Let
\[ g(x) = x^3 + 5x^2 + 10x + 20. \]
Determine the coordinates of the point where the graphs of $y = g(x)$ and $y = g^{-1}(x)$ intersect exactly at one point $(c,d)$.

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


Compute the smallest base-10 positive integer greater than 15 that is a palindrome when written in both base 2 and base 4.


Evaluate the argument $\theta$ of the complex number
\[
e^{11\pi i/60} + e^{31\pi i/60} + e^{51 \pi i/60} + e^{71\pi i /60} + e^{91 \pi i /60}
\]
expressed in the form $r e^{i \theta}$ with $0 \leq \theta < 2\pi$.


Find all $c$ which satisfy $$\frac{c}{5} \le 4 + c < -3(1 + 2c).$$ Express your answer in interval notation, simplifying any fractions which occur in your answer.

Solution: \boxed{[-5, -1)}


Suppose $q(x)$ is a monic cubic polynomial with real coefficients such that $q(5-3i)=0$ and $q(0)=-40$.

Determine $q(x)$ in expanded form.

Solution: \boxed{q(x) = x^3 - \frac{190}{17}x^2 + \frac{778}{17}x - 40}


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


What is the average of all the integer values of \( N \) such that \( \frac{N}{77} \) is strictly between \( \frac{5}{11} \) and \( \frac{2}{7} \)?


Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,12\rbrace$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Find the remainder obtained when $n$ is divided by $1000$.


Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + 2b + 3c = 1.\) Find the minimum value of
\[
\frac{1}{a} + \frac{2}{b} + \frac{3}{c}.
\]


How many integers are solutions to these three inequalities?
\[
\begin{array}{cccc}
(1) & -5x & \geq & 3x+15 \\
(2) & -3x & \leq & 9 \\
(3) & 7x & \leq & -14
\end{array}
\]


Determine the perimeter of a triangle with vertices at $P(2,3)$, $Q(2,9)$, and $R(7,4)$.

Solution: \boxed{6 + 5\sqrt{2} + \sqrt{26}}


Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area of the large square.


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 \(a_n = 5a_{n-1} + 1\) if \(a_{n-1}\) is odd. For how many positive integers \(a_1 \le 1000\) is it true that \(a_1\) is less than each of \(a_2\), \(a_3\), and \(a_4\)?


Let $X$ equal the number of five-digit even numbers. Let $Y$ equal the number of five-digit multiples of 7. Find $X+Y$.


Sixty cards are placed in a box, each bearing a number from 1 through 15, with each number appearing on four cards. Four cards are drawn from the box at random without replacement. Let $p'$ be the probability that all four cards bear the same number. Let $q'$ be the probability that three of the cards bear a number $a$ and the fourth bears a number $b$, with $b \neq a$. What is the value of $\frac{q'}{p'}$?


A high school science club has 10 boys and 12 girls. The school wants to select a team of 8 members to participate in a regional science fair. How many different ways can the team be selected without any restrictions?


Let $n$ be a positive integer. If $b \equiv (5^{2n} + 6)^{-1} \pmod{11}$, what is the remainder when $b$ is divided by $11$?


How many different positive values of $x$ will make the statement true: there are exactly $3$ positive two-digit multiples of $x$?


Two spinners are divided into fifths and sixths, respectively. Each section of the first spinner contains the numbers 3, 6, 5, 10, 15, while the second spinner contains 7, 6, 11, 12, 13, 14. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.


If 9 knicks = 3 knacks and 2 knacks = 5 knocks, how many knicks are equal to 40 knocks?


How many three-digit numbers are multiples of neither 3 nor 4?


Two rays with common endpoint $O$ form a $45^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB = 2$. What is the maximum possible length of $\overline{OB}$?


Let \(a, b, c, d, e, f, g,\) and \(h\) be real numbers such that \(abcd = 16\) and \(efgh = 1.\) Find the minimum value of:
\[(af)^2 + (be)^2 + (ch)^2 + (dg)^2.\]


Determine how many triples $(x, y, z)$ of real numbers satisfy the following relations:
\begin{align*}
x &= 2023 - 2024 \operatorname{sign}(y + z), \\
y &= 2023 - 2024 \operatorname{sign}(x + z), \\
z &= 2023 - 2024 \operatorname{sign}(x + y).
\end{align*}


Let $f(x) = \frac{2ax + b}{cx + 2d}$ where $abcd \neq 0$. If $f(f(x)) = 3x - 4$ for all $x$ in the domain of $f$, what is the value of $a + d$?


The digits $1, 2, 3, 4, 5,$ and $6$ can be arranged to form many different $6$-digit positive integers with six distinct digits. In how many such integers do the digits $1$, $2$, and $3$ appear sequentially (in any order) somewhere in the number?


How many 8-digit positive integers are composed of digits 1 or 2 such that they contain at least one pair of consecutive 1's and at least one pair of consecutive 2's?


What is the area, in square units, of a trapezoid bounded by the lines \(y = 2x\), \(y = 12\), \(y = 6\) and the \(y\)-axis? Express your answer as a decimal to the nearest tenth.


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


Determine the sum of the first 7 terms of the geometric series given by: $\frac{1}{2} - \frac{1}{6} + \frac{1}{18} - \frac{1}{54} + \frac{1}{162} - \frac{1}{486} + \frac{1}{1458}$.


If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?


Solution: \boxed{x = 3^{\frac{7}{2}}}


Lucy has a penchant for strawberry milk and decides to visit a local café for a fortnight to indulge in her favorite beverage. This café, similar to a milk bottling plant, occasionally serves caramel milk instead, with a 3/4 chance on any given day of serving strawberry milk. What is the probability that Lucy gets to drink strawberry milk on exactly 5 out of the 7 days she visits?


Sarah baked 4 dozen pies for a community fair. Out of these pies:
- One-third contained chocolate,
- One-half contained marshmallows,
- Three-fourths contained cayenne pepper,
- One-eighth contained walnuts.

What is the largest possible number of pies that had none of these ingredients?


Roberto has six pairs of trousers, eight shirts, and four jackets. How many different outfits can he put together if an outfit consists of a pair of trousers, a shirt, and a jacket, with the restriction that Jacket 1 cannot be worn with Shirt 1 or Shirt 2?


Find the maximum $y$-coordinate of a point on the graph of $r = \sin 3 \theta.$


Let \( g(x) \) be the function defined on \(-2 \le x \le 2\) by the formula
\[ g(x) = 2 - \sqrt{4 - x^2}. \]
If a graph of \( x = g(y) \) is overlaid on the graph of \( y = g(x) \), then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth?


Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}


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.


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{5},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}.$


A triangle with sides 8, 15, and 17 has an inscribed and a circumscribed circle. Calculate the distance between the centers of these two circles.


Let \( a, b, c \) be positive real numbers different from 1. Find the number of real roots of the equation
\[
\begin{vmatrix} x-1 & c-1 & -(b-1) \\ -(c-1) & x-1 & a-1 \\ b-1 & -(a-1) & x-1 \end{vmatrix} = 0.
\]


If two numbers will be randomly chosen without replacement from $\{3, 4, 6, 8\}$, what is the probability that their product will be a multiple of 8?


Julie works for 48 hours per week for 12 weeks during the summer, earning $\$5000$. If she works for 48 weeks during the school year at the same hourly rate and aims to make $\$8000$, how many hours per week must she work?


Laura has a rectangular prism-shaped tank with dimensions $8$ inches, $10$ inches, and $12$ inches. She fills this tank to one third full with water. Then she places eight ice cubes, each measuring $2$ inches on each side, into the tank. How much of the tank is unoccupied by both water and ice?

Solution: \boxed{576 \text{ cubic inches}}


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


Find the distance between the foci of the ellipse
\[\frac{x^2}{45} + \frac{y^2}{5} = 9.\]


Evaluate both the least common multiple (LCM) and the greatest common divisor (GCD) of 210 and 396.


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

What is the remainder when $14!$ is divided by $17$?


Suppose a real number \( y \) satisfies
\[
\sqrt{64 - y^2} - \sqrt{16 - y^2} = 5.
\]
What is the value of \( \sqrt{64 - y^2} + \sqrt{16 - y^2} \)?


Write the expression $\frac{3+4d}{5}+3$ as a single fraction.


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


Consider a circular public area which consists of three parts: a central circular play area, surrounded by a landscaped garden ring, and an outer running track. The running track is consistently four feet wide, the garden ring is six feet wide, and the play area has a diameter of 14 feet. What is the diameter of the circle that forms the outer boundary of the running track?


Four faucets fill a 50-gallon tub. Given that three faucets can fill a 100-gallon tub in 6 minutes, how long, in seconds, does it take these four faucets to fill up the tub? Assume all faucets dispense water at the same rate.


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


Point $P$ lies on the line $x = -3$ and is 15 units from the point $(7, -3)$. Find the product of all possible $y$-coordinates that satisfy these conditions.


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{2}$, meaning that if the length of the object was originally $l$, then it becomes $l + \frac{1}{2} l.$ On the second day, he increased the new length by $\frac{1}{3}$; on the third day, he increased it by $\frac{1}{4}$; and so on. On the $n^{\text{th}}$ day, Barry’s magic makes the object's length double its original length. Determine the value of $n$.


Define $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ 4 & 0 \end{pmatrix}.$ Find the vector $\mathbf{v}$ such that
\[(\mathbf{A}^7 + \mathbf{A}^5 + \mathbf{A}^3 + \mathbf{A} + \mathbf{I}) \mathbf{v} = \begin{pmatrix} 5 \\ 0 \end{pmatrix}.\]

Solution: \boxed{\begin{pmatrix} -\frac{5}{28899} \\ \frac{1700}{28899} \end{pmatrix}}


Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$?


I have 6 books on a shelf, three of which are identical copies of the same science book, and the other three books are all different from each other and from the science books. In how many ways can I arrange these books?


A gardener plants two pine trees, four oaks, and six birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.


A binary operation \( \diamondsuit \) satisfies \( a \diamondsuit (b \diamondsuit c) = (a \diamondsuit b) \cdot c \) and \( a \diamondsuit a = 1 \) for all nonzero real numbers \( a, b, \) and \( c \). Solve the equation \( 504 \diamondsuit (12 \diamondsuit x) = 50 \).


The third and fifteenth terms of an arithmetic sequence are $\frac{5}{11}$ and $\frac{7}{8}$, respectively. Find the ninth term of the sequence.


