A flagpole is supported by a wire which extends from the top of the flagpole to a point on the ground 4 meters from its base. Rob, who is 1.8 meters tall, walks 3.5 meters from the base of the flagpole toward the point where the wire is attached to the ground, and his head just touches the wire at that point. How tall is the flagpole?

Solution: \boxed{14.4 \text{ meters}}


How many three-digit whole numbers have no 7's, no 8's, and no 9's as digits?


Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\frac{36}{x+4}$. For what value of $x$ does $g(x)=18$?


In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 9$ and $DF = 15$. Find $\cos F$ and $\cos D$.


Solution: \boxed{7 - i}


I have 7 books, three of which are identical copies of a math book and two of which are identical copies of a physics book. How many ways can I arrange them on a shelf?


What is the sum of all the integers from -15 to 5, including only even numbers?


A conference center is setting up chairs in rows for a seminar. Each row can seat $13$ chairs, and currently, there are $169$ chairs set up. They want as few empty seats as possible but need to maintain complete rows. If $95$ attendees are expected, how many chairs should be removed?


Solution: \boxed{5 - i}


The sum of the first 1000 terms of a geometric sequence is 300. The sum of the first 2000 terms is 570. Find the sum of the first 3000 terms.


James has a jar that contains 97 quarters and 173 dimes. Lindsay has a jar containing 141 quarters and 289 dimes. They decide to pool their quarters and dimes and roll them into complete rolls, with each roll of quarters containing 35 quarters and each roll of dimes containing 55 dimes. How much, in dollars, are the leftover quarters and dimes worth?


Find the units digit of the sum $35_8 + 47_8$.


What is the sum of the greatest common divisor of $45$ and $4095$ and the least common multiple of $45$ and $4095$?


Given that the point $(7,10)$ is on the graph of $y=g(x)$, there is one point that must be on the graph of $3y=2g(3x)-3$. What is the sum of coordinates of that point?


How many degrees are there in the measure of angle $Q$?

[Diagram description:
A hexagon with vertices labeled $R, S, T, U, V, W$. Both $V$ and $W$ are adjacent to $R$ where angle $Q$ is located.
Label $120^\circ$ at $S$,
Label $130^\circ$ at $T$,
Label $140^\circ$ at $U$ and
Label $100^\circ$ at $V$,
Label $85^\circ$ at $W$.]


What is the sum of the positive whole number divisors of $462$?


Let \(p\) and \(q\) be the solutions to the equation \(3x^2 - 5x - 8 = 0.\) Compute the value of \((9p^3 - 9q^3)(p - q)^{-1}.\)


Find the vertex of the parabola described by the equation $y = -2x^2 - 16x - 42$. Let the vertex be $(p, q)$. What is $q$?


A line passing through the points $(3, -12)$ and $(k, 22)$ is parallel to the line $4x + 6y = 36$. What is the value of $k$?


A fair 8-sided die is rolled. If I roll \( n \), then I win \( n^3 \) dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent.


Solve for $x$ in the equation $5x - 3 = 15x + 15$ and find the value of $6(x + 5)$.


Given an invertible function $g$ with domain and range covering all real numbers, and some values provided by the table:
$$\begin{array}{c || c | c | c | c | c}
x & -1 & 0 & 2 & 3 & 4 \\
\hline
g(x) & 0 & 1 & 3 & 4 & 6
\end{array}$$
Calculate $g(g(2)) + g(g^{-1}(3)) + g^{-1}(g^{-1}(4))$. If there is not enough information to answer this question, enter "NEI".


Five concentric circles are drawn with radii of 3, 5, 7, 9, and 11. Starting with the innermost circle painted black, the next ring is white, the following black, the next one white, and the outermost ring is black again. What is the ratio of the black area to the white area? Express your answer as a common fraction.


What is the smallest five-digit positive integer which is congruent to 7 (mod 13)?


If we simplify the expression $5(x^2 - 3x + 4) - 9(x^3 - 2x^2 + x - 1)$, what is the sum of the squares of the coefficients of the terms?


The graph of $y = f(x)$ is given as part of a function composition. If the graph of $y = h(x)$ is obtained from the graph of $y = f(x)$ by reflecting it across the y-axis and then shifting it 6 units to the right, express $h(x)$ in terms of $f(x)$.

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


Calculate the sum of all integers from -20 to 10.


A decorative globe must have a diameter of 10 inches. Calculate both the surface area and the volume of the globe before any modifications or decorations. Express your answers as common fractions in terms of \(\pi\).


The function $g(x)$ satisfies the functional equation
\[xg(y) = yg(x)\] for all real numbers $x$ and y. If $g(10) = 15$, find $g(5)$.


Consider the geometric sequence $$3, 9x, 27x^2, 81x^3, \ldots.$$ What is the fifth term in this sequence? Express your answer in terms of $x$.


A function $f$ has domain $[3,7]$ and range $[0,2]$. Consider the modified function
\[ g(x) = 2 - f(2x - 6) \]
Determine the domain and range of $g(x)$ and represent the result as an ordered quadruple \((a, b, c, d)\).

Solution: \boxed{\left(\frac{9}{2}, \frac{13}{2}, 0, 2\right)}


Suppose $\cos Q = 0.6$ in a right triangle $PQR$ where $PQ$ measures 15 units. What is the length of $QR$?

[asy]
pair P,Q,R;
P = (0,0);
Q = (7.5,0);
R = (0,7.5*tan(acos(0.6)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,20));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$15$",Q/2,S);
[/asy]


What is the modular inverse of $13$, modulo $101$?

Express your answer as an integer from $0$ to $100$, inclusive.


What is the smallest positive integer $n$ such that $0\le n<13$ and $$-1234\equiv n\pmod{13}~?$$


There are 30 people in a club. 12 of them are left-handed. 20 of them like rock music. 3 of them are right-handed and dislike rock music. How many club members are left-handed and like rock music? Assume that each person is either left-handed or right-handed.


Let $m$ be the smallest positive, three-digit integer congruent to 6 (mod 13). Let $n$ be the smallest positive, four-digit integer congruent to 6 (mod 13). What is $n - m$?


Let \[
p(x,y) =
\begin{cases}
x + y & \quad \text{if } x \geq 0 \text{ and } y \geq 0, \\
x - 3y & \quad \text{if } x < 0 \text{ and } y < 0, \\
2x + 2y & \quad \text{otherwise but } x+y > 0, \\
x + 4y & \quad \text{otherwise}.
\end{cases}
\]
What is $p(p(2,-3),p(-3,-4))$?


What is the smallest positive integer that can be written in the form $3003m + 66666n$, where $m$ and $n$ are integers?


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


Find the numerical value of $k$ such that
\[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]


Terrell normally lifts two 25-pound weights 10 times. If he decides to use two 20-pound weights instead, how many times must he lift them to match the total weight lifted previously?


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


In a new research experiment, a colony of bacteria triples each day instead of doubling. The experiment starts with 5 bacteria. Determine the first day on which the number of bacteria exceeds 200.


What is the area, in square units, of a triangle whose vertices are at $(3, 2)$, $(3, 7)$, and $(8, 2)$?


Determine the value of $b$ for the polynomial $P(x) = x^3 + ax^2 + bx + c$ given that:
1. The mean of the zeros, the product of the zeros, and the sum of its coefficients ($1+a+b+c$) are all equal.
2. The y-intercept of the polynomial's graph is 5.


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


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


Calculate the area, in square units, of triangle $PQR$ where $P(-4, 0)$, $Q(6, 0)$, and $R(3, -5)$. Also, compute the distance from the midpoint of $PQ$ to vertex $R$.


A portion of the graph of \( f(x) = ax^2 + bx + c \) is shown below. The distance between grid lines on the graph is 1 unit.

What is the value of \( a+b+2c \)?

[asy]
size(150);
rr_cartesian_axes(-4,3,-2,9);
real f(real x) {return -2*(x-2)^2 + 8;}
draw(graph(f,-1,5,operator ..), red);
[/asy]


How many odd two-digit numbers have distinct digits?


Let $(x, y)$ be a solution to the system of equations:
\[
\begin{aligned}
\lfloor x \rfloor + \{y\} &= 3.7, \\
\{x\} + \lfloor y \rfloor &= 8.2.
\end{aligned}
\]
Compute $|x - y|$.


Let $f$ be a function satisfying $f(xy) = f(x)/y$ for all positive real numbers $x$ and $y$. If $f(1000) = 6$, what is the value of $f(800)$?


A fair 8-sided die is rolled. If the roll is even, then you win that amount of dollars. If the roll is odd, you win half that amount of dollars (for example, if you roll a 3, then you win $\$\frac{3}{2}$). What is the expected value of your winnings? Express your answer as a dollar value.


Compute $\sqrt{121} \cdot \sqrt{49} \cdot \sqrt{11}$.


As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $14n+5$ and $9n+4$?


Let $g(n)$ be the base-2 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\frac{g(n)}{\log_{2} 3}$ in terms of $n$.

Solution: \boxed{\frac{n}{\log_{2} 3}}


Let $\omega$ be a non-real root of $z^3 = 1$. Let $b_1, b_2, \dots, b_m$ be real numbers such that
\[
\frac{1}{b_1 + \omega} + \frac{1}{b_2 + \omega} + \dots + \frac{1}{b_m + \omega} = 3 + 8i.
\]
Compute
\[
\frac{2b_1 - 1}{b_1^2 - b_1 + 1} + \frac{2b_2 - 1}{b_2^2 - b_2 + 1} + \dots + \frac{2b_m - 1}{b_m^2 - b_m + 1}.
\]


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


What is the smallest positive integer $n$ such that $5n \equiv 1978 \pmod{26}$?


