \le i < j \le 50} b_i b_j.\]


Calculate the number
\[ e^{11\pi i/60} + e^{23\pi i/60} + e^{35\pi i/60} + e^{47\pi i/60} + e^{59\pi i/60} \]
and express it in the form \( r e^{i \theta} \), where \( 0 \le \theta < 2\pi \). Find the value of \( \theta \).


Compute the exact value of the expression
\[|\pi - |\pi - 9||.\]
Write your answer using only integers and \(\pi\), without any absolute value signs.

Solution: \boxed{9 - 2\pi}


How many two-digit prime numbers can be formed by choosing two different digits from the set $\{1, 3, 4, 6\}$ to be used as the tens digit and the units digit?


A triangle is made of wood sticks of lengths 13, 20, and 25 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. What is the smallest integral length that can be cut from each of the three sticks to prevent them from forming a triangle?


Solution: \boxed{y = -2 \text{ or } y = -\frac{13}{2}}


In a survey of $150$ people, $120$ indicated they liked Vivaldi and $90$ indicated they liked Chopin. What is the minimum number of people who could have said they liked both Vivaldi and Chopin?


Determine how many terms are in the expansion of \[(a+b+c)(x+d+e+f+g+h).\]


Determine the positive value for the slope of an asymptote of the hyperbola described by the equation:
\[
\sqrt{(x-2)^2 + (y-3)^2} - \sqrt{(x-2)^2 + (y+1)^2} = 4
\]


On a triangle ABCD circling around a point, the diagram follows:
```
[point positions remain unchanged as well]
```
$\angle ACB = 60^\circ$ and $\angle CAD = 50^\circ$. Find the sum of the degrees in $\angle CAB + \angle ACD$.


Consider the system of equations:
\begin{align*}
4x - 2y + z &= a, \\
6y - 12x - 3z &= b.
\end{align*}
If this system has a solution, find $\frac{a}{b}$, assuming $b \neq 0.$

Solution: \boxed{\frac{a}{b} = -\frac{1}{3}}


Four of the following test scores are Cyprian's and the other four are Margaret's: 82, 85, 88, 90, 95, 97, 98, 100. Cyprian's mean score is 91. What is Margaret's mean score?


Calculate $3524_6 + 2442_6$ and express your answer in base-6.


The director of a choir wants to arrange the choir members in a formation that includes all members without any gaps. If arranged in a square formation, there are 11 members left over. If they are arranged in a formation with 5 more rows than columns, no members are left over. Determine the maximum number of choir members possible.


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

Solution: \boxed{28x - 3}


Given that $\begin{vmatrix} p & q \\ r & s \end{vmatrix} = 3,$ find the value of
\[\begin{vmatrix} 2p & 2p + 5q \\ 2r & 2r + 5s \end{vmatrix}.\]


Find the number of positive integers $n \le 500$ such that $21n$ is a perfect square.


Let $x$ and $y$ be real numbers such that
\[
  1 < \frac{x-y}{x+y} < 4
\]
and
\[
  \frac{x+y}{x-y} \neq 1.
\]
If $\frac{x}{y}$ is an integer, what is its value?


On her $10^{\text{th}}$ birthday in 2004, Lisa's mother said, "My age is now five times your age." In what year will Lisa's mother be able to say, "My age is now twice your age," on Lisa's birthday?


Two friends, Ben and Carol, share 30 chocolates. Carol eats some number of chocolates that is a positive multiple of the number of chocolates eaten by Ben. If they finish all 30 chocolates, then what is the maximum number of chocolates that Ben could have eaten?


Given $A = 3 - 2i$, $M = -3 + 2i$, $S = 2i$, $P = -1$, and $R = A \cdot M$, compute the value of $A - M + S - P + R$.

Solution: \boxed{2 + 10i}


Let $a$, $b$, $c$, $d$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 = 16 \quad \text{and} \quad a \neq b \neq c \neq d.\]
Find the maximum value of $a^3 + b^3 + c^3 + d^3$.


The length of a rectangle is increased by \( 30\% \), but the width of the rectangle is decreased to keep the area of the rectangle unchanged. By what percent was the rectangle's width decreased?


The line $y=-\frac{2}{5}x+10$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, then what is the value of $r+s$?


A traffic light cycles as follows: green for 40 seconds, then yellow for 5 seconds, followed by red for 40 seconds, and then blue for 5 seconds. Mary picks a random five-second time interval to observe the light. What is the probability that the color changes while she is watching?


What is the largest divisor of 540 that is smaller than 60 and also a factor of 180?


What is the least integer value of $b$ such that $-10$ is not in the range of $y = x^2 + bx + 20$?


Milton spilled some ink on his homework again, obscuring the coefficient of $x$, but he remembers that the equation still has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain now?

[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,black);
label("$x+48=0$",(3,1.5),E);
label("$x^{2}+$",(0,1.5),W);
[/asy]


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

Solution: \boxed{\left(\frac{1}{2}, \frac{4}{5}\right)}


A horse 30 feet from the center of a merry-go-round makes 40 revolutions. A second horse 15 feet from the center will travel the same distance by making how many revolutions? Additionally, if a third horse is 10 feet from the center, how many revolutions would it need to make to travel the same distance?


Given the volume of the parallelepiped determined by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ as 6, find the volume of the parallelepiped determined by the vectors $2\mathbf{a} + \mathbf{b}$, $\mathbf{b} + 2 \mathbf{c}$, and $\mathbf{c} - 3 \mathbf{a}$.


A library has eight identical copies of one type of book and four identical copies of a second type of book. At any given time, some of these books 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 of each type is in the library and at least one of each type is checked out?


A cube is painted so that one side is blue, three sides are red, and two sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube.

Solution: \boxed{2 + 1 = 3}


John wants to buy two small pizzas, one large pizza, and four drinks. The drinks cost $q$ dollars each, the small pizza costs the same as one drink, and the large pizza costs four times as much as one drink. If John started with $50$ dollars, how much money would he have left after making all of his purchases? Give your answer as an expression of $q$.

Solution: \boxed{50 - 10q}


The probability of snow tomorrow is $\frac{1}{4}$. Additionally, if it does not snow, there is a probability of $\frac{1}{3}$ that it will be foggy. What is the probability that it will neither snow nor be foggy tomorrow? Express your answer as a common fraction.


A man labels the squares of a chessboard from $1$ through $64$. On each square $k$, he places $3^k$ grains of sand. How many more grains of sand are placed on the $12^{th}$ square than on the first $10$ squares combined?


Suppose that $a$ and $b$ are positive integers such that $(a+bi)^3 = 2 + 11i$. What is $a+bi$?


A ball was frozen in a pond, and when removed, it left a circular hole 30 cm across at the top and 10 cm deep. What was the radius of the ball (in centimeters)?


The difference when the product of 6 and a number is subtracted from 28 is greater than 14. How many positive integers that are divisible by 3 satisfy the given conditions?


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


Let \( Q = \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \left( 1 - \frac{1}{5} \right) \dotsm \left( 1 - \frac{1}{n} \right) \). What is the value of \( Q \) if \( n = 100 \)? Express your answer as a common fraction.


What is the last digit of the decimal expansion of $\frac{1}{2^{12}}$?


Let $C$ be the circle with equation $x^2 + 6x + 36 = -y^2 - 8y + 45$. Find the coordinates $(a, b)$ of the center of $C$ and its radius $r$, and compute $a + b + r$.

Solution: \boxed{-7 + \sqrt{34}}


Each of the first ten prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number?


Let $C$ be a point not on line $AE$ and $D$ a point on line $AE$ such that $CD \perp AE.$ Meanwhile, $B$ is a point on line $CE$ such that $AB \perp CE.$ If $AB = 6,$ $CD = 10,$ $AE = 7$, and $AC = x$, then what is the length of $CE?$


A gift shop offers free gift wrapping with a new promotional offer. They now have 10 varieties of wrapping paper, 4 colors of ribbon, and 5 types of gift cards. However, due to a supply issue, combinations involving the 10th wrapping paper and any red ribbon are not available. How many valid combinations are possible using one variety of wrapping paper, one color of ribbon, and one type of gift card?


What is the smallest positive value of $x$ if $\frac{x}{7} + \frac{2}{7x} = 1$?

Solution: \boxed{\frac{7 - \sqrt{41}}{2}}


The prime factorization of 1764 is $2^2 \times 3^2 \times 7^2$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy=1764$?


A company needs to hire new employees so that 150 of the new employees will be able to handle customer service, 125 of the new employees will be able to manage technical support, and exactly 50 of the new employees will be able to handle both. (These 50 are included in the 150 and 125 mentioned above.) What is the minimum number of employees that need to be hired?


Determine how many three-digit positive integers \( y \) satisfy \( 4325y + 692 \equiv 1403 \pmod{17} \).


At a university, there are $3000$ students enrolled in language courses. Each student studies either German or Russian, and some study both. The number of students who study German is between $70\%$ and $75\%$ of the total student population, and the number studying Russian ranges from $40\%$ to $50\%$. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Calculate $M-m$.


Let $ a, b, c, x, y, z$ be real numbers that satisfy the three equations
\[
  14x + by + cz = 0, \\
  ax + 24y + cz = 0, \\
  ax + by + 43z = 0.
\]
Suppose that $a \ne 14, b \ne 24, c \ne 43$, and $x \ne 0$. What is the value of
\[
\frac{a}{a - 14} + \frac{b}{b - 24} + \frac{c}{c - 43} \, ?
\]


There are approximately 0.454 kilograms in a pound. To the nearest whole pound, how many pounds does a steer that weighs 350 kg weigh?