Solution: \boxed{\frac{25x^2 - 16x^3}{20}}


At the mall's food court, Crystal and her friend Amy each want to buy a meal consisting of one entree, one drink, and one dessert. The table below lists both Crystal’s and Amy’s favorite foods from the food court. Assuming they could choose from a shared assortment and wanted to try different meals from each other, how many possible distinct pairs of meals can they buy if they follow these conditions?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Entrees} & \textbf{Drinks} & \textbf{Desserts} \\
\hline
Pizza & Lemonade & Frozen Yogurt \\
\hline
Chicken Teriyaki & Root Beer & Chocolate Chip Cookie \\
\hline
Corn Dog & Iced Tea & Apple Pie \\
\hline
Fish and Chips & & \\
\hline
\end{tabular}


Ella made a complete list of the prime numbers between 1 and 50. What is the sum of the smallest prime number and the largest prime number on her list, and what is the product of these two numbers?


In parallelogram $EFGH$, angle $F$ measures $120^\circ$. Diagonal $\overline{EH}$ is drawn, bisecting angle $E$. What are the measures of angle $EFG$ and angle $H$?


Let $C$ equal the number of four-digit numbers that are both odd and divisible by 5. Let $D$ equal the number of four-digit multiples of 3. Find $C+D$.


Given ten distinguishable rings, let $n$ be the number of possible six-ring arrangements on the five fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.


What is the remainder when $11^{2023}$ is divided by $100$?


Find all solutions to $z^5 = -32$, express them in the form $z = r (\cos \theta + i \sin \theta)$ where $r > 0$ and $0^\circ \leq \theta < 360^\circ$, and compute $\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5$, where $z_1, z_2, z_3, z_4, z_5$ are the solutions.


A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 5 meters from its base. When John walks 4 meters from the base of the pole toward the point where the cable is attached to the ground, his head just touches the cable. John is 3 meters tall. How many meters tall is the pole?

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


Solution: \boxed{10 - 4i}


Let $N$ be the sum of the divisors of $180$. What is the largest prime factor of $N$?


Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 5,$ $\|\mathbf{b}\| = 7,$ and $\|\mathbf{a} + \mathbf{b}\| = 10.$  Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$


Let $n$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $n$ is divided by $5!$.


Let $ABCD$ and $EFGH$ be two faces of a cube with side length $10$. A beam of light emanates from vertex $A$ and reflects off face $EFGH$ at point $Q$, which is $4$ units from $\overline{EH}$ and $3$ units from $\overline{EF}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $p\sqrt{q}$, where $p$ and $q$ are integers and $q$ is not divisible by the square of any prime. Find $p+q$.


A new formula is introduced to calculate the crushing load \( L \) of a cylindrical pillar which is represented by \( L = \frac{30T^3}{H \cdot D} \), where \( T \) is the thickness in centimeters, \( H \) is the height in meters, and \( D \) is the diameter in centimeters. If \( T = 5 \), \( H = 15 \), and \( D = 10 \), what is the value of \( L \)?


Find the distance between the planes $x - 2y + 2z = 9$ and $2x - 4y + 4z = 18.$


Determine a set of integers $a$, $b$, and $c$ such that the equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ can be reduced to $\cos ax \cos bx \cos cx = 0$ and find the sum $a + b + c$.


Let $r$ and $s$ denote the two real roots of $x^2 - 2x \sqrt{3} + 2 = 0.$ Determine $r^6 + s^6.$


The graph of $y = g(x)$, which consists of five line segments, is shown in blue. The graph goes through the points (-4, -6), (-2, -2), (-1, -3), (1, 1), (2, 0), and (4, 4).

What is the sum of the x-coordinates of all points where $g(x) = x + 2$?


Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$.


Jackie and Phil each have two fair coins and a third coin that comes up heads with a probability $\frac{2}{5}$. They each flip all three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


Let $R$ be the set of integers between $1$ and $2^{20}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $R$, the probability that it is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Consider the ellipse \(4(x-2)^2 + 16y^2 = 64\). Let \(C\) be one of the endpoints of its major axis, and let \(D\) be one of the endpoints of its minor axis. Find the distance \(CD\).


The probability that Tom has a soccer match today is $\frac{5}{8}$. Additionally, the probability that he has a science test today is $\frac{1}{4}$. What is the probability that Tom neither has a soccer match nor a science test today? Express your answer as a common fraction.


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?


For all integers $n$ that are multiples of 4, what is the largest integer that always divides into the difference between $n$ and the fourth power of $n$?


Let $a$, $b$, $c$, $d$, $e$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 + e^2 = 5.\]
Find the maximum value of $a^3 + b^3 + c^3 + d^3 + e^3.$


There are 6 blue chips, 5 red chips, 4 yellow chips, and 3 green chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


Solve the inequality
\[\frac{x^2 - 16}{x - 4} \leq 0.\]

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


Jeff decides to play with a Magic 8 Ball again, but this time he asks it 8 questions. Each time he asks a question, it still has a 2/5 chance of giving him a positive answer. What is the probability that it gives him exactly 3 positive answers out of the 8 questions?


Given that the determinant of a $2 \times 2$ matrix $A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is 3, find the values of $\begin{vmatrix} 3a & 3b \\ 3c & 3d \end{vmatrix}$ and $\begin{vmatrix} 4a & 2b \\ 4c & 2d \end{vmatrix}$.


Suppose $x-4$ and $y+4$ are multiples of 9.

What is the smallest positive integer, $n,$ for which $x^2 + xy + y^2 + n$ is a multiple of 9?


When a biased coin is flipped seven times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac{m}{n}$, in lowest terms, be the probability that the coin comes up heads in exactly $4$ out of $7$ flips. Find $m+n$.


Find the matrix $\mathbf{N}$ that quadruples the first row and doubles the second row of a matrix. In other words,
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 4a & 4b \\ 2c & 2d \end{pmatrix}.\] If no such matrix $\mathbf{N}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}}


The function \( g(x) \) satisfies
\[ g(x + y) = g(x) + g(y) \] for all real numbers \( x \) and \( y \), and \( g(6) = 7. \) Find \( g(8). \)


Consider the ellipse represented by the equation \[\sqrt{(x-4)^2 + (y-5)^2} + \sqrt{(x+6)^2 + (y-9)^2} = 24\]. Determine the distance between its foci.


What is the sum of the x-coordinates of all the points of intersection of the two circles defined by \(x^2 - 6x + y^2 - 8y + 24 = 0\) and \(x^2 - 4x + y^2 - 8y + 16 = 0\)?


At the mall's food court, Charlie wants to choose a meal consisting of one entree, one drink, and one dessert. The table below lists Charlie's available options in the food court. How many distinct possible meal combinations can Charlie choose from?

\begin{tabular}{ |c | c | c | }
\hline \textbf{Entrees} & \textbf{Drinks}&\textbf{Desserts} \\ \hline
Pasta & Cola & Pecan Pie \\ \hline
Burger & Iced Tea & Ice Cream \\ \hline
Hot Dog & Coffee & \\ \hline
BBQ Ribs & Fruit Punch & \\ \hline
\end{tabular}


Calculate the total volume in cubic feet of a round swimming pool which is 20 feet in diameter and 5 feet deep throughout, with an additional hemispherical bottom. Express your answer in terms of $\pi$.


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

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


A math conference is presenting a lecture series with seven different lecturers. If Dr. Smith's lecture depends on Dr. Jones's lecture so that Dr. Smith must be scheduled sometime after Dr. Jones, and additionally, Dr. Allen's lecture must occur after Dr. Brown's lecture, in how many orders can the seven lecturers be scheduled?


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


Suppose I have a collection of 12 pieces of art, including 4 prints by Picasso. If these pieces are hung in a row randomly, what is the probability that all four Picasso prints are hung consecutively?


Compute the value of the expression $(8a^3b) \cdot (4ab^2) \cdot \left(\frac{1}{(2ab)^3}\right)$.


In triangle $XYZ$, where $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 4\cos Y$. Find the length of $XY$.


Find the smallest solution to the equation \[\frac{1}{x-3} + \frac{1}{x-5} = \frac{4}{x-4}.\]

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


Four prime numbers are randomly selected without replacement from the first twelve prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction.


How many four-digit positive integers have at least one digit that is a 5 or a 7?


Solution: \boxed{x = 18}


I am planning to dine at a new seafood restaurant that also has some vegetarian options. I am a vegan and allergic to soy. They have three dishes without any animal products, and that's just one tenth of their entire menu. Two of these vegan dishes contain soy. What fraction of the items on the menu can I eat?


Samantha creates a triangular pyramid for an art project. The height of her pyramid is 15 inches, and it has an equilateral triangular base with each side measuring 12 inches. She needs to find the smallest cube-shaped box to accommodate her pyramid so that she can transport it upright to her art show. What is the volume of this cube-shaped box, in cubic inches?


The triangle $\triangle ABC$ is an isosceles right triangle with $AB = 6$. The circumcenter of $\triangle ABC$ is denoted by $O$. What is the length of $BO$?


Let $a,$ $b,$ $c$ be nonnegative real numbers such that $a + b + c = 2.$ Find the maximum value of
\[\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c}.\]


Find the matrix \(\mathbf{N}\) that triples the second column of a matrix. In other words,
\[\mathbf{N} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & 3b \\ c & 3d \end{pmatrix}.\]
If no such matrix \(\mathbf{N}\) exists, then enter the zero matrix.

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


What is the smallest prime factor of 2023?


Samson derives utility according to the relation $$\text{Utility} = (\text{hours of math done} + 2) \times \text{hours of frisbee played}.$$ On one particular day, he plays $s$ hours of frisbee and spends $10 - 2s$ hours doing math. The following day, he manages to get the same amount of utility while spending $3-s$ hours playing frisbee and $2s+4$ hours on math. Find $s$.


Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + b + c = 5\). Find the minimum value of
\[
\frac{9}{a} + \frac{16}{b} + \frac{25}{c}.
\]


John drove continuously starting from 8:15 a.m. and took a 30-minute break at noon. He then resumed driving and reached his destination at 2:45 p.m. on the same day, covering a total distance of 240 miles. What was his average speed in miles per hour?


A regular octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 96 centimeters. What is the number of square centimeters in the area of the octagon?

Solution: \boxed{448 \text{ cm}^2}


**What is the value of $\sqrt{25 - 10\sqrt{6}} - \sqrt{25 + 10\sqrt{6}}$?**


Find all solutions to the equation $\sqrt[3]{6 - x} = -\frac{5}{2}$.


How many positive real solutions are there to $x^{12} + 5x^{11} - 3x^{10} + 2000x^9 - 1500x^8 = 0$?


Point $O$ is the center of an ellipse with major axis $\overline{AB}$ and minor axis $\overline{CD}$. Point $F$ is one focus of the ellipse. If $OF = 8$ and the diameter of the inscribed circle of triangle $OCF$ is 3, compute the product $(AB)(CD)$.


A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be an odd number? Express your answer as a common fraction.


Compute the sum and the product of all the roots of the equation $(3x+2)(x-5) + (3x+2)(x-8) = 0$.