Find the number of integer values of $x$ where the square root of $x$ is greater than 3 and less than 5.


A number $x$ is equal to $5\cdot18\cdot36$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?


Either increasing the radius or the height of a cylinder by five inches will result in the same volume. The original height of the cylinder is three inches. What is the original radius in inches?

Solution: \boxed{3 + 2\sqrt{6}}


Suppose $b$ is a positive integer greater than or equal to $2.$ When $125$ is converted to base $b$, the resulting representation has $3$ digits. What is the number of possible values for $b$?


Write $0.\overline{726}$ as a simplified fraction.


How many positive integers smaller than $500,000$ are powers of $3$, but are not powers of $9$?


In a class of 32 students, each student is randomly paired with another student for a project. What is the probability that Margo is paired with her best friend Irma, and simultaneously, another student, Joe, is paired with his best friend, Pete? Express your answer as a common fraction.


A right triangle with integer leg lengths is called "super cool'' if the number of square units in its area is equal to the number of units in the sum of the lengths of its legs. Find the sum of all the different possible areas of "super cool" right triangles.


Let $a$ and $b$ be real numbers. One of the roots of $x^3 + ax + b = 0$ is $2 + i \sqrt{7}$. Find $a + b$.


In the diagram, two identical smaller circles touch both the larger circle and each other inside the larger circle. The radius of the larger circle is \(8\). Calculate the area of the shaded region, assuming the smaller circles touch each other at the center of the larger circle.

[asy]
size(100);
import graph;
filldraw(Circle((0,0),3),mediumgray);
filldraw(Circle((-1.5,0),1.5),white);
filldraw(Circle((1.5,0),1.5),white);
[/asy]


An urn contains $6$ green balls and $4$ blue balls. A second urn contains $10$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.50$. Find $N$.


Yan is between his home and the library. To get to the library, he can either walk directly to the library or walk home and then ride his bicycle to the library. He rides 5 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the library?


For how many integers $n$ between 1 and 200 is the greatest common divisor of 21 and $n$ equal to 3?


A five-digit integer is chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit is an even number and less than 6? Express your answer as a common fraction.


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 \( 13! \) is divided by \( 17 \)?


Express the value of the following as a common fraction:
\[
\left(1-\frac12\right)^2\cdot\left(1-\frac13\right)^2\cdot\left(1-\frac14\right)^2 \dotsm \left(1-\frac1{50}\right)^2
\]


Round 1278365.7422389 to the nearest hundred.


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

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


What is the value of $\frac13\cdot\frac92\cdot\frac1{27}\cdot\frac{54}{1}\cdot\frac{1}{81}\cdot\frac{162}{1}\cdot\frac{1}{243}\cdot\frac{486}{1}$?


What is the area enclosed by the graph of $|x| + |3y|$ = 12?


The values of \(f\), \(g\), \(h\), and \(j\) are 6, 7, 8, and 9, respectively, but not necessarily in that order. Compute the largest possible value of the sum of the four products \(fg\), \(gh\), \(hj\), and \(fj\).


How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-25^2)} = 0\]have for $x$?


Let \(\theta\) be the angle between the line
\[\frac{x - 2}{3} = \frac{y + 1}{5} = \frac{z - 1}{4}\]
and the plane \(5x - 3y + 4z = 6.\) Find \(\sin \theta.\)


How many perfect squares less than 1,000,000 can be represented as the difference of two consecutive perfect squares?


What is the coefficient of \(a^2b^3\) in \((a+b)^5\left(c+\dfrac{1}{c}\right)^8\)?


Find the quadratic polynomial \( q(x) \) such that \( q(-3) = 0, \) \( q(6) = 0, \) and \( q(7) = -45. \)

Solution: \boxed{q(x) = -4.5x^2 + 13.5x + 81}


Malcolm can run a race at a speed of 4 minutes per mile on flat terrain, while Joshua can run at 6 minutes per mile. If the race is 15 miles long but the last 5 miles are uphill, slowing Malcolm down by an additional 2 minutes per mile and Joshua by 3 minutes per mile, how many minutes after Malcolm crosses the finish line will Joshua cross the finish line if they start the race together?

Solution: \boxed{35 \text{ minutes}}


A room measures 10 feet by 15 feet, and it is to be tiled with tiles measuring 3 inches by 5 inches. How many tiles are needed to entirely cover the floor?


Find the product of $345_7 \cdot 3_7$. Express your answer in base 7.


Let $P,$ $Q,$ and $R$ be points such that $\frac{PR}{PQ} = \frac{RQ}{PQ} = \frac{1}{2}.$ If $Q = (2, 5)$ and $R = (0, -10)$, then what is the sum of the coordinates of $P$?


Evaluate the sum $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{39}\rceil$$
Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$


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


The equations of the asymptotes of a hyperbola are $y = 3x + 4$ and $y = -3x + 2$. Given that the hyperbola passes through the point $(1, 8)$, determine the constants $h$ and $k$ for the standard form of the equation of the hyperbola, which is \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\] where $a,$ $b$, $h,$ and $k$ are constants with $a, b > 0.$ Find the value of $a + h$.


In a library, there are $1450$ packages, each containing $42$ pamphlets. The librarian needs to redistribute them so that there are $45$ pamphlets in each new package. Determine how many pamphlets will remain after repacking as many full new packages as possible.


When three numbers are added two at a time, the sums are 35, 47, and 52. What is the sum of all three numbers, and what is their product?


How many possible distinct arrangements are there of the letters in the word STARS?


What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 8$, $y = 3$, and the $y$-axis? Express your answer as a decimal to the nearest tenth.


What is the degree measure of the smaller angle between the hour hand and the minute hand of a clock at exactly 4:45 p.m. on a 12-hour analog clock?


What is the remainder when 790,123 is divided by 15?


In the equation, $\frac{1}{4}+\frac{7}{x}=\frac{13}{x}+\frac{1}{9}$, what is the value of $x$?


How many terms of the arithmetic sequence 100, 96, 92, $\dots$ appear before the number 1 appears?


Calculate the product of the divisors of $60$.


The quadratic $x^2 + 800x + 500$ can be written in the form $(x+d)^2 + e$, where $d$ and $e$ are constants. What is $\frac{e}{d}$?


Thirty gremlins and twenty imps attend the Fantasy Creatures Gala. The gremlins shake hands with each other and all the imps. Each imp shakes hands with exactly 5 other imps, ensuring no repeats, and with every gremlin. Calculate the total number of handshakes at the event.


John borrows $500 from Alice, who charges an interest of $6\%$ per month (which compounds monthly). What is the least integer number of months after which John will owe more than twice as much as he borrowed?


A student must choose a program of four courses from an expanded list of courses consisting of English, Algebra, Geometry, History, Art, Latin, and Physics. The program must include English and at least one science course (either Physics, Algebra, or Geometry). In how many ways can this program be constructed?


Compute \[ \left\lfloor \dfrac {1005^3}{1003 \cdot 1004} - \dfrac {1003^3}{1004 \cdot 1005} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


Lucas collects minerals for his geological study. He arranges the same number of minerals on each shelf, and then places these shelves into three different cabinets. The first cabinet contains a total of 924 minerals, the second contains 1386 minerals, and the third contains 462 minerals. What is the largest number of minerals that Lucas could be placing on each shelf?


A fair coin is flipped 9 times. What is the probability that at least 6 of the flips result in heads?


Solution: \boxed{-2 - 12z + 5y}


Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that the penny, dime, and half-dollar all come up the same?


Compute $11^{-1} \pmod{1021}$. Express your answer as a residue from $0$ to $1020$, inclusive.


Let t(x) has domain \{-2, -1, 0, 1\} and range \{-4, -2, 0, 2\}. Let u(x) has domain \{0, 1, 2, 3\} and is defined by u(x) = x - 1. What is the sum of all possible values of u(t(x))?


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


Evaluate the determinant:
\[
\begin{vmatrix} 1 & x & y+z \\ 1 & x+y & z \\ 1 & x+z & y \end{vmatrix}.
\]


Define a function $g$ by $g(x)=7x-4$. If $g(x)=f^{-1}(x)-5$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$, determine the value of $7a+3b$.


Let \(\mathbf{N}\) be a matrix such that
\[
\mathbf{N} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} -2 \\ 4 \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \end{pmatrix}.
\]
Compute \(\mathbf{N} \begin{pmatrix} 7 \\ 0 \end{pmatrix}.\)

Solution: \boxed{\begin{pmatrix} 14 \\ 0 \end{pmatrix}}


If $\tan a + \tan b = 17$ and $\cot a + \cot b = 40$, what is $\tan(a+b)$?


If the three points $(2,c,d),$ $(c,4,d),$ $(c,d,6)$ are collinear, determine the value of $c + d$.


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


What is the minimum number of times you must throw four fair six-sided dice to ensure that one particular sum is rolled at least three times?