Solution: \boxed{771\text{ pounds}}


2 diagonals of a regular hexagon (a 6-sided polygon) are chosen. What is the probability that they intersect inside the hexagon?


What is the sum of all integers in the set $\{1,2,3,4,5,6\}$ that are primitive roots $\pmod{13}$?


In the diagram, if $\angle PQR = 60^\circ$, what is the measure of $\angle PMN$?
[asy]
size(6cm);
pair p = (0, 0); pair m = dir(180 - 30); pair n = dir(180 + 30); pair r = 1.3 * dir(30); pair q = 2 * 1.3 * Cos(60) * dir(-30);
label("$M$", m, N); label("$R$", r, N); label("$P$", p, 1.5 * S); label("$N$", n, S); label("$Q$", q, SE);
draw(m--q--r--n--cycle);
add(pathticks(m--p, s=4));
add(pathticks(n--p, s=4));
add(pathticks(r--p, 2, spacing=0.9, s=4));
add(pathticks(r--q, 2, spacing=0.9, s=4));
[/asy]


If 2 cards from a standard deck are selected randomly, what is the probability that either two queens or at least 1 king occurs? (There are 4 Kings, 4 queens, and 52 total cards in a standard deck.)


12 coins are simultaneously flipped. What is the probability that heads are showing on at most 3 of them?


A pet store has 20 puppies, 10 kittens, and 12 rabbits. Emily, John, and Lucy each want to buy a pet, ensuring they all get different types of pets. How many ways can Emily, John, and Lucy buy pets and leave the store satisfied?


When the product $(5x + 3y + 2)(2x + 5y + 4)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?


Find the sum of the distinct prime factors of $7^7 - 7^3$.


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'' where the digits $A$, $B$, $C$, and $D$ are not necessarily distinct. Each of the two-digit primes is less than 60. Determine how many such house numbers are possible.


Points \( A(1, 3) \) and \( B(5, 8) \) are the endpoints of a diameter of a circle graphed in a coordinate plane. After translating the circle by the vector \( \mathbf{u} = (2, -1) \), what is the equation of the new circle?

Solution: \boxed{(x - 5)^2 + (y - 4.5)^2 = \frac{41}{4}}


Compute $\tan \left(\tan^2 2025^\circ\right)$.

Solution: \boxed{\tan 1^\circ}


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


All the numbers 3, 4, 5, 6, 7, 8 are assigned to the six faces of a cube, one number to each face. There is an additional condition that 3 and 4, which are the smallest numbers, must be on opposite faces. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that meet at that vertex. What is the greatest possible value of the sum of these eight products?


Jenny's graduating class has 360 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 18 students in each row, then there can be $x$ students in each row. What is the sum of all possible values of $x$?


A function $g(x)$ is defined for all real numbers $x$. For all non-zero values of $x$, we have
\[3g\left(x\right) + g\left(\frac{1}{x}\right) = 7x + 6\]
Let $T$ denote the sum of all of the values of $x$ for which $g(x) = 2010$. Compute the integer nearest to $T$.


Find the focus of the parabola $x = -\frac{1}{8}y^2.$

Solution: \boxed{(-2, 0)}


What is the sum of all positive integers $n$ that satisfy $$\mathop{\text{lcm}}[n,120] = \gcd(n,120) + 300~?$$


Triangle $PQR$ has side-lengths $PQ = 15, QR = 30,$ and $PR = 22.$ 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 $\triangle PXY?$


What is the ratio of the area of an equilateral triangle inscribed in a semicircle with radius \(r\) to the area of an equilateral triangle inscribed in a circle with radius \(r\)? Express your answer as a common fraction.


Find the values of $a$ and $b$ that satisfy the equation $x^2 - 4x + 9 = 25$, where $a \geq b$, and then calculate the value of $3a + 2b$.

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


Tori's mathematics test contained 80 questions split into three categories: 15 arithmetic, 25 algebra, and 40 geometry problems. She correctly answered 60% of arithmetic, 50% of algebra, and 70% of geometry questions. To pass the test, she now needs to obtain 65% correct answers overall. How many additional correct answers would Tori need to pass the test?


**Problem Statement**:
Five standard six-sided dice are rolled. It is known that there are two pairs of dice showing the same number, but different from each other. These four dice are set aside, and the remaining die is re-rolled. What is the probability that after re-rolling this die, at least three of the five dice show the same value?


How many integers from 1 through 999, inclusive, do not contain any of the digits 0, 3, 4, 5, or 8?


Determine the number of $x$-intercepts and $y$-intercepts for the graph of the equation $y = -x^2 + 3x - 2$.


Given that $b$ is a multiple of $780$, find the greatest common divisor of $5b^4 + 2b^3 + 6b^2 + 5b + 65$ and $b$.


If \( 3x - 2y = 8 \) and \( x + 3y = 7 \), what is the value of \( x \)?


Find the remainder when the sum of the numbers assigned to eight consecutive days starting from day number 100,045 is divided by 5.


The Everton Youth Soccer Team consists of 25 players, including 4 goalies. In a training session, each goalie will defend penalty kicks from every other team member including the other goalies. How many penalty kicks are needed to ensure each player has a chance to shoot against each goalie?


Piravena needs to make a trip from city $D$ to $E$, then from $E$ to $F$, then from $F$ to $D$. The cities form a right-angled triangle with $F$ as the right angle. The distances are as follows: from $D$ to $F$ is 2000 km, and from $D$ to $E$ is 4500 km. The cost per kilometer for a bus is $\$0.18$, and for an airplane, there's a $\$120$ booking fee plus $\$0.12$ per kilometer. Determine the total distance Piravena travels.

Solution: \boxed{10531 \text{ km}}


In a certain sequence, the first term is $a_1 = 2010$ and the second term is $a_2 = 2011$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \geq 1$. Determine $a_{1000}$.


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


What is the smallest positive integer \( x \) that, when multiplied by \( 900 \), produces a product that is a multiple of \( 1152 \)?


At Logical Learning, to pass the final exam which includes both algebra and geometry sections, a student must score at least \(85\%\) in each section separately to pass. If the algebra section consists of 40 problems and the geometry section has 30 problems, what is the greatest number of problems a student can miss in each section and still pass?


The mean of the set of numbers $\{91, 89, 95, 90, 92, y\}$ is 92. What is the median of this set of six numbers?


If I roll a fair, regular six-sided die five times, what is the probability that I will roll the number $1$ exactly three times and the number $2$ exactly two times?


Mary and James each sit in a row of 12 chairs. They choose their seats at random. What is the probability that they don't sit next to each other?


A cube has a side length of $8$. Its vertices are alternately colored red and blue. Calculate the volume of the tetrahedron formed by the blue vertices of the cube.


What is the sum of all of the positive even factors of $720$?


Find the number of distinct numbers in the list
\[\left\lfloor \frac{1^2}{2000} \right\rfloor, \ \left\lfloor \frac{2^2}{2000} \right\rfloor, \ \left\lfloor \frac{3^2}{2000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{2000} \right\rfloor.\]


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


Let \( g(x) = \frac{1}{cx+d} \) where \( c \) and \( d \) are nonzero constants. Find all solutions to \( g^{-1}(x) = -2 \). Express your answer in terms of \( c \) and/or \( d \).

Solution: \boxed{\frac{1}{-2c + d}}


The area of a rectangular plot of land is 180 square meters. If the length of the rectangle is three times the width, what is the perimeter of the rectangle?


The letters of the alphabet are given numeric values based on the two conditions below.

$\bullet$  Only the numeric values of $-3,$ $-2,$ $-1,$ $0,$ $1,$ $2,$ and $3$ are used.

$\bullet$  Starting with A and going through Z, a numeric value is assigned to each letter according to the following pattern: \[
1, 2, 0, 1, 0, -1, -2, 0, -1, 0, 1, 2, 0, 1, 0, -1, -2, 0, -1, 0,\ldots
\]
This pattern has a cycle of length $10$. The letter A has a value of $1$, B of $2$, C is $0$, ..., J is $0$, and Z cycles back in a similar pattern starting again from the list.

Calculate the sum of the numeric values of the letters in the word ``harmonics."


In a right triangle, medians are drawn from points $A$ and $B$ to the midpoints of segments $\overline{BC}$ and $\overline{AC}$, respectively. The lengths of these medians are 5 and $3\sqrt{5}$ units, respectively. Calculate the length of segment $\overline{AB}$.


John has two numbers, \( a \) and \( b \). When he divides \( a \) by 84, he gets a remainder of 77. When he divides \( b \) by 120, he gets a remainder of 113. What remainder does he get when he divides \( a+b \) by 42?


How many 12-digit positive integers have all digits either 1 or 2, and have at least two consecutive digits that are the same?


How many integers between 1 and 300 are multiples of both 5 and 7 but not of either 6 or 8?


Given the equation of a parabola $x^2 - 4x + 3y + 10 = 0$. Find the vertex of the parabola.

Solution: \boxed{(2, -2)}


Regular hexagon $ABCDEF$ and regular square $AFGH$ are drawn on opposite sides of line segment $AF$ such that they are coplanar. What is the degree measure of exterior angle $GFD$?