Square EFGH has its center at $(10, -6)$ 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 $(2,2)$ 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? Provide your answer as an ordered pair.

Solution: \boxed{(32, -28)}


We have a triangle $\triangle XYZ$ with a point $M$ on $YZ$ such that $XM$ is an altitude of $\triangle XYZ$. Given that $XZ = 15$, $YM = 9$, and $YZ = 20$, what is the area of $\triangle XYZ$?


Fifty teams play a tournament where each team plays every other team exactly once. There are no draws, and each team has a $50\%$ chance of winning any game. The probability that no two teams win the same number of games is given by $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Calculate $\log_2 n$.


Isosceles triangle $ABE$ of area 180 square inches is cut by line segment $\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 135 square inches. If the altitude of triangle $ABE$ from $A$ is 24 inches, what is the length of $\overline{CD}$?

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


Calculate the following sum:
\[
\sum_{N = 1}^{2048} \lfloor \log_2 N \rfloor.
\]


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

Solution: \boxed{g(10) = -48}


The solutions to $z^6 = 64i$ can be expressed in the form
\begin{align*}
z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4), \\
z_5 &= r_5 (\cos \theta_5 + i \sin \theta_5), \\
z_6 &= r_6 (\cos \theta_6 + i \sin \theta_6),
\end{align*}
where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ.$  Find $\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5 + \theta_6,$ in degrees.


The letters of the alphabet are assigned numeric values based on two conditions:

$\bullet$ Numeric values used are $-3, -2, -1, 0, 1, 2, 3$.

$\bullet$ Starting with A through Z, a numeric value is assigned to each letter according to this pattern:
$$
1, 2, 3, 0, -1, -2, -3, 0, 1, 2, 3, 0, -1, -2, -3, 0, \ldots
$$

The pattern repeats every 8 letters. Determine the sum of the numeric values of the letters in the word "mathematics."


How many distinct arrangements of the letters in the word "balloon" are there?


Altitudes $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at $H$. If $\angle BAC = 60^\circ$ and $\angle ABC = 55^\circ$, then what is $\angle AHB$?


Zoey wants to buy cookies that come in packages of 5, and she buys boxes of milk that come in packages of 7. What is the smallest number of cookie packages Zoey can buy in order to be able to buy an equal number of cookies and milk boxes?


Find how many ordered pairs of positive integers $(x,y)$, with $x<y$, is the harmonic mean of $x$ and $y$ equal to $1024$.


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


Consider a function $g(x)$ such that its values are described in the table below:
\[
\begin{tabular}{|c||c|c|c|c|c|} \hline
$x$ & 1 & 2 & 3 & 4 & 5 \\ \hline
$g(x)$ & 4 & 5 & 1 & 2 & 3 \\ \hline
\end{tabular}
\]
If the inverse $g^{-1}$ of $g$ exists, what is $g^{-1}(g^{-1}(g^{-1}(5)))$?


Five six-sided dice are rolled. It is known that after the roll, there are two pairs of dice showing the same number, and one odd die. The odd die is rerolled. What is the probability that after rerolling the odd die, the five dice show a full house?


Calculate the area of trapezoid EFGH with vertices E(2,1), F(2,4), G(7,1), and H(7,-3).


Suppose Acme Corporation has released a modified version of their alphabet soup where each vowel (A, E, I, O, U) appears with different frequencies: A appears 6 times, E appears 4 times, I appears 5 times, O appears 3 times, and U appears 2 times. How many five-letter words can be formed if the word must contain exactly two A's and at least one O?


Let $p$, $q$, $r$ be the sides of a triangle, and let $\theta$, $\phi$, $\psi$ be the angles opposite these sides respectively. Suppose that $p^2 + q^2 = 2001r^2$. Find the value of
\[\frac{\cot \psi}{\cot \theta + \cot \phi}.\]


Express $7.\overline{3}$ as a common fraction.


How many positive two-digit integers have an even number of positive factors?


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


The number $1043$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$?


Determine the first term of the geometric sequence where the third term is 18 and the fifth term is 72.


Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be vectors such that $\|\mathbf{u}\| = \|\mathbf{v}\| = 1$, $\|\mathbf{u} + \mathbf{v}\| = 2$, and
\[\mathbf{w} - \mathbf{u} - 3\mathbf{v} = 2 (\mathbf{u} \times \mathbf{v}),\]
find $\mathbf{v} \cdot \mathbf{w}$.


Suppose $q(x)$ is a function such that $q(x) + (2x^6 + 5x^4 + 10x^2) = (8x^4 + 30x^3 + 40x^2 + 5x + 2)$. Express $q(x)$ as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{-2x^6 + 3x^4 + 30x^3 + 30x^2 + 5x + 2}


Each of the numbers \( b_1, b_2, \dots, b_{99} \) is \( \pm 1 \). Find the largest possible value of:
\[
\sum_{1 \le k < l \le 99} b_k b_l.
\]


A uniquely designed die is shaped such that when tossed, the probability of a face labelled "Star" showing up is $\frac{2}{5}$, while the face labelled "Moon" comes up $\frac{3}{5}$. If you win $\$4$ upon the die landing on "Star" and lose $\$3$ when it lands on "Moon", what is the expected value, in dollars, of one roll of the die?


Suppose $m$ and $n$ are positive 4-digit integers such that $\gcd(m,n)=5$. What is the smallest possible value for $\mathop{\text{lcm}}[m,n]$?


Juan measures both the diameter of a circle and the side length of a square. The actual diameter of the circle is 30 cm, and the side length of the square is 30 cm as well, but Juan's measurement tool has an error of up to $30\%$. What is the largest possible percent error in Juan's computed area of the circle and the square?


Spinners $C$ and $D$ are spun. Each spinner has numbers 1 through 6, where spinner $C$ has numbers {1, 1, 2, 3, 3, 4} and spinner $D$ has numbers {1, 2, 2, 3, 4, 4}. Calculate the probability that the product of the two spinners' numbers is even. Express your answer as a common fraction.


Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$


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)}


My friend Lily is fascinated with numbers that are divisible by 4. How many different last two digits are possible in numbers that Lily likes?


On "Modulo $m$ graph paper," graph the congruences $$y \equiv 3x + 6 \pmod{20}$$ and $$y \equiv 7x + 18 \pmod{20}.$$ Determine the sum of the $x$-coordinates of the points where these graphs overlap.


Let \( a, b, c \) be the roots of the polynomial \( x^3 - 8x^2 + 10x - 1 = 0 \). Define \( p = \sqrt{a} + \sqrt{b} + \sqrt{c} \). Compute \( p^4 - 16p^2 - 8p \).


The Jackson High School Soccer Team has 24 players on the team, including four goalkeepers. They are conducting a drill where each goalkeeper practices saving penalty kicks. Each player, including the other goalkeepers, takes a turn to shoot against each goalkeeper. How many penalty kicks must be taken to ensure that every player has shot against each of the goalkeepers?


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


A ball is dropped from 500 feet high and always bounces back up two-thirds the distance it just fell. After how many bounces will the ball first reach a maximum height less than 2 feet?


Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $RP=9$. Two ants start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$?


How many positive integers, not exceeding 150, are multiples of 3 or 5 but not 6?


If \(a \div b = 3\) and \(b \div c = \frac{2}{5}\), what is the value of \(c \div a\)? Express your answer as a common fraction.


In triangle $DEF,$ $d = 8,$ $e = 15,$ and $f = 17.$ Let $J$ be the incenter.

Find the barycentric coordinates (x, y, z) such that
\[\overrightarrow{J} = x \overrightarrow{D} + y \overrightarrow{E} + z \overrightarrow{F},\]
where $x + y + z = 1.$

Solution: \boxed{\left( \frac{8}{40}, \frac{15}{40}, \frac{17}{40} \right)}


Convert \(2\sqrt{3} e^{17 \pi i/6}\) to rectangular form.

Solution: \boxed{-3 + i\sqrt{3}}


Each row of a seating arrangement seats either 7 or 9 people. Fifty-eight people are to be seated. How many rows seat exactly 9 people if every seat is occupied?


Suppose that \( f(x)=\frac{2}{3x+c} \). For what value of \( c \) does \( f^{-1}(x) = \frac{3 - 6x}{x} \)?


If two numbers are randomly chosen without replacement from the set $\{4, 5, 6, 9\}$, what is the probability that their product will be a multiple of 10? Express your answer as a common fraction.


Let $O$ be the origin. A variable plane has a distance of 2 from the origin, and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the centroid of triangle $ABC.$ Find
\[\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}.\]


How many odd three-digit numbers are there such that the sum of the tens and units digits is 12?


How many integers are between $(12.1)^3$ and $(12.2)^3$?


In $\triangle ABC,$ $AB=AC=20$ and $BC=24.$ Points $G, H,$ and $I$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $AGHI$?


The sequence $(a_n)$ satisfies
\[a_1 + a_2 + a_3 + \dots + a_n = n^2 a_n\] for all $n \ge 2.$ If $a_{50} = 2,$ find $a_1.$


A ball is dropped from a height of $324$ cm and it bounces back up $\frac{3}{4}$ of the height from which it falls. After how many bounces does the ball first rise less than $40$ cm?


Let $O$ be the origin.  A variable plane has a distance of 3 from the origin, and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$  Let $(p,q,r)$ be the centroid of triangle $ABC.$  Calculate $\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}.$


A regular polygon has a perimeter of 180 cm and each side has a length of 15 cm. How many sides does the polygon have, and what is the measure of each interior angle?


What is the largest $2$-digit prime factor of the integer $n = {210 \choose 105}$?


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

Solution: \boxed{\left(\frac{18}{7}, \frac{18}{7}\right)}


A block of modeling clay is a right rectangular prism measuring six inches by two inches by one inch. How many whole blocks are needed to create a cylindrical sculpture that is twelve inches high and has a diameter of six inches?


A hexagon $ABCDEF$ has internal angles where $\angle A = 100^\circ$ and $\angle B = 120^\circ$. Angles $C$ and $D$ are equal, and angle $E$ is $30^\circ$ more than the average of angles $C$, $D$, and $F$. Find the measure of the largest angle in the hexagon.


Find the remainder when \(x^3 + 2x^2\) is divided by \(x^2 + 3x + 2.\)

Solution: \boxed{x + 2}


How many positive integers less than $900$ are either a perfect cube or a perfect square?


In Joanne's design for a quilt, she sketches a large square and inside it, nine smaller identical circles such that each circle is tangent to its neighbors and to the nearest sides of the square. Each side of the square measures 16 inches. Calculate the total area of the square that is not covered by the circles.

Solution: \boxed{256 - 64\pi}


A sphere is cut into five congruent wedges. The circumference of the sphere is $15\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\pi$.