Lark has forgotten her locker combination. It is a sequence of four numbers. The first number is in the range from 1 to 20 and must be odd. The second number is in the range from 1 to 40 and must be even. The third number is in the range from 1 to 30 and must be a multiple of 3. The last number is in the range from 1 to 50 and must be a multiple of 5. How many combinations could possibly be Lark's?


Let $d$ be a positive number such that when $144$ is divided by $d$, the remainder is $9$. Compute the sum of all possible two-digit values of $d$.


Determine the sum of all prime numbers \( p \) for which there exists no integer solution in \( x \) to the congruence \( 5(10x+2) \equiv 3 \pmod p \).


Find the smallest positive integer $b$ for which $x^2 + bx + 1176$ factors into a product of two polynomials, each having integer coefficients.


If the angle between the vectors \(\mathbf{c}\) and \(\mathbf{d}\) is \(60^\circ,\) what is the angle between the vectors \(-2\mathbf{c}\) and \(3\mathbf{d}\)?


A frustum of a right circular cone is formed by cutting a small cone off the top of a larger cone. The frustum has an altitude of 18 centimeters, the area of its lower base is $144\pi$ sq cm, and the area of its upper base is $36\pi$ sq cm. Find the altitude of the small cone that was cut off.


How many numbers in the set $\{7, 17, 27, 37, \ldots\}$ can be written as the sum of two prime numbers?


A jar contains 15 jellybeans (six red, three blue, and six white). If you pick four jellybeans from the jar at random and without replacement, what is the probability that at least three will be red? Express your answer as a common fraction.


Consider the Lucas sequence, where the first term is 1, the second term is 3, and each term thereafter is the sum of the previous two terms. Determine the remainder when the $52^{\mathrm{th}}$ term of the sequence is divided by 5.


Determine the number of integers that are multiples of 30 and lie between the smallest positive multiple of 30 that is a perfect square and the smallest positive multiple of 30 that is a perfect cube, inclusive.


A circle has a radius of 6. What is the area of the smallest square that can entirely contain this circle, and what is the circumference of the circle?


Each of the $1500$ students at a school studies either Spanish or French, and some study both. The number who study Spanish is between $70\%$ and $75\%$ of the school population, and the number who study French is between $35\%$ and $45\%$. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Determine $M-m$.


Find the sum of the first five prime numbers that have a units digit of 1 or 7.


John wants to enclose a rectangular garden using 270 feet of fencing. The fencing will cover three sides of the garden (the fourth side is bordered by John's house). What is the maximum area that John can enclose with this garden?


A parabola has vertex $V = (0, -2)$ and focus $F = (0, -1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 50.$ Find $P.$

Solution: \boxed{(14, 47)}


The base-10 numbers 328 and 57 are multiplied. The product is then written in base-7. What is the units digit of the base-7 representation?


What is the base-10 integer 648 when expressed in base 7?


Define a function $g(x)$, for positive integer values of $x$, by
\[ g(x) = \left\{\begin{aligned}
3 \log_3 x & \quad \text{if } \log_3 x \text{ is an integer}, \\
1 + g(x + 1) & \quad \text{otherwise}.
\end{aligned} \right.\]
Compute $g(100)$.


Give an example of a quadratic function that has zeroes at \(x=1\) and \(x=5\), and that takes the value \(8\) when \(x=3\).

Solution: \boxed{-2x^2 + 12x - 10}


Determine the number of real solutions for \(x\) in the equation \( (2x - 3x + 7)^2 + 2 = |x| + 1 \).


There are four points that are $7$ units from the line $y = 15$ and $15$ units from the point $(10,15)$. Find the sum of the $x$- and $y$-coordinates of all four of these points.


Let $\alpha$ and $\beta$ be complex numbers such that $\alpha + 2\beta$ and $i(\alpha - 3 \beta)$ are both positive real numbers. If $\beta = 2 + 3i,$ compute $\alpha.$

Solution: \boxed{6 - 6i}


What is the smallest four-digit palindrome that is divisible by 5?


Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is three times the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?


Three siblings, Andy, Alexa, and Aaron, bake 36 cookies. Aaron eats a number of cookies that is twice the number of cookies eaten by his brother Andy. Alexa eats the remaining cookies. If the siblings finish all 36 cookies, what is the maximum number of cookies that the brother, Andy, could have eaten?


What is the remainder when \( 98^{24} \) is divided by 100?


Simplify the expression:

$$7x + 3 - 2x + 15 + y$$

Solution: \boxed{5x + y + 18}


A "super ball" is dropped from a window 20 meters above the ground. On each bounce, it rises $\frac{2}{3}$ of the distance of the preceding high point. Additionally, a steady wind reduces each ascending height by 10%. The ball is caught when it reaches the high point after hitting the ground for the third time. Calculate to the nearest meter, how far has it traveled?


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


Simplify the sum: \(i^0 + i^1 + \cdots + i^{2011}\).


For any positive integer \(n\), the value of \(n!\) is the product of the first \(n\) positive integers. Calculate the greatest common divisor of \(8!\) and \(10!\).


Calculate the matrix $\mathbf{Q}$ such that for any vector $\mathbf{v}$ with integer components, $\mathbf{Q} \mathbf{v}$ is the projection of $\mathbf{v}$ onto the vector $\begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} \frac{1}{11} & \frac{3}{11} & -\frac{1}{11} \\ \frac{3}{11} & \frac{9}{11} & -\frac{3}{11} \\ -\frac{1}{11} & -\frac{3}{11} & \frac{1}{11} \end{pmatrix}}


It is currently $6\!:\!00\!:\!00 \text{ a.m.}$ What time will it be in $9876$ seconds? (Enter the time in the format "HH:MM:SS", using the 24-hour clock format.)


Liam makes a snowman by stacking four snowballs with radii 4 inches, 6 inches, 3 inches, and 7 inches. Assuming all snowballs are perfectly spherical, calculate the total volume of snow he uses, in cubic inches, and express your answer in terms of $\pi$.


The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction.


A right circular cone with base radius $r$ and height $h$ lies on its side on a flat table. It rolls on the table surface without slipping, with the meeting point of the cone's base and table tracing a circular arc centered at the vertex touching the table. The cone returns to its initial position after completing $15$ full rotations. Determine the ratio $h/r$, expressed in the form $m\sqrt{n}$, where $m$ and $n$ are integers, and $n$ is not divisible by the square of any prime. Calculate $m + n$.


The matrix for projecting onto a certain line $\ell,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{5} & -\frac{2}{5} & 0 \\ -\frac{2}{5} & \frac{4}{5} & 0 \\ 0 & 0 & 0 \end{pmatrix} \renewcommand{\arraystretch}{1}.\]
Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$

Solution: \boxed{\begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix}}


Suppose the function $g(x)$ is defined on the domain $\{x_1, x_2, x_3\}$ such that the graph of $y=g(x)$ consists of just three points. Those three points form a triangle of area $45$. Determine the area of the triangle formed by the graph of $y = 3g(4x)$ which also consists of just three points.


The faces of a decagonal die are labeled with digits $1$ through $10$. What is the probability, expressed as a common fraction, of rolling a sum of $17$ with a pair of such decagonal dice?


When simplified, what is the value of $(\sqrt{2} \times 2^{1/2} \times 2) + (18 \div 3 \times 2) - (8^{1/2} \times 4)$?

Solution: \boxed{16 - 8\sqrt{2}}


What is the coefficient of \( x^3 \) when \( 3x^4 - 4x^3 + 2x^2 - 3x + 2 \) is multiplied by \( 2x^2 - 3x + 5 \)?


Determine the 15th term in the geometric sequence where the first term is $32$ and each subsequent term is multiplied by $\frac{1}{4}$.


A pirate is counting the loot he has stolen from the ship "Golden Marauder," where everything is accounted in base 7. The treasure chest includes $4516_{7}$ dollars worth of gems, $3216_{7}$ dollars worth of silver coins, $654_{7}$ dollars worth of silks, and $301_{7}$ dollars worth of spices. What is the total dollar amount that this pirate has accumulated? Express your answer in base 10.


Find a monic quartic polynomial, in $x$, with rational coefficients such that $3 + \sqrt{5}$ and $2 - \sqrt{7}$ are roots of the polynomial.

Solution: \boxed{x^4 - 10x^3 + 25x^2 + 2x - 12}


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


Find the greatest common divisor of $8!$ and $(6!)^2.$


Calculate the coefficient of \(x^3\) when the expression \(2(x^2 - 2x^3 + x) + 4(x + 3x^3 - 2x^2 + 2x^5 + 2x^3) - 3(2 + x - 5x^3 - x^2)\) is simplified.


Solution: \boxed{\frac{80}{3} \text{ miles per hour}}


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

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


A certain species of algae in a lake grows in such a way that its covered area quadruples every two days. The lake was fully covered by the algae on day $24$. Determine on what day the lake was $6.25\%$ covered by algae.


A number $N$ is such that its base-$10$ logarithm is between two consecutive integers. Given that $\log_{10}2500 < \log_{10}N < \log_{10}10000$, find the sum of these two integers.


Two different integers are randomly chosen from the set $$\{ -6, -3, 0, 2, 5, 9 \}.$$ What is the probability that their product is non-positive? Express your answer as a common fraction.


Calculate the sum of the angles $\theta_1 + \theta_2 + \theta_3 + \theta_4$ in degrees for the equation $z^4 = 16i$.


How many terms of the arithmetic sequence 100, 95, 90, $\dots$ appear before the number 50 appears?