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


Let $p$ and $q$ be relatively prime positive integers such that \(\dfrac{p}{q} = \dfrac{1}{2^1} - \dfrac{2}{3^2} + \dfrac{3}{2^3} - \dfrac{4}{3^4} + \dfrac{5}{2^5} - \dfrac{6}{3^6} + \cdots\), where the numerators increase by 1 each term, and the denominators alternate between powers of 2 and 3, with exponents also increasing by 1 for each subsequent term. Assume the series converges. Compute \(p+q\).


Three cubes, each with faces numbered from 1 to 6, are tossed. What is the probability that the sum of the numbers shown on the top faces of all three cubes is divisible by 3? Express your answer as a common fraction.


Two 8-sided dice are rolled. What is the probability that the sum rolled is a perfect square?


On a modulo $17$ graph paper, graph the congruences $$y \equiv 6x + 3 \pmod{17}$$ and $$y \equiv 13x + 1 \pmod{17}.$$ Determine the sum of the $x$-coordinates of points that lie on both graphs.


At a mixed doubles tennis tournament, there were four teams consisting of one man and one woman each. After the tournament, each player shook hands once with every player of the opposite gender except their own partner. How many such handshakes occurred?


As $p$ ranges over the primes greater than $7$, how many different remainders can $p^2$ leave upon division by $210$?


Let $M$ denote the number of $6$-digit positive integers whose digits are in non-decreasing order. Determine the remainder obtained when $M$ is divided by $1000$. (Digits may repeat.)


Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g$, and the $x$-coordinate of the vertex of $g$ is a root of $f$. Suppose both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(50, -200),$ what is the value of $a + c$?


A regular octagon is inscribed in a circle of radius 4 units. In square units, what is the area of the octagon? Express your answer in simplest radical form.


Let $s(\theta) = \frac{1}{1+\theta}$. Evaluate $s(s(s(s(s(s(50))))))$, where $s$ is applied 6 times.


A cylindrical container has a height of 10 inches and the circumference of its base is 15 inches. A spiral stripe is painted on the container that winds around the cylinder exactly twice while it reaches from the bottom of the container to the top, appearing directly above its starting point. What is the length in inches of the stripe?


Find the value(s) of $x$ such that $10xy - 15y + 3x - 4.5 = 0$ is true for all values of $y$.

Solution: \boxed{x = \frac{3}{2}}


How many positive perfect square integers are factors of the product $\left(2^{12}\right)\left(3^{15}\right)\left(5^{18}\right)\left(7^8\right)$?


A student, Leo, needs to earn 30 study points for a special credit. For the first 6 points, he needs to complete 1 project each. For the next 6 points, he needs 2 projects each; for the next 6 points, 3 projects each, and so on. Determine the minimum number of projects Leo needs to complete to earn 30 study points.


Find the sum of the first eight terms in the geometric sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$.


Define a function $g$ from the positive integers to the positive integers with the following properties:
(i) $g$ is increasing.
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.
(iii) If $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Compute all possible values of $g(88).$


Determine the number of real solutions for the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \ldots + \frac{50}{x - 50} = x.
\]


The Fibonacci numbers $F_n$ are given by the recursion $F_0 = 0, F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$. Find the units digit of $F_{F_{15}}$.


The real number \(y\) satisfies \(y^2 - 7y + 12 < 0.\) Find all possible values of \(y^2 + 7y + 12.\)

Solution: \boxed{(42, 56)}


Consider an even composite number greater than 12. What is the smallest even abundant number that exceeds 12?


Simplify the expression $\dfrac{45}{28} \cdot \dfrac{49}{75} \cdot \dfrac{100}{63}$.


How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8?


How many different 9-digit positive integers exist, where the number does not start with 0?


Determine the number of ordered tuples \((a, b, c, d, e, f, g, h, i)\) of real numbers such that:
\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}^{-1}
=
\begin{pmatrix}
\frac{1}{a} & \frac{1}{d} & \frac{1}{g} \\
\frac{1}{b} & \frac{1}{e} & \frac{1}{h} \\
\frac{1}{c} & \frac{1}{f} & \frac{1}{i}
\end{pmatrix}.
\]


A student must choose a program of five courses from a list consisting of English, Algebra, Geometry, History, Art, Latin, Biology, and Chemistry. This program must contain English, at least one mathematics course, and at least one science course. How many ways can the student choose this program?


Let integer $n$ be between $50$ and $120$ and be a multiple of $8$. When $n$ is divided by $12,$ the remainder is $4$. Similarly, when $n$ is divided by $7,$ the remainder is also $4$. What is the value of $n$?


A water channel with an inside diameter of 8 inches is designed to transport water from a lake to a village. Assuming no water loss due to friction or turbulence, how many pipes each with an inside diameter of 1.5 inches and of the same length are required to transport an equivalent volume of water?


Given a cylinder of fixed volume \( V \), the total surface area (including the two circular ends and the lateral surface) is minimized for a radius \( R \) and height \( H \). If the volume is increased to \( 2V \), find the new ratio \( \frac{H}{R} \) that minimizes the surface area.


In $\triangle ABC$,  point $D$ is the midpoint of side $BC$.  Point $E$ is on $AC$ such that $AE:EC = 2:1$. Point $F$ is on $AD$ such that $AF:FD=2:1$. If the area of $\triangle DEF$ is 15, determine the area of $\triangle ABC$.


A particle moves according to the position function $(3t^2 + t + 1, 6t + 2).$ Find the speed of the particle at time $t$.

Solution: \boxed{\sqrt{36t^2 + 12t + 37}}


Let $z$ be a complex number such that $|z| = 1.$ Find the largest possible distance between $(1 + 2i)z^2$ and $z^4$ when plotted in the complex plane.

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


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


What is the sum of the digits of the base-2 expression for $345_{10}$?


The function
\[ g(z) = \frac{(1 - i \sqrt{3}) z + (2 \sqrt{3} + 12i)}{3} \] represents a rotation around some complex number \( d \). Find \( d \).

Solution: \boxed{\frac{16 \sqrt{3}}{7} + \frac{18i}{7}}


Thirty gremlins and twenty imps attend the Annual Mischief Convention. Due to some alliances, exactly five imps are willing to shake hands with each other but refuse to shake hands with the remaining imps. All imps shake hands with all gremlins. Meanwhile, being sociable, all gremlins shake hands with each other as well as all the imps. Calculate the total number of handshakes at the convention.


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


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


Compute the sum $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor$.


Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 1001 \left( \frac{1}{x} + \frac{1}{y} + 1000 \right)\] for all $x,$ $y > 0.$

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


What is the minimum number of colors needed to color a hexagonal tessellation grid, such that no two tiles sharing a vertex are the same color?


What is the greatest common divisor of 1729 and 1337?


Two distinct integers, \(x\) and \(y\), are randomly chosen from the set \(\{1,2,3,4,5,6,7,8,9,10,11,12\}\). What is the probability that \(xy-x-y\) is even?


Let $g(x)$ be a polynomial with real, nonnegative coefficients. If $g(4) = 16$ and $g(16) = 1024$, find the largest possible value of $g(8)$.


What is the remainder when \( 4x^3 - 12x^2 + 16x - 20 \) is divided by \( x - 3 \)?


Find the square root of $\dfrac{10!}{210}$.


Let \(x\), \(y\), and \(z\) be non-zero complex numbers such that \(x + y + z = 10\) and
\[
(x - y)^2 + (x - z)^2 + (y - z)^2 = xyz.
\]
Find \(\frac{x^3 + y^3 + z^3}{xyz}\).


Let \( x_1, x_2, x_3 \) be positive real numbers such that \( 2x_1 + 3x_2 + 4x_3 = 100 \). Find the smallest possible value of
\[
x_1^2 + x_2^2 + x_3^2.
\]


The first term of a given sequence is 3, and each successive term is obtained by doubling the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 10000?


Calculate the sum of the first 100 natural numbers and find the sum from 51 to 100.


How many integer values of \( n \) satisfy \( -100 < n^3 < 100 \)?


Evaluate the sum $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \cdots + \lceil\sqrt{40}\rceil$$


I have created a new game where for each day in May, if the date is a prime number, I walk three steps forward; if the date is composite, I walk one step backward. If I stop on May 31st, how many steps long is my walk back to the starting point?


Terrell typically does 10 repetitions lifting two 25-pound weights. If he switches to lifting two 20-pound weights, how many repetitions must he complete to lift the same total weight?


What real number equals the expression \(3 + \frac{9}{2 + \frac{9}{3 + \frac{9}{2 + \cdots}}}\), where the \(2\)s and \(3\)s alternate?

Solution: \boxed{\frac{3 + 3\sqrt{7}}{2}}


Find the remainder when the polynomial $x^{888}$ is divided by the polynomial $(x^2 - x + 1)(x + 1).$


In 1950, there were 600,000 cases of measles reported in the U.S. In 2000, there were 200 cases reported. How many cases of measles would have been reported in 1990 if the number of cases reported from 1950 to 2000 decreased linearly?


How many units are in the sum of the lengths of the two longest altitudes in a right triangle with sides $9$, $40$, and $41$?


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2012} \times 7$?


In a bowling tournament, 6 bowlers are competing for the top prizes with a playoff system. First, #6 bowls against #5, and the winner enters the main playoff with bowlers #1 through #4. The winner of #6 vs. #5 then bowls #4, the winner of which bowls #3, and so on, until they finally bowl #1. The matches continue in a knockout format where the loser of each match receives the next available lowest prize starting from 5th prize down to 2nd, after all matches. How many different orders can the bowlers receive the prizes?