The lengths of the three sides of a triangle are \( 10 \), \( y+5 \), and \( 3y-2 \). The perimeter of the triangle is \( 50 \). What is the length of the longest side of the triangle?


Find a positive integer that is divisible by 24 and whose cube root is a number between 8 and 8.2.


Factor completely: $x^8 - 4x^6 + 6x^4 - 4x^2 + 1$.


Given vectors $\mathbf{a}$ and $\mathbf{b}$ such that $\|\mathbf{a}\| = 5,$ $\|\mathbf{b}\| = 7,$ and $\|\mathbf{a} + \mathbf{b}\| = 10.$  Find $\cos \theta,$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$


What is the smallest positive number that is prime and $6$ less than a perfect square?


What is the smallest positive integer value of $x$ such that $(3x)^2 + 2 \cdot 41 \cdot 3x + 41^2$ is a multiple of 53?


For how many integer values of $n$ between 1 and 539 inclusive does the decimal representation of $\frac{n}{770}$ terminate?


A book with 63 pages numbered 1 to 63 has its pages renumbered in reverse, from 63 to 1. For how many pages do the new page number and old page number share the same units digit?


A fly starts on the edge of a circular room with a radius of 100 feet. It first walks along the ceiling, passing through the center, to the opposite edge of the circle. It then progresses to a third point on the circle's edge making a straight line but not passing through the center again. The path from this third point back to the starting point is 120 feet. Determine the total feet the fly traveled.

Solution: \boxed{480 \text{ feet}}


For positive integers $n$, define $g(n)$ to return the smallest positive integer $k$ such that $\frac{1}{k}$, when expressed in base $6$, has exactly $n$ digits after the radix point. Determine the number of positive integer divisors of $g(2023)$.


To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$?


In her previous seven attempts, Jenny recorded times, in seconds, of 102, 108, 110, 99, 104, 107, and 113 for swimming 50 meters. After her eighth attempt, she managed to decrease her median time to 106 seconds. What was her time, in seconds, for her eighth attempt?


What is the largest divisor of 456 that is less than 60 and also a factor of 108?


In a friendly 4-team football tournament, played over a weekend, each team plays one match on Saturday. Team E plays Team F, and Team G plays Team H. A draw in any match leads to a penalty shootout to determine a winner. Then on Sunday, the winners of the Saturday matches play in the final for the first and second place, while the losers play for third place. Given this setup, where matches could end in a draw but must produce a winner in a shootout, what is the total number of possible four-team ranking sequences at the end of the tournament?


The equation $\sin^2 x + \sin^2 2x + \sin^2 5x + \sin^2 6x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\] for some positive integers $a,$ $b,$ and $c.$  Find $a + b + c.$


Find the phase shift of the graph of \( y = 2 \sin \left( 3x + \frac{\pi}{4} \right) - 2 \).


The polynomial
\[px^4 + qx^3 + rx^2 + sx + t = 0\]
has coefficients that are all integers, and roots $-3$, $4$, $6$, and $\frac{1}{2}$. If $t$ is a positive integer, find its smallest possible value.


What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)(n+11)(n+13)
\] for all positive even integers \( n \)?


Let \(f : \mathbb{R} \to \mathbb{R}\) be a function such that \(f(1) = 3\) and
\[f(x^2 - y^2) = (x - y) (f(x) + f(y))\]for all real numbers \(x\) and \(y.\)

Let \(n\) be the number of possible values of \(f(3)\), and let \(s\) be the sum of all possible values of \(f(3)\). Find \(n \times s\).


Sarah tosses four coins. What is the probability that she will get at least two heads?


In a three-dimensional rectangular coordinate system, what is the number of units in the distance from the origin to the point (8, -15, 17)?


How many even three-digit numbers are there such that the sum of the hundreds and tens digits is 9?


The measures of the interior angles of a convex pentagon form an increasing arithmetic sequence. How many such sequences are possible if the pentagon is not equiangular and all of the angle degree measures are positive integers less than $120$ degrees?


Determine the domain of the real-valued function $$f(x)=\sqrt{-8x^2-10x+12}.$$ The end points of the domain should be expressed as fractions.

Solution: \boxed{\left[-2, \frac{3}{4}\right]}


How many different four-digit numbers can be formed by arranging the digits in 5003?


In a survey, $82.5\%$ of respondents believed that mice were dangerous. Of these, $52.4\%$ incorrectly thought that mice commonly caused electrical fires. Given that these 27 respondents were mistaken, determine the total number of people surveyed.


The ecology club at a school has 30 members: 12 boys and 18 girls.  A 4-person committee is to be chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?


If Ben rolls a standard four-sided die until he rolls the same number on consecutive rolls, what is the probability that his 12th roll is his last roll? Express your answer as a decimal to the nearest thousandth.


Joe now wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property?


Calculate the sum
\[
\text{cis } 80^\circ + \text{cis } 90^\circ + \text{cis } 100^\circ + \dots + \text{cis } 160^\circ
\]
and express it in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.


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


If $x+y=10$ and $xy=12$, what is the value of $x^3-y^3$?


Find the focus of the parabola given by the equation \( y = -\frac{1}{16} x^2 \).

Solution: \boxed{(0, -4)}


The sides of a triangle have lengths of $24$, $24$, and $24\sqrt{2}$. Find the length of the shortest altitude.


The four zeros of the polynomial \(x^4 + px^2 + qx - 144\) are distinct real numbers in arithmetic progression. Compute the value of \(p.\)


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


Given that $b$ is an even multiple of $3883$, find the greatest common divisor of $4b^2 + 35b + 56$ and $3b + 8$.


The circumference of a hemisphere is 36 cm. In square centimeters, what is the total surface area of the hemisphere? Express your answer as a common fraction in terms of $\pi$.


How many numbers are in the even-numbered rows (rows that can be expressed as $2k$ for some integer $k$) of Pascal's Triangle from the $0$th row to the $20$th row?


A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 5-primable positive integers are there that are less than 1000?


Let the equation $x^3 - 6x^2 + 11x - 6 = 0$ have three real roots $a$, $b$, and $c$. Find $\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}$.


The integers $m$ and $n$ are randomly selected where $0 \leq m \leq 7$ and $2 \leq n \leq 9$. What is the probability that the product $2m \times n$ is a multiple of $4$? Express your answer as a common fraction.


The lattice shown is continued for $10$ rows, and in each row, there are now $7$ numbers. What will be the fourth number in the $10$th row?
\begin{tabular}{rccccc}
Row 1: & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
Row 2: & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\
Row 3: & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\
Row 4: & 22 & 23 & 24 & 25 & 26 & 27 & 28
\end{tabular}


What is the 58th digit after the decimal point in the decimal representation of $\frac{1}{17}$?


Let $m' = \underbrace{33333333}_{\text{8 digits}}$ and $n' = \underbrace{555555555}_{\text{9 digits}}$.

What is $\gcd(m',n')$?


A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching?


Let $\mathbf{R}$ be the matrix for reflecting over the vector $\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ in $\mathbb{R}^3$. Find $\mathbf{R}^2$.

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


In the adjoining diagram, \(AB\) is a diameter of the circle, \(CD\) is a chord parallel to \(AB\), and \(AC\) intersects \(BD\) at \(E\), with \(\angle AED = 90^\circ - \alpha\). Determine the ratio of the area of \(\triangle CDE\) to that of \(\triangle ABE\).

Solution: \boxed{\sin^2 \alpha}


A hollow cylindrical pipe has a total surface area $\text{SA}$ of $50\pi$ square feet. The pipe has an outer radius of $5$ feet and an inner radius of $3$ feet. Determine the height of the pipe if there are no top or bottom faces.


One leg of a right triangle is 15 inches, and the measure of the angle opposite that leg is $60^\circ$. What is the number of inches in the hypotenuse of the triangle?


In how many ways can four people sit in a row of seven chairs?


In trapezoid $PQRS$ with $\overline{QR}\parallel\overline{PS}$, let $QR = 1500$ and $PS = 2800$. Let $\angle P = 45^\circ$, $\angle S = 45^\circ$, and $T$ and $U$ be the midpoints of $\overline{QR}$ and $\overline{PS}$, respectively. Find the length $TU$.


The graph of the line $x+y=b$ is the perpendicular bisector of the line segment from $(2,5)$ to $(10,11)$. What is the value of $b$?


Mr. Lee has 12 grandchildren. Assuming that the gender of each child is determined independently and with an equal likelihood of male and female, what is the probability that Mr. Lee has more than 6 grandsons or more than 6 granddaughters?


Bonnie makes the frame of a cube out of 12 pieces of wire that are each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to make a collection of smaller cube frames that are not connected to each other. The total volume of Roark’s cubes is the same as the volume of Bonnie’s cube. What is the ratio of the total length of Bonnie’s wire to the total length of Roark’s wire?


Solve the congruence $13n \equiv 9 \pmod{47}$, as a residue modulo 47. (Provide an answer between 0 and 46.)


Solution: \boxed{|w| = 2}


Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect fourth power, and all the multiples of 30 between them. How many integers are in Cameron's list?


Compute $4 \begin{pmatrix} -3 \\ 5 \end{pmatrix} - 3 \begin{pmatrix} -2 \\ 6 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} -6 \\ 2 \end{pmatrix}}


On the first day, Barry Sotter used his magic to make an object's length increase by $\frac{1}{3}$, so if the original length was $x$, it became $x + \frac{1}{3} x$. On the second day, he increased the new length by $\frac{1}{4}$; on the third day by $\frac{1}{5}$; and so on. On the $n^{\text{th}}$ day, Barry made the object's length exactly 50 times its original length. Find the value of $n$.


If $\mathbf{v} \times \mathbf{w} = \begin{pmatrix} 5 \\ -2 \\ 4 \end{pmatrix},$ then find $(2\mathbf{v} - \mathbf{w}) \times (3\mathbf{w} - \mathbf{v}).$

Solution: \boxed{\begin{pmatrix} 25 \\ -10 \\ 20 \end{pmatrix}}


If the two roots of the quadratic \(4x^2 + 17x + m\) are \(\frac{-17 \pm i\sqrt{471}}{8}\), what is \(m\)?


Four points $C, D, F, M$ are on a straight line. The point $H$ is not on the line such that $\angle CDH = 130^\circ$ and $\angle HFM = 70^\circ.$ If the reflex angle at $H$ is $y^\circ,$ what is the value of $y$?


Wei decides to modify the logo for his new company by using a larger square and additional circles. The new design has six circles, each tangent to two sides of the square and adjacent circles, set within a square frame. If the side length of the new square is 24 inches, determine how many square inches will be shaded.