Alex is designing a fence around a rectangular playground. He has exactly 400 feet of fencing material. The playground must be enclosed on all four sides. Regulations require that the length of the playground must be at least 100 feet and the width must be at least 60 feet. Alex wants to maximize the playground area to include various play equipment and sitting areas. Determine the optimal area, in square feet.


How many different four-digit numbers can be formed by arranging the digits in 2101?


Tim has a more colorful selection in his sock drawer which now contains 12 gray-bottomed socks, 10 white-bottomed socks, and 6 blue-bottomed socks. Every morning, he picks three socks at random. What is the probability that Tim picks either a matching pair of socks or three socks each of different colors?


If \( x \) is a real number and \( \lceil x \rceil = 14 \), how many possible values are there for \( \lceil x^2 \rceil \)?


Let \( D, E, \) and \( F \) be points on a circle of radius \( 25 \). If \( \angle EFD = 120^\circ \), what is the length of the circumference of the minor arc \( DE \)? Express your answer in terms of \( \pi \).


Solve the inequality and equation system where
\[ |x - 3| + |x + 4| < 8 \]
and
\[ x^2 - x - 12 = 0. \]

Solution: \boxed{x = -3}


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


Consider the logarithm $\log_{2}{3456}$. Determine the integers $c$ and $d$ such that $\log_{2}{3456}$ is between $c$ and $d$. Find $c+d$.


Alexio now has 150 cards numbered 1-150, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 4, 5, or 7? Express your answer as a common fraction.


Find the remainder when $3x^5 + 2x^3 - 5x^2 + 9$ is divided by $x^2 + 3x + 2$.

Solution: \boxed{122x + 121}


Given the function $y = h(x)$ described by the following segments:
- $y = -3 - x$ for $x \in [-4, 0]$,
- $y = \sqrt{9 - (x-3)^2} - 3$ for $x \in [0, 6]$,
- $y = 3(x - 3)$ for $x \in [3, 4]$.

For constants $p, q, r$, define a new function $k(x) = ph(qx) + r$. The graph of $y = k(x)$ is obtained by stretching $h(x)$ horizontally by a factor of 3 and shifting it down by 5 units. Determine the ordered triple $(p, q, r)$.

Solution: \boxed{(1, \frac{1}{3}, -5)}


The theme park records the duration of each of the 19 most popular attractions in minutes and seconds. The durations are represented in a stem-and-leaf plot where, for example, "1 15" represents 1 minute, 15 seconds. Determine the median duration in seconds.

\begin{tabular}{c|cccccc}
0 & 45 & 50 & 55 & 58 & & \\
1 & 05 & 10 & 22 & 35 & 40 & 45\\
2 & 00 & 10 & 15 & 20 & 30 & 45\\
3 & 05 & 10 & 15 & & & \\
\end{tabular}


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


All squares have the same value, and all hexagons have the same value. What is the sum of three hexagons?
\begin{align*}
\square + \hexagon + \square + \hexagon + \square&= 18\\
\hexagon + \square+\hexagon+\square+\hexagon &= 20\\
\hexagon + \hexagon + \hexagon &= \ ?
\end{align*}


Suppose that $\{b_n\}$ is an arithmetic sequence with
$$
b_1+b_2+ \cdots +b_{150}=150 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{300}=450.
$$
What is the value of $b_2 - b_1$? Express your answer as a common fraction.


Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 73 \times 83 \times 93$ is divided by $5$.


Calculate $98 \cdot 202 \equiv m \pmod{75}$, where $0 \le m < 75$.


Let \(p, q, r\) be the roots of the polynomial \(x^3 - x^2 + x - 3 = 0\). Find \(p^4 + q^4 + r^4\).


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


A bowl contains 15 jellybeans (five red, three blue, four white, and three green). If you pick four jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red and one is green? Express your answer as a common fraction.


Determine an expression for the area of $\triangle QCA$ in terms of $p$, where the coordinates of point $A$ are now $(3, 12)$. [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 12); pair b = (12, 0);
pair a = (3, 12); pair t = (3, 0); pair c = (0, 9);

draw((-2, 0)--(15, 0), Arrow);
draw((0, -2)--(0, 15), Arrow);
draw(q--a--b);
draw(a--c--b);

label("$Q(0, 12)$", q, W);
label("$A(3, 12)$", a, NE);
label("$B(12, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (15, 0), E);
label("$y$", (0, 15), N);
label("$C(0, p)$", c, W);
[/asy]


Which of the following points is closest to the origin: $(2,3)$, $(4,0)$, $(1,-1)$, $(-3,4)$, $(0,0,5)$?


Two spinners are divided into halves and fifths, respectively. Assume the left spinner is quartered with values assigned as 5, 6, 7, and 8, and the right spinner divided into fifths containing the values 1, 2, 3, 4, and 5. What is the probability that the product of the results of the two spins will be a multiple of 3? Express your answer as a common fraction.


What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $C(-3,0)$ and $D(4,5)$?


Solution: \boxed{3xyz + 6xz - 15yz - 30z + 12xy + 24x - 60y - 120}


Eleanor has 8 daughters and no sons. Some of her daughters have 7 daughters each, and the rest have none. Eleanor has a total of 43 daughters and granddaughters, and no great-granddaughters. How many of Eleanor's daughters and granddaughters have no daughters?


Julia has an unfair eight-sided die. The probability that it rolls an 8 is $\frac{3}{8}$, and the probability that it rolls any other number from 1 to 7 is the same for each number. What is the expected value of the number shown when this die is rolled?


Find the coefficient of the $x^3$ term in the expansion of the product $$(3x^3 + 4x^2 + 5x + 6)(7x^2 + 8x + 9).$$


Find the product of all real values of $r$ for which $\frac{1}{3x} = \frac{r-x}{6}$ has exactly one real solution.


One of the asymptotes of a hyperbola has equation $y=4x-3.$ The foci of the hyperbola have the same $x-$coordinate, which is $3.$ Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".

Solution: \boxed{y = -4x + 21}


Calculate $5.47 + 2.58 + 1.95$ and write the sum as a decimal.


A bin has 10 black balls and 8 white balls. 4 of the balls are drawn at random. What is the probability of drawing 2 black balls and 2 white balls, given that the first ball drawn is black?


A child has a set of 128 distinct blocks. Each block can be made of 2 materials (plastic, wood), 4 sizes (small, medium, large, extra-large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). Determine how many blocks in the set differ from the 'plastic medium red circle' in exactly 2 ways.


In a club end-of-year officer election, 8 equal officer positions are to be filled. There are 20 candidates, 10 of whom have previously held positions. Determine how many potential officer line-ups contain at least 3 former officers.


What is the domain of the function $f(x) = \log_3(\log_5(\log_7x))$?

Solution: \boxed{(7, \infty)}


How many three-digit numbers are multiples of neither 4 nor 6?


Find the smallest possible value of \(x\) in the simplified form \(x=\frac{a+b\sqrt{c}}{d}\) if \(\frac{7x}{8}-1=\frac{4}{x}\), where \(a, b, c,\) and \(d\) are integers. What is \(\frac{acd}{b}\)?


What is $\log_{5}{3100}$ rounded to the nearest integer?


Keisha's basketball team is still deciding on a new uniform. This time, the seventh-graders can pick from three colors (black, gold, or blue) for the shorts, and the eighth-graders have the same three choices (black, white, or gold) for the jerseys. The groups won't discuss their choices with each other. Assuming each color choice is equally likely, what is the probability that the shorts and jersey will be different colors?


How many integers between $200$ and $300$ have three different digits in increasing order?


A new solid $T$ is defined by the set of all points $(x, y, z)$ in space such that $|x| + |y| + |z| \leq 2$. Find the volume of solid $T$.


Using the digits 1, 2, 3, 4, 5, 6, how many even three-digit numbers less than 600 can be formed if each digit can be used more than once?


How many zeroes does $15!$ end with, when $15!$ is written in base 12?


How many positive perfect square integers are factors of the product $\left(2^{12}\right)\left(3^{16}\right)\left(7^{18}\right)\left(11^7\right)$?


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le16$, then what is the largest possible value of $a+b+c$?


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

Solution: \boxed{\cos F = \frac{3}{5}}


The mean of $7, 10, 15$, and $21$ is equal to the mean of $18$ and $z$. What is the value of $z$?


What is the smallest possible value of the sum $\lvert x + 3\rvert + \lvert x + 6\rvert + \lvert x + 7\rvert$?


The algae in the Johnson's backyard pond tripled every two days and the pond was completely covered in algae by the $28^{th}$ day of the month. On what day of that month was the pond $88.89\%$ algae-free?


A right circular cylinder is inscribed in a hemisphere so that one base of the cylinder is coplanar with the hemisphere's base. The radius of the hemisphere is increased to 7, while the radius of the cylinder remains 2. Calculate the height of the cylinder if the top base of the cylinder coincides with the top of the hemisphere.


Calculate how many positive integer divisors of $2310^{2310}$ are divisible by exactly 2310 positive integers.


Given the partial fraction decomposition of
\[
\frac{x^2 - 13}{x^3 - 3x^2 - 4x + 12}
\]
is expressed as
\[
\frac{A}{x-2} + \frac{B}{x+2} + \frac{C}{x-3}.
\]
Find the product $ABC$.


Evaluate the expression \((c^c - c(c-1)^c)^c\) when \( c = 4 \).