Both $a$ and $b$ are positive integers, and $b > 1$. When $a^b$ is the greatest possible value less than 500, what is the sum of $a$ and $b$?


A marathon is $26$ miles and $385$ yards. One mile equals $1760$ yards.

John has run fifteen marathons in his life. If the total distance John covered in these marathons is \(m\) miles and \(y\) yards, where \(0\le y<1760\), what is the value of \(y\)?


What is the smallest positive integer that can be written in the form $3030m + 50505n$, where $m$ and $n$ are integers?


The length of the median to the hypotenuse of an isosceles, right triangle is $15$ units. Calculate the perimeter of the triangle. Express your answer in simplest radical form.

Solution: \boxed{30 + 30\sqrt{2}}


Find the remainder when $3x^6 - x^5 + 2x^3 - 8$ is divided by $x^2+3x+2$.

Solution: \boxed{-206x - 212}


Find how many integers $n$ satisfy $150 < n < 300$ and $n$ has the same remainder whether it is divided by $7$ or by $9$.


Ramon now sells four enchiladas and five tacos for $\$4.00$, and sells five enchiladas and four tacos for $\$4.40$. Assuming a fixed price per item, what is the cost, in dollars, of four enchiladas and five tacos? Express the answer as a decimal to the nearest hundredth.


Add $5_6 + 23_6.$ Express your answer in base $6.$


How many integers fall between $\sqrt{8}$ and $\sqrt{78}$ on a number line?


Two distinct vectors \(\mathbf{p}\) and \(\mathbf{q}\) define a line. A vector of the form
\[k \mathbf{p} + \frac{2}{5} \mathbf{q}\]
is also on the line for some \(k\). Find \(k\).


How many distinct sequences of four letters can be created from the letters in PLANET if each sequence must begin with N, end with T, and no letter can be repeated within a sequence?


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


What is the least five-digit positive integer which is congruent to 6 (mod 17)?


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


The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 130^\circ$ and $\angle AOB = 150^\circ$, as shown. Determine the degree measure of $\angle ABC$.


Find the sum of the angles \(\theta_1, \theta_2, \dots, \theta_6\) for the roots of the equation:
\[ z^6 = -1 - i, \]
where the roots are \( \text{cis } \theta_1, \text{cis } \theta_2, \dots, \text{cis } \theta_6 \) with \(0^\circ \le \theta_k < 360^\circ\) for all \(1 \le k \le 6\). Answer in degrees.


Eight 8-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?


Suppose $a,$ $b,$ and $c$ are three positive numbers such that $abc = 1,$ $a + \frac{1}{c} = 7,$ and $b + \frac{1}{a} = 11.$ Find $c + \frac{1}{b}.$


The coefficients of the polynomial
\[a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0\] are all integers, and its roots $s_1, s_2, \dots, s_6$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^6 + a_1 x^5 + a_2 x^4 + a_3 x^3 + a_4 x^2 + a_5 x + a_6 = 0\] are also $s_1, s_2, \dots, s_6.$ Find the number of possible integer multisets $S = \{s_1, s_2, \dots, s_6\}.$


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 145^\circ & -\sin 145^\circ \\ \sin 145^\circ & \cos 145^\circ \end{pmatrix}^n = \mathbf{I}.\]


Add consecutive prime numbers starting with $2$ in the following order: $2$, $2 + 3$, $2 + 3 + 5$, and so on. Continue this process to obtain the first 15 such sums. How many of these sums are primes?


In the circle with center $O$, the measure of $\angle TIP$ is $60^\circ$ and $OT=15$ cm. $R$ and $S$ are another two points on the circle such that $RS$ intersects $TP$ at angle $I$. $OTI$ forms a straight line. Find the number of centimeters in the lengths of arcs $RP$ and $RS$. Express your answer in terms of $\pi$.


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


Find the least odd prime factor of $1331^6 + 1$.


How many license plates consist of 2 vowels or 2 consonants followed by 3 digits, if the first digit must be odd, and the last two digits must be even?


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

Solution: \boxed{\begin{pmatrix} \frac{16}{5} \\ -\frac{18}{5} \end{pmatrix}}


A rectangular box has a volume of $5400$ cubic inches, a surface area of $1560$ square inches, and 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 $2$ inches?

Solution: \boxed{7208 \text{ cubic inches}}


The Fibonacci sequence is defined as $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 3$.

Suppose Fibonacci numbers $F_a$, $F_b$, $F_d$ form an increasing geometric sequence instead of arithmetic, and have $a + b + d = 3000$. Find the value of $a$ if $b-a = c = 2$ and $d = b+c$.


In the triangle shown, for $\angle A$ to be the largest angle of the triangle, it must be that $p<y<q$. What is the least possible value of $q-p$, expressed as a common fraction? [asy]
draw((0,0)--(1,0)--(.4,.5)--cycle);
label("$A$",(.4,.5),N); label("$B$",(1,0),SE); label("$C$",(0,0),SW);
label("$y+15$",(.5,0),S); label("$y+6$",(.7,.25),NE); label("$4y$",(.2,.25),NW);
[/asy]


Add $8A2_{12} + 3B7_{12}$. Express your answer in base $12$, using 'A' for 10 and 'B' for 11 if necessary.


Square EFGH has its center at $(6, -6)$ and has an area of 16 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at $(0,0)$ and a scale factor of 3, followed by a 90-degree clockwise rotation. What are the coordinates of the vertex of the image of square EFGH that is closest to the origin? Give your answer as an ordered pair.

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


In the diagram, $ABCD$ is a trapezoid with an area of $18.$ $CD$ is three times the length of $AB.$ What is the area of $\triangle ABD?$

[asy]
draw((0,0)--(1,4)--(9,4)--(18,0)--cycle);
draw((9,4)--(0,0));
label("$D$",(0,0),W);
label("$A$",(1,4),NW);
label("$B$",(9,4),NE);
label("$C$",(18,0),E);
[/asy]


Consider triangle $ABC$ where sides $AB$, $BC$, and $CA$ are tangent to a circle with center $D$ as shown. Suppose $\angle ABC = 75^\circ$ and $\angle BCD = 45^\circ$. Calculate $\angle BAC$.


The polynomial \(x^{103} + Ax + B\) is divisible by \(x^2 + x + 1\) for real numbers \(A\) and \(B\). Find \(A + B\).


Let $f(x)$ be a function such that $f(x) = xg(x)$ where $g(x)$ satisfies $g(3x) = g(x)$ for all $x$. If $xf(y) = yf(x)$ for all real numbers $x$ and $y$, and if $f(15) = 45$, find $f(3)$.


Eleven positive integers from a list of fifteen positive integers are $3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$. What is the largest possible value of the median of this list of fifteen positive integers?


Joanna has 10 identical candies and 5 identical boxes. How many ways can she distribute all her candies into the boxes such that each box gets at least one candy and only the number of candies in each box matters?


Let $f(x) = 3x + 5$ and $g(x) = 4x - 3$. Find the value of $\frac{f(g(f(g(3))))}{g(f(g(f(3))))}$. Express your answer in the form $\frac{a}{b}$.


Find the matrix $\mathbf{M}$ that doubles the first column and triples the second column of any 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

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


A sequence $b_1, b_2, \ldots$ of non-negative integers is defined by $b_{n+2} = |b_{n+1} - b_n|$ for $n \geq 1$. If $b_1 = 1024$, $b_2 < 1024$, and $b_{1004} = 1$, how many different values of $b_2$ are possible?


What is the base $2$ representation of $56_{10}$?


Express $\frac{0.\overline{8}}{1.\overline{36}}$ as a common fraction.


Determine the sum of all real numbers \(x\) satisfying
\[(x^2-6x+5)^{x^2-6x+3} = 1.\]


When \(1 + 8 + 8^2 + \cdots + 8^{100}\) is divided by \(500\), a remainder of \(M\) is obtained. Determine the value of \(M\).


How many positive four-digit integers of the form $\_\_75$ are divisible by 15?


Find the number of ordered nonuples \( (a, b, c, d, e, f, g, h, i) \) of real numbers such that:
\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}^{-1} =
\begin{pmatrix}
\frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\
\frac{1}{d} & \frac{1}{e} & \frac{1}{f} \\
\frac{1}{g} & \frac{1}{h} & \frac{1}{i}
\end{pmatrix}.
\]


Lee can make 18 cookies with 2 cups of flour and 1 cup of sugar. How many cookies can he make with 3 cups of flour and 1.5 cups of sugar?


Suppose the function $g$ is graphed, and each small box on the graph has width and height 1. The function values at specific $x$ coordinates are provided as follows: $g(0)=0$, $g(2)=1$, $g(4)=3$, $g(6)=5$, $g(8)=7$, $g(1)=8$, $g(3)=6$, $g(5)=4$, $g(7)=2$, $g(9)=0$. Sara writes the number 4 on her thumb and applies $g$ to 4, writing the output on her next finger. She continues this process of applying $g$ and writing the output on the next finger. What number will Sara write on her tenth finger?


Find the smallest positive integer that is both an integer power of 12 and is not a palindrome.


Six identical rectangles are arranged to form a larger rectangle $PQRS$, and the area of $PQRS$ is $6000$ square units. The three rows of rectangles form $PQ$ while two columns form $PS$. What is the length, $x$, rounded off to the nearest integer, if one side of each rectangle is $x$ and the other is $\frac{2}{3}x$?