Solution: \boxed{576 - 96\pi}


The line segment connecting \((1, -3)\) and \((-4, 6)\) can be parameterized by the equations
\[
x = et + f, \quad y = gt + h,
\]
where \(0 \le t \le 1\), and \(t = 0\) corresponds to the point \((1, -3)\). Find \(e^2 + f^2 + g^2 + h^2\).


In rectangle $ABCD$, $AB = 10$ cm, $BC = 15$ cm, and $DE = DF$. The area of triangle $DEF$ is one-third the area of rectangle $ABCD$. Find the length of segment $EF$ in simplest radical form.

Solution: \boxed{10\sqrt{2} \text{ cm}}


In a row of 10 chairs, Sarah and Tom choose their seats at random. What is the probability that they do not sit next to each other?


When a certain biased coin is flipped seven times, the probability of getting heads exactly twice is equal to that of getting heads exactly three times. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $4$ out of $7$ flips. Find $i+j$.


A triangular region is bounded by the two coordinate axes and the line given by the equation $3x + y = 9$. Check if the point (1,1) lies inside this triangular region and find the region's area in square units.


Consider the function $g(x) = \arcsin x + \arccos x + e^{\arctan x}.$ Find the range of the function $g(x)$ for $x \in [-1, 1]$.

Solution: \boxed{\left[\frac{\pi}{2} + e^{-\frac{\pi}{4}}, \frac{\pi}{2} + e^{\frac{\pi}{4}}\right]}


The value of $y$ varies inversely as $x^2$. If $y=8$ when $x=1$, find the value of $x$ when $y=0.5$.


Let \( c \) be a complex number. Suppose there exist distinct complex numbers \( r \), \( s \), and \( t \) such that for every complex number \( z \), we have
\[
  (z - r)(z - s)(z - t) = (z - 2cr)(z - 2cs)(z - 2ct).
\]
Compute the number of distinct possible values of \( c \).


Identify the third and fourth smallest primes greater than 20 and find their product.


If the operation \(X\) is defined as \(a X b = b + 15a - a^2 - 5b\), what is the value of \(3X7\)?


What is the ratio of the area of the shaded triangle to the area of the square? The square is divided into a 5x5 grid of smaller, equal-sized squares. A triangle is shaded such that it covers half of a square at the center of the grid and three full squares adjacent to this half-covered square. The vertices of the triangle touch the midpoints of the sides of the squares it covers.


Solve for $x$ if $16^x \cdot 16^x \cdot 16^x \cdot 16^x = 256^5$.


Emily is at a cafeteria where she can pick one type of protein, three different sides, and one dessert. If the order of food items is not important, how many different meals can she choose?

$\bullet$ Protein: turkey, fish, tofu, lamb

$\bullet$ Sides: fries, salad, rice, pasta, broccoli

$\bullet$ Dessert: pie, gelato, fruit salad, cheesecake, cookies


In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 30 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. Calculate the area of quadrilateral $EFCD$.

Solution: \boxed{187.5 \text{ square units.}}


Let $n=5!$, and let $p$ be the product of all positive integers less than $n$ which are invertible modulo $n$. Determine the remainder when $p$ is divided by $n$.


If \(x\), \(y\), and \(z\) are positive real numbers with \(xy = 30\), \(xz = 60\), and \(yz = 90\), what is the value of \(x+y+z\)?


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


When \(\frac{5}{3125}\) is written as a decimal, how many zeros are there between the decimal point and the first non-zero digit?


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 and less than 30?


There are 10 sprinters in the Olympic 100-meter finals, with four of them being Canadians. The gold medal is awarded for first place, silver for second, and bronze for third. Calculate the number of ways the medals can be awarded if at most one Canadian sprinter gets a medal.


Calculate how many zeros follow the digit '1' in the decimal expansion of $(100!)^{100}$.


In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XZ = \sqrt{41}$. What is $\tan X$?


Suppose that all four of the numbers \[5 - 2\sqrt{2}, \; -5 - 2\sqrt{2}, \; 2+\sqrt{8}, \; 2-\sqrt{8}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


Two cylindrical cans have the same volume. The height of one can is four times the height of the other. If the radius of the taller can is 16 units, determine the radius of the shorter can in simplest radical form.


Let \( a \), \( b \), \( c \), \( p \), \( q \), and \( r \) be real numbers that satisfy the three equations
\begin{align*}
  17p +  bq +  cr &= 0 \\
   ap + 29q +  cr &= 0 \\
   ap +  bq + 56r &= 0.
\end{align*}
Suppose that \( a \ne 17 \) and \( p \ne 0 \). What is the value of
\[ \frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 56} \, ?\]


Completely factor the following expression: $16x^4 - 4x^2$.

Solution: \boxed{4x^2(2x + 1)(2x - 1)}


Determine how many numbers exist in the following sequence: $$-6, -2, 2, 6, \ldots, 50$$


Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the equations
\[
17x +  by + cz = 0, \quad ax + 29y + cz = 0, \quad ax + by + 50z = 0.
\]
If $ a \ne 17$ and $ x \ne 0$, what is the value of
\[
\frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 50} \,?
\]


Find the remainder when the polynomial $x^{1012}$ is divided by the polynomial $(x^2 + 1)(x - 1).$


Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + 2y + 3z = 1.$ Find the maximum value of $x^2 + y^2 + z^3.$


Alice takes 15 minutes to clean the kitchen. It takes Bob \(\frac{3}{4}\) of the time that Alice takes to clean the living room. How many minutes does it take Bob to clean the living room?


In right $\Delta XYZ$, $\angle XYZ$ is a right angle. Point $N$ is the midpoint of $\overline{XZ}$. The length of $\overline{XY}$ is 5 cm, and the length of $\overline{YZ}$ is 12 cm. What is the number of centimeters in the length of median $\overline{YN}$? Express your answer as a decimal to the nearest tenth.


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


How many positive perfect squares less than \(10^8\) are multiples of 36?


How many different four-digit integers can be formed using the digits 2, 2, 2, 9, and 0?


Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 3 \\ 1 \\ -4 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{9}{26} & \frac{3}{26} & -\frac{12}{26} \\ \frac{3}{26} & \frac{1}{26} & -\frac{4}{26} \\ -\frac{12}{26} & -\frac{4}{26} & \frac{16}{26} \end{pmatrix}}


Find all $x$ which satisfy $$\frac{2x}{5} \leq 3 + x < 4 - \frac{x}{3}.$$

Solution: \boxed{[-5, \frac{3}{4})}


How many four-digit numbers greater than 4999 can be formed such that the product of the middle two digits is an odd number?


If $\tan x = 3,$ then find $\tan \left( x + \frac{\pi}{3} \right).$

Solution: \boxed{-\frac{6 + 5\sqrt{3}}{13}}


Let $O$ be the origin. Determine a scalar $k$ so that whenever points $A$, $B$, $C$, $D$, and $E$ satisfy the equation:
\[2 \overrightarrow{OA} - 3 \overrightarrow{OB} + 4 \overrightarrow{OC} + k \overrightarrow{OD} + 2 \overrightarrow{OE} = \mathbf{0},\]
the points $A, B, C, D, E$ are guaranteed to be coplanar.


In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 20, respectively. The height of the trapezoid from $PQ$ to $RS$ is 6 units. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$?


An $\textit{annulus}$ is the region between two concentric circles. The concentric circles have radii $b$ and $c$, with $b>c$. Let $\overline{OX}$ be a radius of the smaller circle, let $\overline{XZ}$ be tangent to the larger circle at $Z$, and let $\overline{OY}$ be the radius of the smaller circle that contains $Z$. Let $a=XZ$, $d=YZ$, and $e=XY$. What is the area of the annulus? Express your answer in terms of $\pi$ and at most one of the variables $a,b,c,d,e$.

Solution: \boxed{\pi a^2}


Let $\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter $BC$ meets side $AC$ at $D$. If $AD = 3$ and $BD = 6$, what is $CD$?


When the binary number $110111001011_2$ is divided by 8, what is the remainder (give your answer in base 10)?


You roll a fair 8-sided die (sides numbered 1 to 8). If the result of the roll is even, you win double the amount in dollars of the result (e.g., if you roll 4, you win $8). If the roll is odd, you win nothing. What is the expected value of your winnings? Express your answer as a dollar value.


The number $131$ is identified as the second-smallest three-digit palindromic prime. What is the third-smallest three-digit palindromic prime?


What is the greatest integer less than 100 for which the greatest common factor of that integer and 12 is 2?


A point \((x, y)\) is randomly picked from inside the rectangle with vertices  \((0,0)\), \((5,0)\), \((5,2)\), and \((0,2)\). What is the probability that \(x < 2y\)?


In a mathematics competition, there are 30 questions in total. Each correct answer awards 8 points, an incorrect answer results in a deduction of 2 points, and an unanswered question scores 2 points. John decides to answer the first 25 questions and leave the last 5 unanswered. How many questions must John answer correctly to score at least 160 points?


Solution: \boxed{2 \sqrt{2} + 5^{2/3}}


Calculate the sum of the digits of the product of the base eight numbers $35_8$ and $42_8$, and express the result in base ten.


Circle \(D\) has a radius of 8 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle \(D\)?

Solution: \boxed{64 \text{ square centimeters}}


For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $2^{30}$?


Two concentric circles are drawn with a smaller circle inside a larger one. The point \( O \) is the common center. A chord \( AB \) of the larger circle, which is 120 units in length, is tangent to the smaller circle. Find the area of the shaded region between the two circles if the radius of the smaller circle is 40 units. Express your answer in terms of \( \pi \).


In the circle with center $O$, the measure of $\angle SIP$ is $45^\circ$ and $OS=15$ cm. Find the number of centimeters in the length of arc $SP$. Express your answer in terms of $\pi$.


A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 4$?


Find the largest integer $n$ for which $12^n$ evenly divides $30!$.


In a local basketball league, teams have 8 members and each of the 5 teams takes turns hosting events. At each event, each team selects three members of that team to be on the event committee, except the host team, which selects four members. How many possible 13 member event committees are there?


Find the average value of $k$, $2kz$, $4kz$, $8kz$, and $16kz$.

Solution: \boxed{\frac{k(1 + 30z)}{5}}


Sophia constructs a conical lampshade for a design project. The lampshade has a height of 15 inches and a circular base with a diameter of 8 inches. Sophia needs to find the smallest cylindrical container to pack her lampshade in such that the lampshade fits completely inside when placed upright. What is the volume of this cylindrical container, in cubic inches?


Find the product of all positive integral values of $n$ such that $n^2 - 41n + 420 = p$ for some prime number $p$. Note that there is at least one such $n$.


Jason is trying to remember a five-digit combination to his safe. He knows that he only used digits 1 through 6 (possibly repeated), where every even digit was followed by an odd digit, and every odd digit was followed by an even digit. How many possible combinations does Jason need to try?