Consider a rectangular prism with lengths of edges 4 units, 6 units, and 8 units. Calculate the total number of diagonals within this prism, where a diagonal is either a segment joining two vertices on the same face not directly connected by an edge (face diagonal), or a segment joining two vertices that are not on the same face (space diagonal).


Let $P$ be the point on line segment $\overline{AB}$ in a three-dimensional space, where $A$ and $B$ have coordinates $(1,2,3)$ and $(4,5,6)$, respectively, such that the ratio $AP:PB = 5:3$. Find the coordinates of point $P$ using the equation:
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]
for some constants $t$ and $u$. Determine the ordered pair $(t,u)$.

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


Point \( P \) lies on the line \( x = -4 \) and is 13 units from the point \( (3, -1) \). Find the product of all possible \( y \)-coordinates that satisfy the given conditions.


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


Suppose that $b$ is a positive integer greater than or equal to $2$. When $256$ is converted to base $b$, the resulting representation has $4$ digits. What is the number of possible values for $b$?


Cara is sitting at a circular table with her six friends, including her friend Alex. How many different pairs of people could Cara be sitting between where one of the pairs must always be Alex?


A rectangular box has a volume of $5670$ cubic inches and a surface area of $2534$ square inches. The sum of the lengths of its $12$ edges is $252$ inches. What would the volume of the box be, in cubic inches, if its length, width, and height were each increased by one inch?

Solution: \boxed{7001 \text{ cubic inches}}


A certain integer has $5$ digits when written in base $10$. The same integer has $d$ digits when written in base $16$. What is the sum of all possible values of $d$?


Thirty tiles are numbered from 1 to 30 and placed in box $A$. Another thirty tiles, numbered from 15 to 44, are placed in box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ is less than 20, and the tile from box $B$ is either even or greater than 35? Express your answer as a common fraction.


The volume of a given sphere is \( 72\pi \) cubic inches. Calculate the surface area of the sphere in terms of \( \pi \).

Solution: \boxed{36\pi 2^{2/3}}


Find \( g(x) \), with terms in order of decreasing degree, if \[ 4x^5 + 3x^3 - 2x + g(x) = 7x^3 - 5x^2 + 4 \].

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


Jen rode 50 miles at 12 miles per hour, 20 miles at 40 miles per hour, and 10 miles at 20 miles per hour. What was her average speed, in miles per hour, for the entire ride?

Solution: \boxed{\frac{480}{31} \text{ mph}}


Let $X,$ $Y,$ and $Z$ be points on the line such that $\frac{XZ}{ZY} = 3$. If $Y = (2, 6)$ and $Z = (-4, 8)$, determine the sum of the coordinates of point $X$.


If the three points $(2,a,b),$ $(a,3,b),$ $(a,b,4)$ are collinear, what is the value of $a + b$?


Calculate $\sqrt{50x} \cdot \sqrt{18x} \cdot \sqrt{8x}$.

Solution: \boxed{60x \sqrt{2x}}


Jennifer is compiling a report on extracurricular clubs at her college for a project. The college has Chess and Drama clubs. Jennifer has a list of all 30 students who are members of at least one of the clubs. She also knows that 22 students are in the Chess club and 20 are in the Drama club. If Jennifer interviews two students randomly from her list, what is the probability that she will have the opportunity to gather information about both clubs after the interviews? Express your answer as a fraction in simplest form.


Let \( N \) be the largest integer whose square has exactly 4 digits when written in base 8.

What is \( N \), expressed in base 8?


Find the minimum value of
\[(15 - x)(12 - x)(15 + x)(12 + x).\]


Evaluate $45\cdot \left(4\frac{1}{3} - 5\frac{1}{4}\right) \div \left(3\frac{1}{2} + 2\frac{1}{5}\right)$. Express your answer as a mixed number.


In a physical education class, students are divided into four rows for a drill. The class has more than 50 students. Two rows have the same number of students, another two rows each have one student more than those two rows. What is the smallest possible class size for this scenario?


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


Compute $\left(3\sqrt{15625} - 5\right)^2$.


What is the perimeter of the triangle formed by the points $(1, 4)$, $(-7, 0)$, and $(1, 0)$?

Solution: \boxed{4\sqrt{5} + 12}


If $n$ is the smallest positive integer greater than 3 for which there exist positive real numbers $a$ and $b$ such that
\[(a + bi)^n + a = (a - bi)^n + b,\] compute $\frac{b}{a}$.


For positive integers $n$, let $g(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, modifying the digits configuration such that $k$ is constrained to be a power of a prime number. Determine how many positive integer divisors $g(2000)$ has if $g(n)$ is constrained to be the smallest power of $5$.


There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are Americans. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded if at most one American gets a medal?


Let $G$ be the centroid of triangle $PQR.$ If $GP^2 + GQ^2 + GR^2 = 22,$ then find $PQ^2 + PR^2 + QR^2.$


Let the positive integers \(A, B,\) and \(C\) form an arithmetic sequence, and let the integers \(B, C,\) and \(D\) form a geometric sequence. If \(\frac{C}{B} = \frac{7}{4},\) what is the smallest possible value of \(A + B + C + D\)?


What is the smallest positive integer which, when divided by 3, gives a remainder of 2, when divided by 4, gives a remainder of 3, when divided by 5, gives a remainder of 4, when divided by 6, gives a remainder of 5, when divided by 7, gives a remainder of 6, and when divided by 11, gives a remainder of 10?


In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel with lengths of 10 and 24 units respectively, and the altitude is 15 units. Points $G$ and $H$ are the midpoints of sides $AD$ and $BC$, respectively. Determine the area of quadrilateral $GHCD$.


Find the coefficient of \(x\) when \(4(x - 5) + 3(2 - 3x^2 + 6x) - 10(3x - 2)\) is simplified.


A palindrome is an integer that reads the same forward and backward, such as 1221 or 1331. What percent of the palindromes between 1000 and 2000 contain at least one 7?


An equilateral triangle has two vertices at $(1,3)$ and $(9,3)$. If the third vertex is in the first quadrant, what is the y-coordinate?

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


What is the largest three-digit multiple of 7 whose digits' sum is 21?


An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. What is the smallest abundant number that is not a multiple of 5?


What is the remainder when 4873 is divided by 29?


What is the largest integer $n$ for which $\binom{10}{3} + \binom{10}{4} = \binom{11}{n}$?


For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 100$.
(2) the quadratic equation $x^2 + (2a-3)x + (a-1)^2 = 0$ has two integer solutions?


Find the direction vector of a line $\ell$ corresponding to the reflection matrix
\[\begin{pmatrix} \frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & -\frac{3}{5} \end{pmatrix}.\]
Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ and $b$ are integers, $a > 0$, and $\gcd(|a|,|b|) = 1$.

Solution: \boxed{\begin{pmatrix} 2 \\ 1 \end{pmatrix}}


Express as a common fraction:
$$\cfrac{ \frac{3}{7}+\frac{5}{8} }{ \frac{5}{12}+\frac{2}{9}}$$.


Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $n$ is a perfect square and $1 \le n \le 50$, is $f(n)$ prime?


Find the smallest integer $x$ such that $\lvert 3x - 4 \rvert \leq 25$ and $x$ is divisible by 3.


Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = -\frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Find $\sin \theta.$


What is the minimum value of the function $3x^2 - 12x + 7 + 749$?


Update the number of moons for each planetary body and include an additional celestial body, Ceres, with 0 moons. Here is the new tabulation:

\begin{tabular}{c|c}
Planet & $\#$ of Moons \\
\hline
Mercury & 0 \\
Venus & 2 \\
Earth & 2 \\
Mars & 4 \\
Jupiter & 21 \\
Saturn & 25 \\
Uranus & 18 \\
Neptune & 3 \\
Pluto & 2 \\
Ceres & 0 \\
\end{tabular}
What is the median number of moons per planetary body, including now Pluto and Ceres?


Jay, Paul, and Naomi, friends and students, decide to explore different parts of a park. Jay walks east at a speed of 1 mile every 20 minutes, Paul walks west at a rate of 3.2 miles every 40 minutes, and Naomi jogs north at 2 miles every 30 minutes. How far apart in miles is the most distant pair after 2 hours?


Compute $\arccos (\sin 3)$, where all functions are in radians.

Solution: \boxed{3 - \frac{\pi}{2}}


What is the sum of the different prime factors of $192360$?


Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 4$. Find the minimum value of $\frac{1}{x} + \frac{4}{y} + \frac{9}{z}$.


Suppose the number $b$ satisfies the equation $5 = b + b^{-1}$. What is the value of $b^6 + b^{-6}$?


How many even perfect square factors does $2^6 \cdot 7^3 \cdot 3^4$ have?


The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence?


On a $12$-hour clock, an elapsed time of five hours looks the same as an elapsed time of $25$ hours. Because of this, we can say that five hours is "clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $5$ hours and is "clock equivalent'' to its square number of hours?


The equation $x^2 - 6x + 13 = 25$ has two solutions, $a$ and $b$, with $a \geq b$. What is the value of $3a + 2b$?

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


Find the product of the greatest common divisor and the least common multiple of $90$ and $135.$


If \( x = 3 \) and \( y = 4 \), then what is the value of \(\frac{x^5 + 3y^3}{8}\)?