I have a drawer with 8 forks, 5 spoons, and 7 knives. If I reach in and randomly remove four pieces of silverware, what is the probability that I get two forks, one spoon, and one knife?


Let $k, a_2, a_3$ and $k, b_2, b_3$ be two different geometric sequences with nonconstant terms and different common ratios. If \[a_3 + 2a_2 = b_3 + 3b_2,\] what is the sum of the common ratios of these two sequences?


If the eight solutions of $x^8 = 1$ are written in the form $a+bi$, where $a$ and $b$ are real, find the product of those solutions with $a > 0$.


Evaluate and find the integers $a$ and $b$ such that $\log_{2}{2048}$ is between $a$ and $b$. Find the sum $a+b$.


How many integers between $100$ and $999$ have three different digits in strictly decreasing order?


What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,24]=120$?


What is the units digit of the product of all the odd positive integers between 20 and 100, skipping every second odd number?


What is the area, in square units, of a trapezoid bounded by the lines $y = 2x$, $y = 18$, $y = 10$, and the $y$-axis? Express your answer as a decimal to the nearest tenth.


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


If \(x^{3y} = 8\) and \(x = 2\), what is the value of \(y\)? Express your answer as a common fraction.


How many even perfect square factors does $2^6 \cdot 7^{10} \cdot 3^2$ have?


In $\triangle PQR,$ $PQ=PR=17$ and $QR=16.$ Points $S,T,$ and $U$ are on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{ST}$ is parallel to $\overline{PR}$ and $\overline{TU}$ is parallel to $\overline{PQ}.$ What is the perimeter of parallelogram $PSTU$?


In the town of CycleCity, bicycle license plates each contain three letters. The first letter is chosen from the set $\{B, F, J, M, Q\}$, the second from $\{E, U\}$, and the third from $\{K, S, V, Y\}$. CycleCity decided to increase the number of available license plates by adding two new letters. These new letters can either be added to one letter set or split between two sets. What is the largest possible number of ADDITIONAL license plates that can be produced by adding these two letters?


Four circles each of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the larger circle? Express your answer as a common fraction in simplest radical form.

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


Find the positive integer $n$ such that
\[
e^{\frac{1}{n}} + e^{-\frac{1}{n}} = \sqrt{n}.
\]


Gamma and Delta both participated in a two-day science quiz. Each attempted questions totaling 600 points after the second day. On the first day, Gamma scored 210 points out of 350 points attempted, and on the second day scored 150 points out of 250 points attempted. Delta, who also did not attempt 350 points on the first day, scored a positive integer number of points on each of the two days, and Delta's daily success ratio (points scored divided by points attempted) on each day was less than Gamma's on that day. Gamma's overall success ratio for the two days was $360/600 = 3/5$.

Find the largest possible overall success ratio that Delta could have achieved.


A tree tripled its height every year until the fourth year, and then it doubled its height for the next three years. At the end of 7 years, the tree reached a height of 648 feet. What was the height of the tree at the end of 3 years?

Solution: \boxed{27 \text{ feet}}


An integer greater than 99 and less than 1000 is randomly chosen. What is the probability that its digits are different?


What is the area, in square units, of a triangle with sides of $5, 4,$ and $4$ units? Express your answer in simplest radical form.


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


Carla rotated point $A$ 550 degrees clockwise about point $B$, and it landed at point $C$. Later, Devon rotated the original point $A$ $x$ degrees counterclockwise about point $B$ and it also landed at point $C$. If $x < 360$, what is the value of $x$?


The image of the point with coordinates $(2,3)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(10,6)$. Find $m+b$.


There are 40 fifth-graders with an average age of 10. Two classes of fifth-graders have a total of four classroom teachers. The average age of the teachers is 40 years. The parents of these fifth-graders amount to 60 people with an average age of 34. Calculate the average age of all fifth-graders, their teachers, and parents combined.


What is the smallest possible median for the five number set $\{x, 3x, 4, 3, 7\}$ if $x$ can be any integer?


A sequence of three real numbers forms an arithmetic progression with a first term of 5. If 3 is added to the second term and 15 is added to the third term, the three resulting numbers form a geometric progression. What is the largest possible value for the first term of the geometric progression?


In the diagram, triangles $ABC$ and $CBD$ are isosceles. The perimeter of $\triangle CBD$ is $24,$ the perimeter of $\triangle ABC$ is $23,$ and the length of $BD$ is $10.$ What is the length of $AB?$


Find the value of \( k \) so that the line \( 6x - 4y + k = 0 \) is tangent to the parabola \( y^2 = 16x \).


The function $f$ defined by $f(x)= \frac{ax+b}{cx+d}$, where $a$, $b$, $c$ and $d$ are nonzero real numbers, has the properties $f(20)=20$, $f(100)=100$, $f(f(x))=x$ for all values except $\frac{-d}{c}$, and $f'(20) = -1$. Find the unique number that is not in the range of $f$.


Let $T = 1 - 2 + 3 - 4 + \cdots + 2021 - 2022$. What is the residue of $T$, modulo 2022?


The image of the point with coordinates $(2, 3)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(10, -1)$. Find $m + b$.


What is the smallest whole number \( b \) such that 90 can be expressed in base \( b \) using only three digits?


Find the integer $n$, $0 \le n \le 13$, such that \[n \equiv 1729 \pmod{13}.\]


Jar A originally contains 6 green buttons, 3 red buttons, and 9 blue buttons. Carla moves a number of green and blue buttons to an empty Jar B such that the number of blue buttons moved is twice the number of green buttons moved. After this transfer, Jar A contains only half of its original number of buttons. What is the probability that a randomly chosen button from Jar A is blue, and a randomly chosen button from Jar B is a green button? Express your answer as a common fraction.


Find the units digit of the quotient $\frac{7^{1993} + 5^{1993}}{6}$.


Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her three stamp books. One of her stamp books has a total of 945 stamps, the second has 1260 stamps, and the third has 630 stamps. What is the largest number of stamps that Jenna could be putting on each page?


Marius has eight kangaroos of different heights for a photography exhibition. He wants to line them up in a single row, but the tallest and the shortest kangaroos insist on being placed at either the first or last position in the line. How many ways can Marius arrange these kangaroos?

Solution: \boxed{1440\text{ ways}}


Determine the number of positive integers \( n \le 2000 \) such that \( 21n \) is a perfect square.


Consider the cubic polynomial \(x^3 + bx^2 + cx + d = 0\) where \(b\), \(c\), and \(d\) are integers. Determine all possible values of \(m\), where \(m\) is the exact number of distinct integer roots of the polynomial.

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


In a city represented by a Cartesian plane, Dennis is located at $(8, -15)$, Emily is at $(3, 20)$, and they plan to meet at a point that lies equidistant from both of them, then walk together to their friend Felix's location at $(5, 5)$. How many units do Dennis and Emily need to walk upwards to reach Felix?


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


Find the largest value of $n$ such that $6x^2 + nx + 72$ can be factored as the product of two linear factors with integer coefficients.


If \( g(x) = \sqrt[3]{\frac{x+5}{5}} \), for what value of \( x \) will \( g(3x) = 3(g(x)) \)? Express your answer in simplest form.


Among all triangles \(ABC\), find the maximum value of \(\cos A + \cos B \cos C.\)


Let $Q$ be the product of the first $120$ positive odd integers. Find the largest integer $n$ such that $Q$ is divisible by $5^n .$


Find the perimeter of a triangle formed by the points $O(0, 0)$, $A(-3, 10)$, and $B(4, -5)$ in the complex plane.

Solution: \boxed{\sqrt{109} + \sqrt{274} + \sqrt{41}}


Define $\mathbf{B} = \begin{pmatrix} 0 & 2 \\ 4 & 0 \end{pmatrix}.$  Find the vector $\mathbf{u}$ such that
\[(\mathbf{B}^8 + \mathbf{B}^6 + \mathbf{B}^4 + \mathbf{B}^2 + \mathbf{I}) \mathbf{u} = \begin{pmatrix} 0 \\ 32 \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} 0 \\ \frac{32}{4681} \end{pmatrix}}


The real number $x$ satisfies $x^2 - 5x + 6 < 0.$ Find all possible values of $x^3 + 5x^2 + 6x + 1.$

Solution: \boxed{(41, 91)}


Marguerite drove 120 miles in 3 hours. She took a 30-minute stop during her journey. If Sam drove for 3.5 hours at the same average driving rate as Marguerite, but without any stops, how many miles did he drive?

Solution: \boxed{168 \text{ miles}}


How many of the positive divisors of 840 are multiples of 4?


How many odd three-digit integers have the property that their digits, read left to right, are in strictly increasing order (each digit is greater than the previous digit)?


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 190 hours, 45 minutes and 30 seconds the time will be $A:B:C$. What is the value of $A + B + C$?


For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$?


If $x, y, z$ satisfy the system of equations:
\begin{align*}
y + z &= 18 - 4x \\
x + z &= 16 - 4y \\
x + y &= 9 - 4z,
\end{align*}
what is $3x + 3y + 3z$?


Find the polynomial $p(x)$ with real coefficients, such that $p(3) = 10$ and for all real numbers $x$ and $y$:
\[ p(x) p(y) = p(x) + p(y) + p(xy) - 5. \]

Solution: \boxed{x^2 + 1}


The positive integers $X, Y,$ and $Z$ form an arithmetic sequence, while the integers $Y, Z,$ and $W$ form a geometric sequence. If $\frac{Z}{Y} = \frac{7}{4}$, what is the smallest possible value of $X + Y + Z + W$?