I have 7 books, two pairs of which are identical. The first pair consists of two identical math books, and the second pair consists of two identical science books. The remaining three books are all different. In how many ways can I arrange these books on a shelf?


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


It is now 3:00:00 PM, as displayed on a 12-hour digital clock. In 315 hours, 78 minutes, and 30 seconds the time will be $A:B:C$. What is the value of $A + B + C$?


Find \(b\) if the remainder is constant when \(12x^3 - 9x^2 + bx + 8\) is divided by \(3x^2 - 4x + 2\).


Find the equation of the directrix of the parabola $x = -\frac{1}{4} y^2 + 2.$

Solution: \boxed{x = 3}


What is the greatest divisor of 450 that is smaller than 60 and also a factor of 90?


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


At Linda's Local Produce, 4 bananas cost the same as 3 pears, and 8 pears cost the same as 5 mangoes. How many mangoes cost the same as 20 bananas?


Given five members of a group -- Alice, Bob, Carol, David, and Elsa -- in how many ways can three of these five be chosen and assigned to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job?


The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ has the property that
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_6 = 60,$ then find $b_7.$


Determine the greatest common divisor of $45045$ and $30030$.


In triangle $ABC,$ the side lengths $a,$ $b,$ and $c$ satisfy $a + 3b + 3c = a^2$ and $a + 3b - 3c = -4.$  Determine the largest angle of the triangle, in degrees.

Note: $a$ is the side length opposite $\angle A,$ etc.


A circle is tangent to the lines $3x - 4y = 20$ and $3x - 4y = -40$. The center of the circle lies on the line $x - 3y = 0$. Find the center of the circle.

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


Let $\mathbf{B} = \begin{pmatrix} x & 2 \\ -3 & y \end{pmatrix}$ for some real numbers $x$ and $y.$ If $\mathbf{B} + 2\mathbf{B}^{-1} = \mathbf{0}$, find $\det \mathbf{B}$.


Bag A now contains 4 white marbles, 5 black marbles, and 2 red marbles. Bag B has 7 yellow marbles and 5 blue marbles. Bag C has been changed to contain 3 yellow marbles and 7 blue marbles. Introduce Bag D which has 8 yellow marbles and 2 blue marbles. A marble is drawn at random from Bag A. If it is white, a marble is drawn from Bag B; if it is black, a marble is drawn from Bag C; and if it is red, a marble is drawn from Bag D. What is the probability that the second marble drawn is yellow?


A region is bounded by semicircular arcs constructed on the sides of a square, with each side measuring $4/\pi$. What is the perimeter of this region?


A triangular prism has 5 faces, 9 edges, and 6 vertices. One pyramid is attached to one of its triangular faces, and another pyramid is attached to one of its quadrilateral faces. After adding these pyramids, calculate the number of faces, vertices, and edges of the resulting solid. Add these three numbers to find out what their sum is.


What is the smallest integer $b > 3$ for which the base $b$ number $23_b$ is a perfect cube?


At the end of a chess tournament, the top 6 players have a playoff. The seedings are such that #6 plays #5 first. The loser gets the 6th prize. The winner then plays #4, the loser of this game gets the 5th prize. The winner plays #3, the loser gets the 4th prize. This is followed by the winner playing #2, where the loser gets the 3rd prize. Finally, the winner confronts #1 and the loser receives the 2nd prize while the winner receives the 1st prize. Calculate the number of different ways the 6 prizes can be distributed among the players.


Points $P$, $Q$, $R$, and $S$ are in space such that each of $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ is perpendicular to the other two. If $SP = 12$, $SQ = 12$, and $SR = 8$, what is the volume of pyramid $SPQR$?


A choir has 90 members who need to be arranged in rows for a performance, with the same number of members per row. If there must be between 5 and 15 members per row, in how many possible row-lengths can the choir be arranged?


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


Twelve 6-sided dice are rolled. What is the probability that exactly four of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.


Suppose $g(x)$ is a function defined for all real $x$, and suppose $g$ is invertible. Consider the graphs of $y=g(x^3)$ and $y=g(x^6)$. How many points do they intersect?


Points $P$, $Q$, $R$, and $S$ are located in space such that lines $\overline{SP}$, $\overline{SQ}$, and $\overline{SR}$ are each perpendicular to the others. Given that $SP = SQ = 12$ and $SR = 8$, what is the volume of pyramid $SPQR$?


The number $0.83125$ can be expressed as a fraction $\frac{p}{q}$ for positive integers $p$ and $q$. When this fraction is in simplest terms, what is $p+q$?


Square $EFGH$ has its center at $(-3,4)$ and has an area of 16 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at (0,0) and a scale factor of 3. What are the coordinates of the vertex of the image of square $EFGH$ that is farthest from the origin? Give your answer as an ordered pair.

Solution: \boxed{(-15, 18)}


Given that \(9^{-1} \equiv 90 \pmod{101}\), find \(81^{-1} \pmod{101}\), as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)


Solution: \boxed{3x^{28} - 9x^{11} + 3x^8 - 21x^5 + 3x^2}


In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel with lengths 10 and 25 units, respectively, and the altitude of the trapezoid is 15 units. Points $E$ and $F$ are midpoints of sides $AD$ and $BC$, respectively. Diagonals $AC$ and $BD$ intersect at point $G$. Calculate the area of quadrilateral $EFCD$.


A sign painter paints individual numerals for a row of 70 houses. The houses are numbered consecutively from 1 to 70. How many times does the digit "3" appear in these house numbers?


Consider a right triangle with sides $5$, $12$, and $13$. The vertices of this triangle are the centers of three mutually externally tangent circles. Determine the sum of the areas of these circles.


A horse 30 feet from the center of a merry-go-round makes 40 revolutions. Another horse rides on a track that is 15 feet from the center. If the first horse travels at a speed of 3 feet per second and the second horse travels at a speed of 6 feet per second, how long will it take for the second horse to travel the same distance as the first?


Compute \[ \left\lfloor \dfrac {2007^3}{2005 \cdot 2006} - \dfrac {2005^3}{2006 \cdot 2007} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


Let \(S_n\) be the sum of the first \(n\) terms of an arithmetic sequence with a common difference of 5. If \(\frac{S_{3n}}{S_n}\) is a constant that does not depend on \(n\) for all positive integers \(n\), determine the first term of the sequence.


Let $a = \frac{4}{7}$ and let $b = \frac{5}{6}$. Compute $\left( a^3 b^{-2} \right)^2$.


What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 3:20? If it's not the smaller angle, calculate the larger one. Express your answer as a decimal to the nearest tenth.


If $g(x) = \sqrt[3]{\frac{x + 4}{5}}$, find the value of $x$ for which $g(3x) = 3g(x)$.


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


What is the smallest positive integer $n$ such that $890n \equiv 1426n \pmod{30}?$


What is the smallest integer \(b > 4\) for which the base \(b\) number \(34_b\) is a perfect square?


What is the area, in square units, of a trapezoid bounded by the lines $y = 3x$, $y = 10$, $y = 5$ and the $y$-axis? Express your answer as a decimal to the nearest tenth.


A gasoline tank is $\frac{4}{5}$ full. After $18$ gallons have been used, it is $\frac{1}{3}$ full. How many gallons does this tank hold when it is full?


A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 0$ is $\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix},$ and the vector on the line at $t = 4$ is $\begin{pmatrix} 6 \\ 7 \\ -1 \end{pmatrix}.$ Find the vector on the line at $t = 6.$

Solution: \boxed{\begin{pmatrix} 8 \\ 11 \\ -3 \end{pmatrix}}


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 $DE$.


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


What is the smallest prime whose digits sum to $23$?


Determine how many five-letter words can be formed where the first and last letter are identical, and the middle letter is a vowel (A, E, I, O, U).


Given that $b$ is an even multiple of $1187$, find the greatest common divisor of $2b^2 + 31b + 67$ and $b + 15$.


**
Let \( P \) be a point on the parabola \( y = x^2 - 6x + 15 \), and let \( Q \) be a point on the line \( y = 2x - 7 \). Find the shortest possible distance \( PQ \).

**

Solution: \boxed{\frac{6 \sqrt{5}}{5}}


Evaluate $(a^a)^b - (b^b)^a$ for $a = 3$ and $b = 4$.


Angela has $a$ marbles, Brian has three times as many marbles as Angela, Caden has twice as many marbles as Brian, and Daryl has five times the number of marbles Caden has. If in total Angela, Brian, Caden, and Daryl have 120 marbles, what is the value of $a?$


For how many $n=2,3,4,\ldots,99,100$ is the base-$n$ number $24314_n$ a multiple of both $5$ and $7$?


A sphere is inscribed in a cube with edge length 16 feet. Then a smaller cube is inscribed in the sphere. Determine the volume of the inscribed cube and the surface area of the sphere. Express your answer in simplest radical form.

Solution: \boxed{256\pi \text{ square feet}}


The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square, a perfect cube, nor a perfect fifth power? Express your answer as a common fraction.


A pirate is now searching for buried treasure on 7 islands. On each island, there is a $\frac{1}{3}$ chance that the island has buried treasure and no traps, a $\frac{1}{6}$ chance that the island has traps but no treasure, and a $\frac{1}{2}$ chance that the island has neither traps nor treasure. What is the probability that, while searching all 7 islands, the pirate will encounter exactly 4 islands with treasure, and none with traps?


Three dice, each with faces numbered 1 through 8, are tossed and the numbers shown on the top faces are added. What is the probability that the sum is even? Express your answer as a common fraction.


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 $a_1 \le 1000$ is it true that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$?


The arithmetic progressions $\{4, 9, 14, \ldots\}$ and $\{3, 12, 21, \ldots \}$ have some common values. What is the largest value less than 1000 that they have in common?


Find the product of all positive divisors of 24.


Find the total number of ordered pairs of positive integers $(x,y)$ that satisfy the equation $xy=4410$ given that the prime factorization of 4410 is $2 \times 3^2 \times 5 \times 7^2$.


The sum of the squares of two positive integers is 145. The product of the two integers is 40. What is the sum of the two integers?


The increasing sequence consists of all those positive integers which are powers of 4 or sums of distinct powers of 4. Find the $50^{\text{th}}$ term of this sequence.


Solve for \( x \) in the equation \[ \frac{x^2 - x - 2}{x + 2} = x + 3 \].


A biased 8-sided die has the numbers 1 through 8, where the probabilities of rolling numbers 1 through 4 are each $\frac{1}{10}$, while the probabilities for numbers 5 through 8 are each $\frac{3}{20}$. Calculate the expected value of a roll.


The sum of two numbers is 10. The difference of their squares is 24. What is the positive difference of the two numbers?


For how many non-negative real values of \( x \) is \( \sqrt{100-\sqrt[3]{x}} \) an integer?