Suppose we need to divide 15 dogs into three groups, one with 4 dogs, one with 7 dogs, and one with 4 dogs. We want to form the groups such that Fluffy is in the 4-dog group, Nipper is in the 7-dog group, and Daisy is in the other 4-dog group. How many ways can we arrange the remaining dogs into these groups?


A number $x$ is equal to $7\cdot24\cdot54$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?


Two spinners are divided into four and five sectors, respectively. The first spinner is divided into numbers {2, 5, 7, 11} and the second into numbers {3, 4, 6, 8, 10}. If each spinner is spun once, what is the probability that the product of the two spins is an even number? Express your answer as a common fraction.


In parallelogram $EFGH$, $EF = 5x + 7$ cm, $FG = 4y + 1$ cm, $GH = 27$ cm, and $HE = 19$ cm. The area of parallelogram $EFGH$ is given by the equation $A = 2x^2 + y^2 + 5xy + 3$. Determine the values of $x$ and $y$ and compute $A$.

Solution: \boxed{A = 145.25 \text{ cm}^2}


A box contains 5 white balls and 6 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate in color starting with a black ball?


What is the perimeter of hexagon $ABCDEF$ in the following diagram? [asy]
pair cis(real r,real t) { return (r*cos(t),r*sin(t)); }
pair a=(0,0);
pair b=cis(1,-pi/2);
pair c=cis(sqrt(3),-pi/3);
pair d=cis(2,-pi/3+atan(1/sqrt(2)));
pair e=cis(3,-pi/3+atan(1/sqrt(2))+atan(1/sqrt(3)));
pair f=cis(sqrt(10),-pi/6);
dot(a); dot(b); dot(c); dot(d); dot(e); dot(f);
draw(a--b--c--d--e--f--a);
draw(a--c); draw(a--d); draw(a--e);
draw(0.86*b--0.86*b+0.14*(c-b)--b+0.14*(c-b));
draw(0.92*d--0.92*d+0.14*(e-d)--d+0.14*(e-d));
label("$A$",a,NW);
label("$B$",b,SW);
label("$C$",c,SSE);
label("$D$",d,ESE);
label("$E$",e,NE);
label("$F$",f,NE);
label("1",(a+b)/2,W);
label("1",(b+c)/2,S);
label("2",(c+d)/2,SE);
label("2",(d+e)/2,E);
label("1",(e+f)/2,NE);
[/asy]

Solution: \boxed{7 + \sqrt{10}}


Given the polynomial $x^3 + 2x^2 + 3$, find the remainder when it is divided by $x^2 - 2x + 4$.

Solution: \boxed{4x - 13}


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


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


For positive integers $n$, let $g(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n+1$ digits after the decimal point. How many positive integer divisors does $g(2009)$ have?


Solve the equation $$x^2 - 13x + 40 = 0$$ and specify whether the roots are positive, negative, or of different signs.


Kim earned scores of 86, 82, and 89 on her first three mathematics examinations. She is expected to increase her average score by at least 2 points with her fourth exam. What is the minimum score Kim must achieve on her fourth exam to meet this target?


Two circles with a radius of 15 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) in centimeters between these two circles. Express your answer in simplest radical form.


A rectangular field's width is one-third as long as it is wide, and it is completely enclosed by 72 meters of fencing. What is the number of square meters in the area of the field?


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{z}$ where $z$ is an integer such that $0<z\le12$, then what is the largest possible value of $a+b+c$?


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


The solutions to the equation \(5x^2 + 7 = 2x - 6\) can be written in the form \(x = p \pm q i\), where \(p\) and \(q\) are real numbers. What is \(p + q^2\)? Express your answer as a fraction.


Find the $3 \times 3$ matrix $\mathbf{M}$ such that for any $3 \times 3$ matrix $\mathbf{N},$ the matrix multiplication $\mathbf{M} \mathbf{N}$ results in a matrix where the first row of $\mathbf{N}$ is swapped with the third row and the second row of $\mathbf{N}$ is tripled. In other words,
\[ \mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} g & h & i \\ 3d & 3e & 3f \\ a & b & c \end{pmatrix}. \]

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


Let $a$ be the number of positive multiples of 12 that are less than $60$. Let $b$ be the number of positive integers that are less than $60$ and a multiple of $4$ and a multiple of $3$. Compute $(a + b)^2$.


Each of the numbers $b_1,$ $b_2,$ $\dots,$ $b_{97}$ is $\pm 1.$  Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 97} b_i b_j.\]


Find the point on the line defined by
\[
\begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}
\]
that is closest to the point \((3,2,1)\).

Solution: \boxed{\left( \frac{44}{35}, \frac{137}{35}, \frac{85}{35} \right)}


In triangle $XYZ,$ the midpoint of $\overline{YZ}$ is $(2, 3, -2),$ the midpoint of $\overline{XZ}$ is $(-1, 6, 3),$ and the midpoint of $\overline{XY}$ is $(3, 1, 5).$ Find the coordinates of vertex $X.$

Solution: \boxed{(0, 4, 10)}


Each outcome on the spinner below has equal probability. If you spin the spinner four times and form a four-digit number from the four outcomes, where the first outcome is the thousand, the second outcome is the hundreds digit, the third outcome is the tens digit, and the fourth outcome is the units digit, what is the probability that you will end up with a four-digit number that is divisible by 25? Express your answer as a common fraction.

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


Determine the maximum value of the function
\[ g(x) = 4x - x^3 \]
for \(0 \leq x \leq 2\).


A notebook with 75 pages numbered 1 to 75 has its pages renumbered in reverse, from 75 to 1. Determine how many pages have the same units digit in both the original and reversed numbering.


How many positive 3-digit numbers are multiples of 25, but not of 60?


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $18$, and each $x$-coordinate is scaled by a factor of $2$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.


What is the smallest whole number $b$ such that 100 can be expressed in base $b$ using only three digits?


Calculate how many envelopes from the following list must pay an extra $\$0.15$ in postage if the United States Postal Service charges this amount when the length of an envelope, in inches, divided by its height, in inches, is less than $1.2$ or greater than $2.8.$ Here are the envelopes:
\begin{tabular}[t]{ccc}
Envelope & Length in inches & Height in inches\\\hline
E &5 &3\\
F &10 &2\\
G &5 &5\\
H &12 &5\\
\end{tabular}


Determine how many solutions exist for the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-50^2)} = 0\] for \(x\).


Let $g(x) = \left\lceil\dfrac{2}{x+3}\right\rceil$ for $x > -3$, and $g(x) = \left\lfloor\dfrac{2}{x+3}\right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Which integer is not in the range of $g(x)$?


What is the largest number of positive, consecutive integers whose sum is 105?


The operation $\&$ is defined as $a \& b = \frac{(a+b)(a-b)}{2}$. What is the value of $8 \& 5$?


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


Consider two distinct integers, $x$ and $y$, randomly selected from the set $\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$. What is the probability that $xy+x+y$ is even?


Calculate $7\cdot7! + 5\cdot5! + 3\cdot3! + 3!$.


What is $\frac{0.\overline{54}}{0.\overline{18}} \times \frac{1}{2}$? Express your answer as a common fraction in lowest terms.


How many positive two-digit integers are multiples of 3, 5, and 7?


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


Parallelogram $EFGH$ with $E(3,6)$, $F(5,10)$, $G(7,6)$, and $H(5,2)$ is reflected across the $x$-axis to $E'F'G'H'$ and then $E'F'G'H'$ is reflected across the line $y=x+2$ to $E''F''G''H''$. Find the ordered pair of $H''$ in the coordinate plane.

Solution: \boxed{(-4, 7)}


What is the least positive integer that can be added to 625 to make the sum a multiple of 5?


Find the $150^{\text{th}}$ term of the sequence consisting of all positive integers which are either powers of 3 or sums of distinct powers of 3.


In a linear arrangement, points $A$, $B$, $C$, $D$, $E$, and $F$ are aligned in a straight line such that the distances are $AB = 2$, $BC = 4$, $CD = 6$, $DE = 4$, and $EF = 2$. Semi-circles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ form a shape as illustrated. Calculate the area of the shaded region that is structured similarly (largest diameter minus two smaller ones, and plus three other smaller ones).


Let $O$ and $H$ denote the circumcenter and orthocenter of a right triangle $ABC$ with $∠C = 90^\circ$. If $BO = BH$, determine all possible values of $∠B$ (in degrees).


Circle $D$ has a radius of 10 cm. How many square centimeters are in the area of the largest possible inscribed equilateral triangle within circle $D$?


A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $143$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$


I have a drawer with 3 shirts, 7 pairs of shorts, and 4 pairs of socks. If I reach in and randomly remove four articles of clothing, what is the probability that I get two shirts, one pair of shorts, and one pair of socks?


Convert the point $(3 \sqrt{3}, 9, -3)$ from rectangular coordinates to spherical coordinates. Provide your answer in the form $(\rho, \theta, \phi)$, where $\rho > 0$, $0 \leq \theta < 2\pi$, and $0 \leq \phi \leq \pi$.

Solution: \boxed{\left(3\sqrt{13}, \frac{\pi}{3}, \cos^{-1}\left(-\frac{1}{\sqrt{13}}\right)\right)}


Convert the point \((7, -7, 4)\) in rectangular coordinates to cylindrical coordinates. Enter your answer in the form \((r, \theta, z)\), where \(r > 0\) and \(0 \le \theta < 2\pi\).