Quadrilateral $ABCD$ is a parallelogram. The external angle on the extension of side $DC$ at point $C$ is $80^\circ$. What is the degree measure of angle $A$?
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3));
draw(Arc((5,2),1,20,80));
label("$D$",(0,0),SW); label("$C$",(5,2),SE); label("$B$",(6,7),NE); label("$A$",(1,5),NW);

label("$80^\circ$",(6.3,2.8), N);
[/asy]


What is the smallest positive integer $n$ such that $\frac{n}{n+110}$ is equal to a terminating decimal?


A circle centered at $O$ is circumscribed around $\triangle ABC$. The central angles are $\angle AOC = 140^\circ$ and $\angle AOB = 160^\circ$. Calculate the measure of $\angle BAC$.


A rectangle has its length increased by $30\%$ and its width increased by $15\%$. Determine the percentage increase in the area of the rectangle.


A parallelepiped $ABCDEFGH$ is formed by vectors $\overrightarrow{AB}, \overrightarrow{AD},$ and $\overrightarrow{AE},$ where $\overrightarrow{AE}$ is scaled by a factor $\alpha \neq 1$. Calculate:
\[
\frac{AG^2 + BH^2 + CE^2 + DF^2}{AB^2 + AD^2 + AE^2}.
\]


How many distinct digits can appear as the units digit of an integral perfect-square number when considered modulo 7?


Express as a common fraction: $$\frac{\frac{3}{7} + \frac{4}{5}}{\frac{5}{11} + \frac{2}{9}}.$$


Add $956_{12} + 273_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.


Ben rolls five fair 10-sided dice, where each die has faces numbered from 1 to 10. What is the probability that exactly three of the dice show a prime number?


Create an updated problem statement:
A checkerboard consists of 3 rows and 4 columns. Each cell of this checkerboard is a square of size $1 \times 1$. Calculate the total number of different squares that can be formed by the lines of the grid, including all possible square sizes.


In triangle $ABC$, side $AC = \sqrt{29}$ and side $AB = 5$. What is $\tan A$?

[asy]
pair A,B,C;
B = (0,0);
A = (0,5);
C = (sqrt(24),0);
draw(A--B--C--A);
draw(rightanglemark(A,B,C,7));
label("$B$",B,SW);
label("$A$",A,NW);
label("$C$",C,SE);
label("$5$",(A+B)/2,NW);
label("$\sqrt{29}$",(A+C)/2,NE);
[/asy]


A group of 12 friends decides to form a committee of 5. Calculate the number of different committees that can be formed. Additionally, if there are 4 friends who refuse to work together, how many committees can be formed without any of these 4 friends?


In the context of electrical engineering, when analyzing AC circuits, the relationship $V = IZ$ continues to hold where $V$ represents voltage, $I$ represents current, and $Z$ the impedance. Given that $V = 2+3i$ and $Z = 2-i$, find the value of $I$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers.

Solution: \boxed{\frac{1}{5} + \frac{8}{5}i}


If $a$ and $b$ are integers with $a > b$, what is the smallest possible positive value of $\frac{a^2+b^2}{a^2-b^2} + \frac{a^2-b^2}{a^2+b^2}$?


Find the remainder when $8\cdot7^{19} + 1^{19}$ is divided by 9.


Compute the sum of the geometric series $2 - 4 + 8 - 16 + \cdots + 2048$.


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 12 and 18 units, respectively, and the altitude is 15 units. Points $G$ and $H$ are the midpoints of sides $AD$ and $BC$, respectively. What is the area of quadrilateral $GHCD$?


In the diagram, the two smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $9.$ The radius of each smaller circle is one-fourth the radius of the larger circle. What is the area of the shaded region?


What is the value of $x + y$ if the sequence $3, ~9, ~x, ~y, ~30$ is an arithmetic sequence?


Solution: \boxed{32x^2 - 2}


Eight 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?


How many 6-digit palindromes are there such that the middle two digits are even?


Find all values of \(b\) for which the points \((0,0,0),\) \((1,b,0),\) \((0,1,b),\) and \((b,0,2)\) are coplanar.


Evaluate $$\lceil\sqrt{5} + 1\rceil + \lceil\sqrt{6} + 1\rceil + \lceil\sqrt{7} + 1\rceil + \cdots + \lceil\sqrt{49} + 1\rceil$$
Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$


Find the cross product of vectors $\begin{pmatrix} 3 \\ -4 \\ 5 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ 7 \\ 1 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} -39 \\ -13 \\ 13 \end{pmatrix}}


Line segment \(\overline{AB}\) is a diameter of a circle with \(AB = 32\). Point \(C\), not equal to \(A\) or \(B\), lies on the circle. As point \(C\) moves around the circle, the centroid (center of mass) of \(\triangle ABC\) traces out a closed curve missing two points. Determine the exact area of the region bounded by this curve.


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


In \( \triangle ABC \) with \( AB = 15 \), \( BC = 14 \), and \( AC = 16 \), let \( N \) be a point on \( \overline{AC} \) such that the circumcircles of \( \triangle ABN \) and \( \triangle BNC \) have equal radii. If \( AN = x \) and \( NC = 16 - x \), determine \( \frac{x}{16-x} \).


The 5 a.m. temperatures for seven consecutive days were recorded as $-6^{\circ}$, $-3^{\circ}$, $-3^{\circ}$, $-4^{\circ}$, $2^{\circ}$, $4^{\circ}$, and $1^{\circ}$ Celsius. What is the mean 5 a.m. temperature for this week in degrees Celsius?


How many numbers in the list $47$, $4747$, $474747$, $\dots$, up to $n$ terms are prime?


Consider the system of equations:
\begin{align*}
4x - 2y &= a, \\
9y - 18x &= b.
\end{align*}
If this system has a solution, find $\frac{a}{b}$, where $b \neq 0.$


In trapezoid $EFGH$, the sides $EF$ and $GH$ are equal. The lengths of the bases are $EF = 10$ units and $GH = 20$ units. The height from $F$ to base $GH$ is 5 units. Calculate the perimeter of trapezoid $EFGH$.

Solution: \boxed{30 + 10\sqrt{2}}


A rectangular prism measures 15-inches by 25-inches by 20-inches. What is the length, in inches, of the diagonal connecting point A and point B?


In right triangle $ABC$, $\tan{A} = \frac{3}{5}$, $AC = 10$, and $\angle C = 90^\circ$. Find $AB$.


The expression $x^2 + 19x + 88$ can be written as $(x + p)(x + q)$, and the expression $x^2 - 23x + 132$ written as $(x - q)(x - r)$, where $p$, $q$, and $r$ are integers. What is the value of $p + q + r$?


Determine the positive value of $x$ that satisfies the equation:
\[\log_3 (x - 1) + \log_{\sqrt{3}} (x^2 - 1) + \log_{\frac{1}{3}} (x - 1) = 3.\]

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


If $\frac{1}{8}$ of $2^{36}$ equals $8^y$, what is the value of $y$?


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


Place each of the digits 5, 6, 7, and 8 in exactly one square to make the smallest possible product. What is this product?


Five concentric circles are drawn with radii of 1, 3, 5, 7, and 9. The innermost circle is painted black, the ring around it is white, the next ring is black, the subsequent ring is white, and the outermost ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.


Noelle's math teacher revises her strategy: For the first seven homework points Noelle wants to earn, she needs to do one homework assignment for each point; for the next seven points, she needs to do two homework assignments for each point; and so on, such that for every subsequent group of seven points, the number of homework assignments per point increases by one. Calculate the minimum number of homework assignments Noelle needs to complete to earn a total of 28 homework points.


Sequence $C$ is a geometric sequence starting at 3 with a common ratio of 3. Sequence $D$ is an arithmetic sequence starting at 15 with a common increment of 15. Each sequence stops as soon as one of its terms exceeds $450$. What is the least positive difference between a number selected from sequence $C$ and a number selected from sequence $D$?

$\bullet$ Sequence $C:$ $3,$ $9,$ $27,$ $81,$ $243,$ $\ldots$

$\bullet$ Sequence $D:$ $15,$ $30,$ $45,$ $60,$ $75,$ $\ldots$


How many integers $n$ are there such that $5 \leq n \leq 15$ and $232_n$ (the number written as $232$ in base $n$) is a perfect square?


Find the minimum value of
\[\frac{x^2 + 9}{\sqrt{x^2 + 5}}\]
over all real numbers $x$.


Find the smallest value of $d$ such that $5$ is in the range of $g(x) = x^2 + 5x + d$.


Given that $$(x+y+z)(xy+xz+yz) = 27$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=11$$ for real numbers $x$, $y$, and $z$, find the value of $xyz$ under the condition that $x + y + z = 3$.


Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 2$ and
\[f(xy + f(x)) = xf(y) + 2f(x)\] for all real numbers $x$ and $y.$

Let $n$ be the number of possible values of $f \left( \frac{1}{2} \right),$ and let $s$ be the sum of all possible values of $f \left( \frac{1}{2} \right).$  Find $n \times s.$


Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(1 - \omega + \omega^2)^6 + (1 + \omega - \omega^2)^6.\]


The solution of the equation $9^{x+5} = 10^x$ can be expressed in the form $x = \log_b 9^5$. What is $b$?