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


Place trapezoid $ABCD$ in a coordinate plane with vertices $A(0,0)$, $B(3,4)$, $C(15,4)$, and $D(18,0)$. The sides $AB$ and $CD$ are non-parallel. Determine the perimeter of $ABCD$.


If 48 apples are distributed evenly among 7 children, how many children will end up with fewer than 7 apples?


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


Suppose functions $h$ and $s$ have the properties that $h(x) = 4s^{-1}(x)$ and $s(x) = \frac{40}{x+5}$. For what value of $x$ does $h(x) = 20$?


If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which  $343x^3+64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$?


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


Let $PQRS$ be a convex pentagon, and let $H_P,$ $H_Q,$ $H_R,$ $H_S,$ and $H_T$ denote the centroids of the triangles $QRS,$ $RSP,$ $SPQ,$ $PQR,$ and $QRP$, respectively. Find $\frac{[H_PH_QH_RH_S]}{[PQRS]}$.


A play includes three different male roles, five women audition, and seven men audition. Additionally, there are two roles open to either gender. Determine how many ways there are to cast these five roles if roles can only be filled as per gender specifications.


What is the smallest possible area, in square units, of a right triangle if two of its sides measure $6$ units and $8$ units?


An element is selected at random from among the first $20$ rows of Pascal’s Triangle. What is the probability that the element chosen is an edge number in the triangle (either the first or last number in its row)?


In a circle, $AB$ is a diameter, and $CD$ is a chord parallel to $AB$. Point $E$ is the intersection of lines $AC$ and $BD$, and $\angle AED = \beta$, but now $\beta$ is such that $\angle AED + \angle BED = 180^\circ - 2\beta$. The ratio of the area of triangle $\triangle CDE$ to that of triangle $\triangle ABE$ is required.

Solution: \boxed{\cos^2 \beta}


Solve for $y$ in the equation $\frac{1}{6} + \frac{6}{y} = \frac{14}{y} - \frac{1}{14}$.


What is the result when you simplify the expression \( z - 3 + 4z + 5 - 6z + 7 - 8z + 9 \)?

Solution: \boxed{-9z + 18}


Complex numbers $a$, $b$, $c$ form an equilateral triangle with side length 24 in the complex plane. If $|a + b + c| = 48$, find $|ab + ac + bc|$.


Let $y_1, y_2, \ldots, y_n$ be real numbers which satisfy $|y_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|y_1| + |y_2| + \dots + |y_n| = 15 + |y_1 + y_2 + \dots + y_n|.\] What is the smallest possible value of $n$?


Two puppies and two cats together weigh 36 pounds. The first puppy and the larger cat together weigh three times as much as the smaller cat, and the first puppy and the smaller cat together weigh the same as the larger cat. How much does the second puppy weigh if it is known that the second puppy weighs 1.5 times as much as the first puppy?


A notebook contains sequentially numbered pages from 1 to $n$. The sum of the page numbers ended up being $2490$ after one of the pages was accidentally counted twice. Find the page number that was counted twice.


A rectangle has a perimeter of 100 inches and each side has an integer length. Additionally, the length and width must be distinct. How many non-congruent rectangles meet these criteria?


How many three-digit perfect cubes are divisible by 25?


How many numbers are in the list $ -48, -41, -34, \ldots, 65, 72?$


A bag contains three red, three blue, and three green marbles. Four marbles are randomly selected, without replacement. What is the probability that one marble of each color is selected, with an extra marble being red?


150 people were surveyed and asked the question: "Is teal more green or blue?" Of them, 90 believe that teal is "more green," and 40 believe it is both "more green" and "more blue." Another 20 think that teal is neither "more green" nor "more blue." Considering a group of 10 people who were undecided, how many of those 150 people believe that teal is "more blue"?


Solution: \boxed{4096 \sqrt{2}}


Factorize the expression $6y^2 - 31y + 35$ such that it can be written in the form $(Cy - 5)(Dy - 7)$, where $C$ and $D$ are integers. Compute $CD + C$.


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


Zara has 20 yellow balloons, 24 blue balloons, 50 pink balloons, and 102 violet balloons. With no regard to color, Zara gives each of her 9 closest friends an equal number of balloons. She distributes as many balloons as she can evenly amongst them. How many balloons does Zara have left after distributing them?


The graph of $y = \frac{p(x)}{q(x)}$ is a rational function, where $p(x)$ is quadratic and $q(x)$ is quadratic as well. Vertical asymptotes are at $x = -3$ and $x = 2$, and the graph passes through the points $(0,0)$ and $(1,2)$. Find $\frac{p(1)}{q(1)}$.


Express 2023 as a sum of distinct powers of 2 and calculate the least possible sum of the exponents of these powers used in this representation.


For a given positive integer $n > 3^2$, what is the greatest common divisor of $n^3 + 27$ and $n + 3$?


What is the sum of the squares of the coefficients of $3(x^5 + 5x^3 + 2x + 1)$?


In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?


Let $a$ be the number of positive multiples of $4$ that are less than $40$. Let $b$ be the number of positive integers that are less than $40$, and a multiple of $2$ and a multiple of $4$. Compute $(a + b)^2$.


How many positive integers less than 1200 are congruent to 3 (mod 7)?


The values of \(f\), \(g\), \(h\), and \(j\) are 7, 8, 9, and 10, but not necessarily in that order. What is the largest possible value of the sum of the four products \(fg\), \(gh\), \(hj\), and \(fj\)?


Let \(x\) and \(y\) be positive integers such that \(x - y = 8\) and \(\text{gcd}\left(\frac{x^3+y^3}{x+y}, xy\right) = 16\). Find the smallest possible value of \(y\).


The lengths of the sides of a triangle with positive area are $\log_{10}20$, $\log_{10}45$, and $\log_{10}m$, where $m$ is a positive integer. Find the number of possible values for $m$.


In a diagram, lines \( CP \) and \( CQ \) trisect \( \angle ACB \). Line \( CM \) bisects \( \angle PCQ \). Determine the ratio of the measure of \( \angle MCQ \) to the measure of \( \angle ACQ \).


Jessica now has a 3 by 3 grid and two colors to use. If non-overlapping regions that share a side cannot have the same color, how many possible ways are there for Jessica to color the grid?


A bag contains 6 red marbles, 8 white marbles, and 9 blue marbles. Four marbles are drawn from the bag (without replacement). What is the probability that the first three marbles drawn are all the same color?


Convert \(2\sqrt{3} e^{13\pi i/6}\) to rectangular form.

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


Evaluate the product \[ (a-12) \cdot (a-11) \cdot \dotsm \cdot (a-1) \cdot a \cdot 3, \] where $a=3$.


What is the average of all the integer values of $M$ such that $\frac{M}{42}$ is strictly between $\frac{5}{14}$ and $\frac{1}{6}$?


Calculate the units digit of the sum of the ten terms in the extended sequence \( 1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 9! + 9, \, 10! + 10 \).


Find the radius of the circle with equation $4x^2 - 8x + 4y^2 - 16y + 20 = 0.$


What is the smallest positive value of $x$ such that $x + 2345$ results in a palindrome?


If $x$ is a real number such that $x + \frac{1}{x} = 5$, find the value of $S_6 = x^6 + \frac{1}{x^6}$.


A hyperbola is defined by the equation \[\frac{x^2}{144} - \frac{y^2}{81} = 1.\] Determine the equations of its asymptotes and find the coordinates of its foci.

Solution: \boxed{(\pm15, 0)}


Solve the quadratic equation $x^2 + 16x = 96$. If one of the solutions is of the form $\sqrt{a} - b$ for positive integers $a$ and $b$, find the value of $a+b$.


The product of two positive integers plus their sum is 71. The integers are relatively prime, and each is less than 15. What is the sum of the two integers?


How many distinct sequences of five letters can be made from the letters in "PROBLEMS" if each letter can be used only once and each sequence must begin with "S" and not end with "M"?


Suppose that $\{b_n\}$ is an arithmetic sequence with
$$
b_1 + b_2 + \cdots + b_{50} = 150 \quad \text{and} \quad
b_{51} + b_{52} + \cdots + b_{100} = 250.
$$
What is the value of $b_2 - b_1$?


In triangle $ABC$, point $D$ is placed on segment $AB$ such that $AD = 15$ and $DB = 10$. Point $E$ bisects angle $ACB$. Find the ratio of the area of $\triangle BCE$ to the area of $\triangle ACE$. Given the lengths of the segments are $AC = 50$ and $BC = 45$.


Suppose $x$, $y$, and $z$ are positive numbers satisfying:
\[
x^2/y = 2, \\
y^2/z = 5, \text{ and} \\
z^2/x = 7.
\]
Find the value of $x$.


Marie has 75 raspberry lollipops, 132 mint lollipops, 9 blueberry lollipops, and 315 coconut lollipops. She decides to distribute these lollipops equally among her 13 friends, distributing as many as possible. How many lollipops does Marie end up keeping for herself?


Find the smallest integer satisfying the following conditions:

$\bullet$ I. The sum of the squares of its digits is $85$.

$\bullet$ II. Each digit is larger than the one on its left.

What is the product of the digits of this integer?


What is the least integer greater than $\sqrt{500}$?


If $x$ is a real number and $\lceil x \rceil = 12,$ how many possible values are there for $\lceil x^2 \rceil$?


A right triangle has legs of length 12 and 9. It is similar to another right triangle with one leg of length 6. What is the length of the other leg in the second triangle, called $y$?


Find $y$ such that $\lfloor y \rfloor + y = \dfrac{17}{4}$. Express $y$ as a common fraction.


Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{16} = 8.\]


I have two 10-sided dice where each die has 3 gold sides, 4 silver sides, 2 diamond sides, and 1 rainbow side. If I roll both dice, what is the probability that they come up showing the same color or pattern?


A geometric sequence of positive integers starts with a first term of 5. The third term in this sequence is 120. What is the fifth term of the sequence?


Lead used to make twelve solid lead balls, each with a radius of 2 cm, is reused to form a single larger solid lead sphere. What is the radius of this larger sphere?


How many odd three-digit integers have the property that their digits, read left to right, are in strictly decreasing order?


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


Let $\star (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{T}$ is the set of positive integers such that for all elements $n$ in $\mathcal{T}$, we have that $\star (n)=10$ and $0\le n< 10^6$. If $p$ is the number of elements in $\mathcal{T}$, compute $\star(p)$.


Let $ABCD$ be a tetrahedron with edges $AB$, $AC$, and $AD$ mutually perpendicular at $A$. Let the lengths of $AB=a$, $AC=b$, and $AD=c$. Given that the areas of triangles $ABC$, $ACD$, and $ADB$ are $s$, $t$, and $u$ respectively, find the area of triangle $BCD$.