Solution: \boxed{(7\sqrt{2}, \frac{7\pi}{4}, 4)}


Given that \(x^2 + y^2 + z^2 = 1\), what is the largest possible value of \(|x| + |y| + |z|\)?


Consider a 4x4 grid of points (equally spaced). How many rectangles, of any size, can be formed where each of its four vertices are points on this grid?


Convert the point $(-3,4)$ in rectangular coordinates to polar coordinates. Enter your answer in the form \((r,\theta),\) where \( r > 0 \) and \( 0 \leq \theta < 2\pi \).

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


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


Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit primes ``$WX$'' and ``$YZ$'' where the digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible?


Let the function $g(x)$ be defined for all real numbers $x$ and satisfy the conditions $g(1) = 1$ and
\[ g(x + y) = 4^y g(x) + 3^x g(y) \]
Find the function $g(x)$.

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


The Fibonacci sequence is defined as 1, 1, 2, 3, 5, $\ldots$ where each term after the first two is the sum of the previous two terms. What is the remainder when the $75^{th}$ term of the sequence is divided by 9?


Triangle $ABC$ has vertices $A(0,10)$, $B(4,0)$, $C(10,0)$. A vertical line intersects line segment $\overline{AC}$ at $R$ and $\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\triangle RSC$ is 15, determine the positive difference of the $x$ and $y$ coordinates of point $R$.

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


What is the smallest value of $x$ that satisfies the equation $10x^2 - 48x + 44 = 0$? Express your answer as a decimal.


Fifty 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 $\log_2 n.$


Calculate the square root of $\dfrac{9!}{90}$.


At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make?


Let's imagine Emma is baking oatmeal raisin cookies for a school event. If she divides them equally among 13 of her friends, she'll have 7 cookies left. If she divides them equally among 8 of her friends, she'll have 5 cookies left. Assuming that Emma made fewer than 150 cookies, what is the sum of the possible numbers of cookies that she could have made?


If \( z \) is a complex number such that
\[
  z + z^{-1} = 1,
\]
what is the value of
\[
  z^{1000} + z^{-1000} \, ?
\]


In $\triangle ABC$, we have $AC=BC=7$ and $AB=4$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=9$. What is $BD$?


Points \( P, Q, R, \) and \( S \) lie on a line, in that order. If \( PQ = 3 \) units, \( QR = 6 \) units, and \( PS = 20 \) units, what is the ratio of \( PR \) to \( QS \)?


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


Solution: \boxed{75x^7(1 - 4x^6)}


Zenobia is planning a cookout and wishes to buy hot dogs, which come in packages of 10, and buns, which come in packages of 12. She wants to buy an equal number of hot dogs and buns while also minimizing the total packaging used. Each package of hot dogs generates 0.4 units of waste, and each package of buns produces 0.3 units of waste. What is the minimal number of hot dog packages she must buy and the least waste generated?


Consider the following set of equations in a plane:
1. \( 3y - 2x = 3 \)
2. \( x + 2y = 2 \)
3. \( 6x - 9y = 6 \)

Determine the number of distinct points formed at the intersection of at least two of these lines.


What is the modular inverse of $17$, modulo $1200$?

Express your answer as an integer from $0$ to $1199$, inclusive.


If triangle $ABC$ has sides of length $AB = 8,$ $AC = 7,$ and $BC = 5,$ then find the value of:
\[\frac{\cos \frac{A + B}{2}}{\sin \frac{C}{2}} - \frac{\sin \frac{A + B}{2}}{\cos \frac{C}{2}}.\]


We flip a fair coin 12 times. What is the probability that we get heads in exactly 9 of the 12 flips, given that the first flip is a tail?


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


In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC =2:3$. Point $F$ is on $AD$ such that $AF:FD=2:1$. If the area of $\triangle DEF$ is 24, determine the area of $\triangle ABC$.


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


Palindromic primes are two-digit prime numbers where both the number and the number formed by reversing its digits are prime. Additionally, each digit of the number itself must be a prime number. What is the sum of all such palindromic primes that are less than 100?


What is the twelfth term in the arithmetic sequence \(1, \frac{3}{2}, 2, \dots\)?


Three different integers are randomly chosen from the set $$\{ -5, -8, 7, 4, -2, 3 \}$$. What is the probability that their product is negative? Express your answer as a common fraction.


Calculate the number of even perfect square factors of the number \(2^6 \cdot 7^{12} \cdot 3^2\).


Determine the number of possible increasing arithmetic sequences that can represent the interior angles of a convex octagon, where no angle is exactly $180^\circ$ and all angles are positive integers less than $160^\circ$ degrees. Assume the octagon is not equiangular.


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


Rationalize the denominator of $\displaystyle \frac{45}{\sqrt[3]{45}}$.


In rhombus $EFGH$, angle $F$ measures $130^\circ$. What is the number of degrees in the measure of angle $H$?


The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 41$, $b_{10} = 101$, and for all $n \ge 3$, $b_n$ is the arithmetic mean of the first $n-1$ terms. Find $b_2$.


The perimeter of a parallelogram is 60 units, and the length of one side is always three times the length of the other side. What is the largest possible area of this parallelogram?


Given that \(a\) is a multiple of \(720\), find the greatest common divisor of \(a^2 + 8a + 18\) and \(a + 6\).


When $\sqrt[4]{2^8\cdot3^2\cdot5^3}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?


Given the function $g$ defined by the points:
\[ g(1)=8, g(2)=7, g(3)=6, g(4)=5, g(5)=4, g(6)=3, g(7)=2, g(8)=1, g(9)=0 \]
Alex writes the number 2 on his thumb. He then applies $g$ to 2 and writes the resulting output on his index finger. If Alex continues this process of applying $g$ to the previously written number and writing the output on a new finger, what number will Alex write on his twelfth finger?


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


Let \(PQRS\) be a tetrahedron where edges \(PQ\), \(PR\), and \(PS\) are mutually perpendicular. Let the areas of triangles \(PQR\), \(PRS\), and \(PQS\) be represented by \(a\), \(b\), and \(c\), respectively. Calculate the area of triangle \(QRS\) in terms of \(a\), \(b\), and \(c\).

Solution: \boxed{\sqrt{a^2 + b^2 + c^2}}


Let $A = 3 - 4i$, $M = -3 + 2i$, $S = 2i$, and $P = -1$. Find the result of $A - M + S + P$.

Solution: \boxed{5 - 4i}


Find the value of $\frac{\frac{1}{4} + \frac{1}{6}}{\frac{3}{8} - \frac{1}{3}}$. Express your answer as a fraction in simplest form.


Find the number of ordered 19-tuples \((a_1, a_2, a_3, \dots, a_{19})\) of integers, such that the square of any number in the 19-tuple is equal to the sum of the squares of the other 18 numbers.


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


Suppose $\cos S = 0.5$ in the diagram below. What is $ST$?

[asy]
pair P,S,T;
P = (0,0);
S = (6,0);
T = (0,6*tan(acos(0.5)));
draw(P--S--T--P);
draw(rightanglemark(S,P,T,18));
label("$P$",P,SW);
label("$S$",S,SE);
label("$T$",T,N);
label("$10$",S/2,S);
[/asy]


How many positive integers, not exceeding 150, are multiples of 3 or 4 but not 6?


How many ways are there to put 5 indistinguishable balls into 3 indistinguishable boxes, with the condition that each box must contain at least one ball?


A domino set is modified such that each domino still consists of two squares, but now each integer from 0 to 12 is paired with every other integer from 0 to 12 exactly once, forming a complete set. A $\textit{double}$ is a domino that has the same integer on both squares. What is the probability that a domino randomly selected from this set will be a $\textit{double}$? Express your answer as a common fraction.


You are given that $x$ is directly proportional to $y^3$, and $y$ is inversely proportional to $z^2$. If the value of $x$ is 2 when $z$ is $4$, what is the value of $x$ when $z$ is equal to $48$? Express your answer as a common fraction.

Solution: \boxed{x = \frac{1}{1492992}}


In a right triangle with integer length sides, the hypotenuse has length 39 units. Determine the lengths of the legs.


Rectangle \( EFGH \) is 10 cm by 6 cm. \( P \) is the midpoint of \( \overline{EF} \), and \( Q \) is the midpoint of \( \overline{GH} \). Calculate the area of region \( EPGQ \).

**


Given a hexagon with certain angles given as $135^\circ$, $105^\circ$, $87^\circ$, $120^\circ$, and $78^\circ$, what is the measure of the unknown angle $Q$?


Express $\sin 3x + \sin 7x$ as a product of trigonometric functions.

Solution: \boxed{2 \sin 5x \cos 2x}


All squares have the same value, and all triangles have the same value. What is the sum of three squares?
\[
\bigtriangleup + \square + \bigtriangleup + \square + \bigtriangleup = 27
\]
\[
\square + \bigtriangleup + \square + \bigtriangleup + \square = 25
\]
\[
\square + \square + \square = \ ?
\]


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


How many three-digit numbers are multiples of neither 4 nor 9?


Let \( x \) be a real number such that \( x + \frac{1}{x} = 5 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_6 \).


In $\triangle ABC$, $AC = BC = 10$ and $AB = 4$. Suppose $D$ is a point on line $AB$ such that $B$ is between $A$ and $D$, and $CD = 12$. Find the length of $BD$.