If $e^{i \theta} = \frac{4}{5} + \frac{3}{5} i$ and $e^{i \phi} = -\frac{5}{13} - \frac{12}{13} i,$ and knowing that $\sin(\theta + \phi) = \sin(\theta) \cos(\phi) + \cos(\theta) \sin(\phi),$ find $\sin(\theta + \phi).$


Eleven distinct points, $P$, $Q$, $R$, and $S$, are to be selected from $2023$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $\overline{PQ}$ intersects the chord $\overline{RS}$?


Let \(\mathbf{p} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\), \(\mathbf{q} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}\), and \(\mathbf{r} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}\). Find the area of the parallelogram with vertices \(\mathbf{0}, \mathbf{p}, \mathbf{q},\) and \(\mathbf{p} + \mathbf{q}\). Then, find the area of the triangle formed by \(\mathbf{q}, \mathbf{r},\) and \(\mathbf{q} + \mathbf{r}\).


How many points of intersection are there between the graphs of the following equations:
\[
y = |3x + 6|, \quad y = -|4x - 1|
\]


Find the phase shift of the graph of $y = 2 \cos \left( 2x + \frac{\pi}{3} \right) + 1$.


A bag contains 4 red, 3 blue, and 8 green marbles. What is the probability, expressed as a fraction, of drawing two marbles sequentially without replacement where both marbles are either red or blue?


Lucas and Maria went to a stationery store. Lucas bought 36 notebooks and Maria bought 60 notebooks. If each package of notebooks contains the same number of notebooks and costs 3 dollars per package, what is the largest possible number of notebooks per package and the total cost for one such package of notebooks?


What is the value of \(x\) for which \((2012 + x)^2 = x^2\)?


A point $(x,y)$ is a distance of 15 units from the $x$-axis. It is a distance of 13 units from the point $(2,7)$. It is a distance $m$ from the origin. Given that $x>2$, what is $m$?

Solution: \boxed{\sqrt{334 + 4\sqrt{105}}}


A cone and a cylinder have the same height of 6 inches. The cone's radius is 1.5 inches. If both solid shapes have the same volume, what is the radius of a sphere that also has this same volume?

Solution: \boxed{1.5\text{ inches}}


In $\triangle PQR$, $PQ=PR=17$ and $QR=16$. Points $S$, $T$, and $U$ are on sides $\overline{PQ}$, $\overline{QR}$, and $\overline{PR}$ respectively, such that $\overline{ST}$ and $\overline{TU}$ are parallel to $\overline{PR}$ and $\overline{PQ}$ respectively. What is the perimeter of parallelogram $PSUT$?


Evaluate the determinant:
\[
\begin{vmatrix} x + 3 & x & x \\ x & x + 3 & x \\ x & x & x + 3 \end{vmatrix}.
\]

Solution: \boxed{27x + 27}


What is the value of the following expression: $2 - 5 + 8 - 11 + 14 - \cdots - 47 + 50 - 53 + 56$ ?


Evaluate the cube root of the expression: $\sqrt[3]{\frac{5}{16}}$.


The average of $x+8$, $7x+3$, and $3x+9$ is $5x-10$. What is $x$?


The four digits $a$, $b$, $c$, and $d$ are in an arithmetic sequence with a common difference of $k$. They are used to form the four-digit numbers $abcd$ and $dcba$. Determine the greatest common divisor of all numbers of the form $abcd+dcba+k$.


A beetle is on the edge of a circular platform with a radius of 75 feet. The beetle walks straight across the platform to the opposite edge, passing through the center of the circle. Following this, it makes another straight journey back to the center of the circle, then walks straight to its original starting point. If the final part of the journey from the center to the starting point was 75 feet, how many total feet did the beetle travel?


A rectangle having integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area?


If $p$ and $q$ are positive 4-digit integers such that $\gcd(p,q)=5$, what is the smallest possible value for $\mathop{\text{lcm}}[p,q]$?


In a right cone with a base radius of \(15\) cm and a height of \(30\) cm, a sphere is inscribed. The radius of the sphere can be expressed as \(b\sqrt{d} - b\) cm. Find the value of \(b + d\).


I have 7 books, three of which are identical copies of a novel, and two are identical copies of a science book, with the remaining two books being unique. In how many ways can I arrange these books on a shelf?


In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 16. What is the greatest possible perimeter of the triangle?


Let $x$ and $y$ be positive real numbers such that $4x + 5y < 90$. Find the maximum value of
\[xy(90 - 4x - 5y).\]


What is the remainder when \(8x^4 - 6x^3 + 17x^2 - 27x + 35\) is divided by \(2x - 8\)?


Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 12x - 8y - 56$. What is $x+y$?


Find the modulo 9 value of $$1+21+333+4444+55555+666666+7777777+88888888+999999999$$ where each number's digits are summed and their sum is then reduced modulo 9.


Find the length of a segment whose endpoints are (1,2) and (9,14). Also, determine the coordinates of the midpoint of this segment.


There are four math teams, with 4, 5, 6, and 7 students respectively. Each team has three co-captains. If I randomly select one team, and then randomly select two members of that team to give them each a book titled "Introduction to Geometry", what is the probability that both members selected are co-captains?


Centered at each lattice point in the coordinate plane are circles with radius $\frac{1}{4}$ and squares with sides of length $\frac{1}{2}$, with sides parallel to the coordinate axes. Calculate how many such squares and circles are intersected by the line segment from $(0,0)$ to $(503, 503)$.


Express $\sqrt{a} \div\sqrt{b}$ as a common fraction, given:

$$\frac{ {\left( \frac{1}{3} \right)}^2 + {\left( \frac{1}{4} \right)}^2 }{ {\left( \frac{1}{5} \right)}^2 + {\left( \frac{1}{6} \right)}^2} = \frac{25a}{61b} $$


The first $40$ numbers of an arrangement are described as follows:
- Row 1: $2$ numbers, each $2$
- Row 2: $4$ numbers, each $4$
- Row 3: $6$ numbers, each $6$
- Row 4: $8$ numbers, each $8$
- This pattern continues similarly in the subsequent rows. What is the value of the $80^{\mathrm{th}}$ number if the arrangement were continued?


Consider a square quilt made from 16 small squares, where some of these squares are divided into two right triangles. The quilt has a pattern such that one entire row and one entire column are shaded. What fraction of the entire quilt is shaded? Express your answer as a common fraction.


What numeral is in the 150th decimal place in the decimal representation of $\frac{13}{17}$?


What is the largest square number that is a multiple of $18$ and also less than $500$?


Find all solutions to the inequality \[\sqrt[3]{x}+ \frac{5}{\sqrt[3]{x}+ 3} \le 0,\] giving your answer in interval notation.

Solution: \boxed{(-\infty, -27)}


A digit is written to the right of the units digit of $864$. If the resulting four-digit number is divisible by $4$, how many possibilities are there for the digit that was written?


Let \( g(n) \) be the product of the proper positive integer divisors of \( n \). For how many values of \( n \) does \( n \) not divide \( g(n) \), given that \( 2 \le n \le 100 \)?


For what values of $b$ is $-3$ not in the range of the function $f(x) = x^2 + bx + 3$? Express your answer in interval notation.

Solution: \boxed{(-2\sqrt{6}, 2\sqrt{6})}


In the new diagram, there are four triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? The diagram is as follows:

[asy]
draw((0,0)--(2,0)--(1,2)--cycle,linewidth(1)); // Large triangle ABC
draw((1,0)--(1,2)--(0,0)--cycle,linewidth(1)); // Triangle ADB
label("A", (1,2), N);
label("B", (2,0), SE);
label("C", (0,0), SW);
label("D", (1,0), S);
filldraw((0,0)--(1,0)--(1,2)--cycle, gray, black); // Shaded triangle ADC
[/asy]


Simplify $\sqrt[3]{40a^5b^8c^{14}}$. What is the sum of the exponents of the variables that are outside the radical?


Solution: \boxed{(-\infty, 1) \cup (2, 3) \cup (4, 5) \cup (6, \infty)}


If Marco can paint a room in 60 minutes, and Carla helps him such that together they take only 40 minutes, what fractional part of the room does Marco paint alone in 32 minutes?


For a certain hyperbola,
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]
where \(a \neq b\), the angle between the asymptotes is \(90^\circ\). Find \(\frac{a}{b}\).


A basketball player made the following number of free throws in 10 successive games: 20, 12, 22, 25, 10, 16, 15, 12, 30, 10. Calculate the median and the mean number of successful free throws.


Two cylindrical cans have the same volume. The height of one can is double the height of the other. If the radius of the narrower can is 8 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form.


Find the difference between $2333_7$ and $1111_7$ in base $7$.


Factor the following expression: $74a^2 + 222a + 148$.

Solution: \boxed{74(a + 2)(a + 1)}


Define the munificence of a polynomial \(p(x)\) as the maximum value of \(|p(x)|\) on the interval \([-2, 2]\). Find the smallest possible munificence of a monic cubic polynomial of the form \(p(x) = x^3 + ax^2 + bx + c\).


$\textbf{Elena's Vintage Postcard Collection}$

Elena organizes her postcard collection by region and by the decade in which they were issued. The prices she paid for them at an antique store were: Asia and Africa, $7$ cents each, Europe $5$ cents each, and North America $6$ cents each. (Asia and Africa are grouped under Eastern Hemisphere, Europe and North America are Western Hemisphere.)