Solution: \boxed{\sqrt{s^2 + t^2 + u^2}}


Solution: \boxed{2^x = 39}


In how many ways can 6 people be seated around a round table if two people, say A and B, must always sit next to each other?


What is the sum of the $x$-values that satisfy the equation $6 = \frac{x^3 - 3x^2 - 10x}{x+2}$?


Point $P$ is the center of an ellipse with major axis $\overline{XY}$ and minor axis $\overline{WZ}.$ Point $G$ is one focus of the ellipse. If $PG = 8$ and the diameter of the inscribed circle of triangle $PWG$ is 4, compute the product $(XY)(WZ).$


Let $a, b, c, d$ be positive real numbers such that $a + b + c + d = 10$. Find the minimum value of
\[
\frac{1}{a} + \frac{9}{b} + \frac{25}{c} + \frac{49}{d}.
\]


In a certain base $b$, the square of $12_b$ is $144_b$. What is $b$?


A wizard is creating a magic ink. He has four different types of magical herbs and six different essences to choose from. However, one particular essence is incompatible with three of the herbs. How many valid combinations can the wizard use to make the magic ink?


As $m$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $13m+4$ and $7m+2$?


Ten teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$


Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x - 6 = 0$.

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


For a certain weekend, the weatherman predicts that it will rain with a $60\%$ probability on Saturday and a $40\%$ probability on Sunday. If it rains on Saturday, the probability that it will also rain on Sunday increases to $80\%$. Calculate the probability that it rains on at least one day over the weekend. Express your answer as a percentage.


Let $f(x)$ be a function piecewise defined as \[f(x) = \left\{
\begin{array}{cl}
-2x & x\le 0, \\
3x-50 & x>0.
\end{array}
\right.\] If $b$ is negative, find $b$ so that $f(f(15))=f(f(b))$.

Solution: \boxed{b = -10}


I read at a speed of one-fifth of my friend's reading speed. If it takes me 3 hours to read two volumes of a series, how many hours will it take my friend to read just one volume of the same series?

Solution: \boxed{0.3 \text{ hours}}


If $\sin \theta = \frac{1}{4},$ then find $\sin 2 \theta.$


Positive integers \( d, e, \) and \( f \) are chosen such that \( d < e < f \), and the system of equations
\[ 2x + y = 2010 \quad \text{and} \quad y = |x-d| + |x-e| + |x-f| \]
has exactly one solution. What is the minimum value of \( f \)?


A parallelogram has vertices with coordinates \((1, 2)\), \((4, 6)\), \((10, 6)\), and \((7, 2)\). Find the sum of the perimeter and the area of the parallelogram.


Find the sum of the series \( 1_7 + 2_7 + 3_7 + \cdots + 33_7 \). Express your answer in base \( 7 \).


Find the product of the divisors of \(72\).

Solution: \boxed{2^{18} \times 3^{12}}


Determine the sum of all possible values of $b$ for which the cubic equation $4x^3 + 3x^2 + bx + 27 = 0$ has exactly one real solution.


How many license plates consist of 3 letters followed by 2 digits, if one of the digits must be odd and the other must be even?


\( ABCD \) is a regular tetrahedron. If \( N \) is the midpoint of \( \overline{BC} \), what is \( \cos \angle AND \)?


Let $a$ and $b$ be the roots of the equation $x^2 - 6x + 8 = 0$. Compute:
\[a^2 + a^5 b^3 + a^3 b^5 + b^2.\]


What common fraction, reduced to its lowest terms, is equivalent to $0.4\overline{37}$?


If the six digits 1, 2, 4, 5, 7, and 5 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by 10? Express your answer as a common fraction.


A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation?


On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 10 kilometers per hour of speed or fraction thereof (e.g., a car traveling 45 kilometers per hour will be five car lengths behind the back of the car in front of it). A sensor near the road counts the number of cars that pass in one hour. Assuming each car is 5 meters long and that the cars can travel at any speed, let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Find the quotient when $N$ is divided by 10.


If the least common multiple of $A$ and $B$ is $240$, and the ratio of $A$ to $B$ is $5:6$, then what is their greatest common divisor?


A convex pentagon has interior angles with measures \(2x-8\), \(3x+12\), \(4x+8\), \(5x-18\), and \(x+6\) degrees. Calculate the measure of the largest angle.


Determine the product of the real parts of the complex solutions for the equation $x^2 + 2x = -i$.


A circular paper with a radius of 6 inches has a section removed to form a \(240^\circ\) sector. This sector is then used to form a right circular cone by bringing the two radii together. Find the circumference of the base of the cone in terms of \(\pi\).


I have 5 shirts, 5 pairs of pants, and 5 hats. Each item comes in the same 5 colors (so that I have one of each item of each color). I refuse to wear an outfit in which two or more items are the same color. How many choices for outfits do I have?


Points $A$, $B$, $C$, $D$, and $E$ are located on $\overline{AB}$ such that $AB = 4AD = 8BC$ and $BE = 2CE$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $E$?


Let \[f(x) = \left\{
\begin{array}{cl}
-x + 4 & \text{if } x \le 1, \\
3x - 7 & \text{if } x > 1.
\end{array}
\right.\] How many solutions does the equation $f(f(x)) = 6$ have?


How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression, and the common difference $d$ is a multiple of $5$?


At Bella's Blossom Shop, a bouquet containing 18 tulips costs $\$36$. If the price of a bouquet is directly proportional to the number of tulips it contains until 40 tulips, after which a 10% discount on the total price is applied, how much will a bouquet of 50 tulips cost?


Simplify $\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{5}}{\sqrt{9}} \cdot \frac{\sqrt{6}}{\sqrt{8}}$ and rationalize the denominator of the resulting fraction.


Suppose \( f(x) \) is a function defined for all real \( x \), and suppose \( f \) is invertible, meaning \( f^{-1}(x) \) exists for all \( x \) in the range of \( f \).

If the graphs of \( y = f(x^2) \) and \( y = f(x^3) \) are drawn, at how many points do they intersect?


Let $A = (2,0)$ and $B = (8,6).$ Let $P$ be a point on the parabola $y^2 = 8x.$ Find the smallest possible value of $AP + BP.$


A line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + t \begin{pmatrix} 4 \\ 5 \end{pmatrix}.
\]
A second line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -3 \\ 9 \end{pmatrix} + u \begin{pmatrix} 2 \\ -1 \end{pmatrix}.
\]
If \(\theta\) is the acute angle formed by the two lines, then find \(\cos \theta.\)


A maintenance team needs to reach the center \( P \) of a circular platform. They place a 20-foot plank from point \( X \) to point \( Y \) and then a 12-foot plank from \( Z \) to \( P \), where \( Z \) is the midpoint of \( \overline{XY} \). Determine the area of the circular platform. Express your answer in terms of \( \pi \).

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


I am playing an extended version of a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \le n \le 50$, I take two steps forward if $n$ is prime and three steps backwards if the number is composite. After all 50 moves, I stop and walk back to my original starting point. How many steps long is my walk back?


How many solutions does the equation $\tan x = \tan (\tan x)$ have in the interval $0 \le x \le \tan^{-1} 1000$? Assume $\tan \theta > \theta$ for $0 < \theta < \frac{\pi}{2}$.


How many positive five-digit integers are divisible by 9?


Let $a,$ $b,$ $c$ be three distinct positive real numbers such that $a,$ $b,$ $c$ form a geometric sequence where $b^2 = ac$, and $\log_c a, \ \log_b c, \ \log_a b$ form an arithmetic sequence. Find the common difference of the arithmetic sequence.


Let $x_1, x_2, x_3$ be the roots of the equation $\sqrt{2023}x^3 - 4047x^2 + 2 = 0$ with $x_1 < x_2 < x_3$. Find the value of $x_2(x_1 + x_3)$.


How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-50^2)} = 0\] have for \(x\)?


For some complex number $z$ with $|z| = 3,$ there is some real $\lambda > 1$ such that $z,$ $z^2,$ and $\lambda z$ form an equilateral triangle in the complex plane. Find $\lambda.$

Solution: \boxed{\frac{1 + \sqrt{33}}{2}}


Factor \( y^6 - 64 \) as far as possible, where the factors are monic polynomials with real coefficients.

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


Assume a deck of 27 cards where each card features one of three symbols (star, circle, or square), each symbol painted in one of three colors (red, yellow, or blue), and each color applied in one of three intensities (light, medium, or dark). Each symbol-color-intensity combination is unique across the cards. A set of three cards is defined as complementary if:
i. Each card has a different symbol or all have the same symbol.
ii. Each card has a different color or all have the same color.
iii. Each card has a different intensity or all have the same intensity.
Determine the number of different complementary three-card sets available.


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


A circle is inscribed in a rectangle with a length twice its width. Inside this circle, a square is inscribed. Finally, a circle is inscribed inside this square. What is the ratio of the area of the smaller innermost circle to the area of the original rectangle?


Compute the smallest base-10 positive integer greater than 7 that is a palindrome when written in both base 3 and 5.


Josh had recorded lap times, in seconds, of 102, 108, 96, 110, 104, and 98 for running 400 meters. After another attempt, he lowered his median lap time to 102 seconds. What was his time, in seconds, for his latest attempt?

Solution: \boxed{102 \text{ seconds}}


Determine the integer \( n \) such that \( 0 \leq n < 151 \) and \( 150n + 3 \equiv 45 \pmod{151} \).


Suppose we modify Larry's process using a new function \( g \) derived from another graph. This graph contains the following points: (0,1), (2,3), (4,5), (6,7), (8,9), (1,0), (3,2), (5,4), (7,6), (9,8). Larry starts by writing the number 4 on his pinky finger. He then applies \( g \) to 4 and writes the output on his ring finger. If Larry continues this process of applying \( g \) and writing the output on a new finger, what number will Larry write on his tenth finger?


What is the smallest integer $n$, greater than $1$, such that $n^{-1} \pmod{2310}$ is defined?


If $0 \leq p \leq 1$ and $0 \leq q \leq 1$, define $F(p, q)$ by
\[
F(p, q) = -4pq + 5p(1-q) + 2(1-p)q - 6(1-p)(1-q).
\]
Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ in the interval $0 \leq q \leq 1$. What is the value of $p$ (in the interval $0 \leq p \leq 1$) that minimizes $G(p)$?


Ana, Bella, and Chloe take turns rolling a standard six-sided die in their respective order. The first one to roll a '6' wins. What is the probability that Chloe will win? Express your answer as a common fraction.


Find the quotient and the remainder when $x^6 - 5x^4 + 3x^3 - 7x^2 + 2x - 8$ is divided by $x - 3$.


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.


What is the number of square units in the area of trapezoid EFGH with vertices E(0,0), F(0,3), G(5,0), and H(5,7)?


What is the minimum value of $3x^2 - 18x + 729$?