In her collection, the distribution of postcards by decade and by region for the 1950s through the 1980s is as follows:

| Decade     | Asia | Africa | Europe | North America |
|------------|------|--------|--------|---------------|
| 1950s      | 5    | 3      | 7      | 4             |
| 1960s      | 8    | 6      | 5      | 10            |
| 1970s      | 14   | 11     | 8      | 12            |
| 1980s      | 9    | 7      | 6      | 15            |

How much did her postcards from the Eastern Hemisphere issued before the 1970s cost her?


After their morning meeting, Jay and Paul start walking in opposite directions. Jay walks 1 mile every 20 minutes and Paul jogs 3 miles every 40 minutes. In miles, how far apart are they after 2 hours?

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


Let \( p, q, r \) be the roots of the cubic polynomial \( x^3 - 2x^2 - x + 2 = 0 \). Calculate
\[ p(q - r)^2 + q(r - p)^2 + r(p - q)^2. \]


What is the side length and the diagonal, in units, of a cube whose surface area is 864 square units?

Solution: \boxed{12\sqrt{3} \text{ units}}


For what value of $k$ is $7 \times 6 \times 4 \times k = 9!$?


Add $12_8 + 157_8.$ Express your answer in base 8.


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


The curve $y = \sin x$ intersects with the line $y = \sin 60^\circ$ on a coordinate plane. Determine the ordered pair of relatively prime positive integers $(p, q)$ that represents the ratio of the segments created by these intersections, assuming $p < q$.

Solution: \boxed{(1, 5)}


Express $0.\overline{2}+0.\overline{03}+0.\overline{0004}$ as a common fraction.


Five consecutive positive integers have a product of 15120. What is the largest of the five integers?


Find the square root of $\dfrac{10!}{210}$.


What is the smallest positive integer $n$ such that $19n \equiv 1456 \pmod{11}?$


Let $p$ and $q$ be real numbers. One of the roots of the equation $x^3 + px + q = 0$ is $2 + i$. Find $p + q$.


Consider two lines: line $l$ parametrized as
\[
\begin{align*}
x &= 2 + 5t,\\
y &=  3 + 4t,
\end{align*}
\]
and line $m$ parametrized as
\[
\begin{align*}
x &= -6 + 5s,\\
y &=  7 + 4s.
\end{align*}
\]
Let $C$ be a point on line $l$, $D$ be a point on line $m$, and let $Q$ be the foot of the perpendicular from $C$ to line $m$.

Then $\overrightarrow{QC}$ is the projection of $\overrightarrow{DC}$ onto some vector $\begin{pmatrix} u_1\\u_2\end{pmatrix}$ such that $u_1+u_2 = 3$. Find $\begin{pmatrix}u_1 \\ u_2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} -12 \\ 15 \end{pmatrix}}


For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$?


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


Suppose that $x, y,$ and $z$ are positive integers satisfying $(x+y+z)^3 - x^3 - y^3 - z^3 = 300$. Find $x+y+z$.


Max has drawn a pair of concentric circles. He begins to draw chords $\overline{PQ}$, $\overline{QR}$, and so on, of the larger circle, each chord being tangent to the smaller circle. Given that $\angle PQR = 60^\circ$, how many sections will Max draw before returning to his starting point at $P$?


A sequence begins with 3, and each subsequent term is triple the sum of all preceding terms. Determine the first term in the sequence that exceeds 15000.


Calculate $\frac{4 \cdot 6! + 24\cdot 5!}{7!}$.


Evaluate $\sqrt{\frac{25}{49} - \frac{16}{81}}$ as a common fraction and express your answer in simplest form.


Determine how many positive integers $x$ satisfy $10 < x^2 + 6x + 9 < 50$.


At a painting club meeting, 7 friends are present. They need to create two separate teams: one team of 4 members, and another team of 2 members for different competitions. How many distinct ways can they form these teams?


What is the smallest positive four-digit integer equivalent to 4 mod 9?


In trapezoid $EFGH$, sides $\overline{EF}$ and $\overline{GH}$ are parallel, $\angle E = 3\angle H$, and $\angle G = 2\angle F$. Find $\angle F$.

Solution: \boxed{\angle F = 60^\circ}


In triangle $ABC$, $BC = 20 \sqrt{2}$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$.


Square EFGH has its center at $(4, 4)$ and has an area of 36 square units. The right side of the square is vertical. The square is then dilated with the dilation center at $(0,0)$ and a scale factor of 3. What are the coordinates of the vertex of the image of square EFGH that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(21, 21)}


Albert has a bag with 10 marbles numbered 1 to 10. Lisa has a bag with 8 marbles numbered from 1 to 8. Lisa draws one marble, and Albert draws three marbles (where the order of his choices matters). How many ways can they draw the marbles such that the product of the numbers on Albert's three marbles equals twice the number on Lisa's marble?


Given that $a$, $b$, $c$, and $d$ have the values 2, 3, 4, and 5, but not necessarily in that order, determine the largest possible value of the sum of the four products $ab$, $ac$, $bd$, and $cd$.


Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. If $DP= 24$ and $EQ = 32$, then what is ${DE}$?


Call a set of integers "widely spaced" 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 spaced?


Add 34.2791 to 15.73684 and round to the nearest hundredth.


There are 10 sprinters in the Olympic 100-meter finals, including four Canadians. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded if at most one Canadian wins a medal?


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


My five friends and I play doubles badminton every weekend. Each weekend, two of us play as a team against another two, while the remaining two rest. How many different ways are there for us to choose the two teams and the resting pair?


Consider the parametric equations as $t$ varies over all real numbers:
\begin{align*}
x &= t^2 + 3, \\
y &= t^3 - 6t + 4.
\end{align*}
Determine the coordinates of the point(s) where this curve intersects itself.

Solution: \boxed{(9, 4)}


The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth.


Find the sum of the first five prime numbers that have a units digit of 3.


Five people can mow a lawn in 8 hours. If one person is unavailable, how many additional people will be needed to mow the lawn in 3 hours, assuming each person mows at the same rate?


Suppose that $a$ and $b$ are positive integers such that $(a+bi)^3 = 2 + 11i$. What is $a+bi$?

Solution: \boxed{2 + i}


In a spelling bee, 60% of the students were eliminated after the first round. Of those who proceeded to the second round, only 25% were still in the contest after that round. Finally, $\frac{1}{4}$ of the remaining students were still in the contest after the third round. If 15 students were still in the contest after the third round, how many students began the contest?


A composite number is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such numbers?


Find four consecutive positive prime numbers such that their sum is divisible by 5. What is the smallest possible value of this sum?


Find the scalar $r$ such that for vectors $\mathbf{a} = \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$, the equation
\[\begin{pmatrix} 6 \\ 0 \\ -3 \end{pmatrix} = p \mathbf{a} + q \mathbf{b} + r (\mathbf{a} \times \mathbf{b})\]
holds true for some scalars $p$ and $q$.


A penny, two nickels, a dime, a quarter, and a half-dollar are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents, rounded to the nearest tenth of a cent.


When $\sqrt[4]{2^6 \cdot 3^5 \cdot 5^2}$ is fully simplified, the result is $x\sqrt[4]{y}$, where $x$ and $y$ are positive integers. What is $x+y$?


A jar contains four red marbles, five green marbles, and eleven white marbles. Three marbles are randomly drawn from this jar without replacement. What is the probability that the first marble drawn will be red, the second will be green, and the third will be white? Express your answer as a common fraction.


For each value of $x,$ $g(x)$ is defined to be the minimum value of the three numbers $3x + 3,$ $\frac{1}{3} x + 2,$ and $-\frac{1}{2} x + 8.$ Find the maximum value of $g(x).$


Lucy has $20$ dollars and $75$ cents. She wants to buy packs of cookies priced at $1.75 each. Each pack contains 2 cookies. How many cookies can Lucy buy with her money?


Let \[p(x,y) =
\begin{cases}
x + 2y &\quad \text{if } x > 0 \text{ and } y \ge 0, \\
2x - 3y &\quad \text{if } x \leq 0 \text{ and } y > 0, \\
x^2 + 2y &\quad \text{if } x < 0 \text{ and } y \leq 0, \\
2x + y &\quad \text{otherwise}.
\end{cases}\]
What is $p(p(2,-1),p(-3,1))$?


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


The asymptotes of a hyperbola are \(y = 2x + 3\) and \(y = -2x + 1\). The hyperbola also passes through the point \((2, 1)\). Find the distance between the foci of the hyperbola.


Given the functions \( p(x) = x^2 - 4 \) and \( q(x) = -|x| \), evaluate \( q(p(x)) \) for \( x = -3, -2, -1, 0, 1, 2, 3 \) and find the sum of these values.


The graphs of four functions, labelled (1) through (4), are shown. The domain of function (2) is $$\{-6, -5, -4, -3, -2, -1, 0, 1\}.$$ Determine the product of the labels of the functions that are invertible. Assume:
- Function (1): $y = x^3 - 3x$
- Function (2): Discrete points shown
- Function (3): $y = -\tan(x)$
- Function (4): $y = 5/x$


Calculate the value of the expression \[(2^{1002} + 5^{1003})^2 - (2^{1002} - 5^{1003})^2\] and express it in the form $k \cdot 10^{1002}$, where $k$ is some integer.


How many odd numbers between $100$ and $999$ have distinct digits, with the condition that the hundred's place digit must be even?