Solution: \boxed{BD = 4\sqrt{3} - 2}


Given that $y$ is a positive integer less than 150, determine how many solutions does the congruence $y + 21 \equiv 79 \pmod{46}$ have?


In rectangle $ABCD$, $AB = 10$ cm, $BC = 5$ cm, and $DG = DH$. The area of triangle $DGH$ is one-fifth the area of rectangle $ABCD$. Determine the length in centimeters of segment $GH$. Express your answer in simplest radical form.

[asy]
draw((0,0)--(0,50)--(100,50)--(100,0)--cycle);
draw((40,50)--(100,20));
label("$A$",(0,50),W);
label("$B$",(0,0),W);
label("$C$",(100,0),E);
label("$D$",(100,50),E);
label("$G$",(40,50),N);
label("$H$",(100,20),E);
[/asy]

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


Find the smallest positive integer $x$ that, when divided by 5, 7, and 8, has remainders of 4, 6, and 7 respectively.


Solution: \boxed{-1 + \frac{1}{8388608}}


The sequence $b_1, b_2, b_3, \dots$ is defined such that $b_1 = 10$, $b_9 = 45$, and for all $n \geq 3$, each $b_n$ is the geometric mean of the first $n-1$ terms. Find $b_2$.


A math teacher has adjusted the homework policy for Noelle. For the first three homework points, she does not need to do any homework assignments. Starting from the fourth point, for each of the next five homework points, Noelle has to do one homework assignment; for each of the next five points, she needs to do two homework assignments; and so on, so that to earn the $n^{\text{th}}$ homework point, she has to do $(n-3)\div5$ (rounded up) homework assignments. For instance, when she has 11 points, it will take $(12-3)\div5=1.8\rightarrow2$ homework assignments to earn her $12^{\text{th}}$ point. What is the smallest number of homework assignments necessary to earn a total of 25 homework points?


Determine the equation of the oblique asymptote for the function $\frac{3x^2 + 8x + 12}{3x + 4}$.

Solution: \boxed{y = x + \frac{4}{3}}


In right triangle $BCD$ with $\angle D = 90^\circ$, we have $BC = 15$ and $BD = 6$. Find $\sin B$ and the area of triangle $BCD$.


There are two sets of containers. Set 1 includes Container A holding 2 red balls and 8 green balls; Containers B and C each hold 8 red balls and 2 green balls. Set 2 includes Container A holding 8 red and 2 green balls, and Containers B and C each holding 2 red and 8 green balls. A fair coin is flipped to determine which set of containers will be used. A container within that set is then randomly selected and a single ball is then drawn from the selected container. Calculate the probability that the ball drawn is green. Express your answer as a common fraction.


In the triangle below, \( \angle A\) needs to be the largest angle. For this to happen, the side lengths are given by \( 2x + 5\), \(4x - 3\), and \(x + 15\), where \(2x+5\) corresponds to \(AB\), \(4x-3\) corresponds to \(AC\), and \(x+15\) corresponds to \(BC\). Find the least value of the difference \(n-m\) so that \(m < x < n\), expressed as a common fraction.


For each value of $x$, let $f(x)$ be the minimum value of the three different expressions $3x + 1$, $x + 3$, and $-x + 9$. Find the maximum value of $f(x)$.


What is the largest four-digit number whose digits add up to 19?


What is the largest digit $N$ for which $5789N$ is divisible by 6?


The equation \(x^2 - 6x + 11 = 23\) has two solutions, \(a\) and \(b\), with \(a \geq b\). What is the value of \(a + 3b\)?

Solution: \boxed{12 - 2\sqrt{21}}


Express $\frac{0.\overline{812}}{2.\overline{406}}$ as a common fraction.


How many positive 3-digit even numbers are divisible by 9?


$f(x)$ has domain $\{0, 1, 2, 3\}$ and range $\{1, 3, 5, 7\}$. $g(x)$ has domain $\{2, 3, 4, 5\}$ and is defined by $g(x) = x + 2$. What is the sum of all possible values of $g(f(x))$?


Sandy is planning to weatherproof her 10' by 7' backyard shed consisting of a single 10 ft by 7 ft wall and two slanted 10 ft by 6 ft roof sections. The siding she desires is available only in 10-foot by 14-foot sheets, each costing $50. How much will Sandy spend for the siding?


Find the distance and the midpoint between the points $(-3, -1)$ and $(9, 4)$.

Solution: \boxed{(3, 1.5)}


Find the greatest common divisor of $51457$ and $37958$.


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?


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


Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=15$. What is the maximum value of \[2A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A?\]


A circle is centered at point \( O \) and is circumscribed around triangle \( DEF \) as shown:
- \( \angle DOE = 130^\circ \)
- \( \angle EOD = 90^\circ \)

What is the measure of \( \angle DEF \) and \( \angle EDF \), in degrees?


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 17 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of $n$?


A palindrome is a number that reads the same forward and backward. What is the second smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base? Give your response in base 2.


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


A pet store has 15 puppies, 10 kittens, and 8 hamsters. Alice, Bob, and Charlie each want to buy a pet. For variety, they each prefer to buy a different type of pet. How many ways can Alice, Bob, and Charlie buy their preferred pets and leave the store satisfied assuming they all pick a different type of pet?


If $x-y=17$ and $xy=6$, what is the value of $x^2+y^2$?


In a new game show $\text{\emph{Wheel of Fortune Plus}}$, the wheel is divided into six equal sections with the values: Bankrupt, $\$2000$, $\$500$, $\$6000$, $\$700$, and $\$300$. What is the probability that the contestant will earn exactly $\$2700$ in their first three spins?


Joe and JoAnn each bought 14 ounces of coffee in a 20-ounce cup. Joe drank 3 ounces of his coffee and then added 4 ounces of cream. JoAnn added 4 ounces of cream, stirred the coffee well, and then drank 4 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? Express your answer as a fraction.


Let \( a \bowtie b = a + \sqrt{b + \sqrt{b + 1 + \sqrt{b + 1 + \sqrt{b + 1 + \ldots}}}} \). If \( 5 \bowtie h = 8 \), find the value of \( h \).

Solution: \boxed{9 - \sqrt{10}}


Three cards are dealt sequentially from a standard deck of 52 cards. What is the probability the first card is a $\diamondsuit$, the second card is a $\spadesuit$, and the third card is a $\heartsuit$?


In triangle $ABC$, $\angle C=90^\circ$, $AC=8$ and $BC=12$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED=90^\circ$. If $DE=6$, then what is the length of $BD$?


Rodney is trying to guess a secret number again with updated clues:
\begin{tabular}{ c }
It is a two-digit integer.\\
The tens digit is even.\\
The units digit is odd.\\
The number is greater than 75.
\end{tabular}
If Rodney guesses a number that has each of these properties, what is the probability that Rodney will guess the correct number? Express your answer as a common fraction.


If $9:x^3 = x:27$, what is the value of $x$?


Points $P$ and $Q$ are midpoints of adjacent sides of a rectangle with length $l$ and width $w$. What fraction of the interior of the rectangle is shaded if a white triangle is formed by vertex $A$ (one of the corners of the rectangle) and points $P$ and $Q$?

[asy]
filldraw((0,0)--(4,0)--(4,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,1)--(2,2)--(0,2)--(0,1)--cycle,white,linewidth(1));
label("P",(0,1),W);
label("Q",(2,2),N);
label("A",(0,2),NW);
[/asy]


For how many non-negative real values of $x$ is $\sqrt{225-\sqrt[3]{x}}$ an integer?


Let \(a,\) \(b,\) and \(c\) be angles such that
\begin{align*}
\cos a &= \tan b, \\
\cos b &= \tan c, \\
\cos c &= \tan a.
\end{align*}
Find the largest possible value of \(\sin a.\)

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


Let $C$ be the circle with equation $x^2 - 4y - 36 = -y^2 + 12x + 16$. Find the values of $a$, $b$, and $r$ where $(a, b)$ is the center of $C$ and $r$ is its radius. What is $a + b + r$?

Solution: \boxed{8 + 2\sqrt{23}}


How many rows of Pascal's Triangle contain the number $47$?


If the product $(4x^2 - 6x + 5)(9 - 3x)$ can be expanded and written in the form $ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are real numbers, then compute the value of $9a + 3b + 2c + d$.


Mady has an infinite number of balls and empty boxes available to her. Each empty box can now hold eight balls, and the boxes are again arranged in a row from left to right. At the first step, she places a ball into the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. Determine the total number of balls in the boxes as a result of Mady's $2500$th step, given that each box now corresponds to a digit in nonary (base 9) representation.


A particle is located at $(10,0)$ on the coordinate plane. Define a move for the particle as a counterclockwise rotation of $\frac{\pi}{3}$ radians about the origin, followed by a translation of $6$ units in the positive $x$-direction. Determine the particle's position after $12$ moves.

Solution: \boxed{(10, 0)}


The base of a rectangular piece of paper $ABC$ is $18\text{ cm}$ long. When the paper is folded from the top vertex, the crease line $DE$ is parallel to the base. The area of the triangle formed below the base after the fold is $9\%$ of the original triangle $ABC$. Calculate the length of $DE$.

Solution: \boxed{5.4\text{ cm}}


How many prime numbers have squares between 8000 and 12000?


The pages of a book are numbered from 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 2550. What was the number of the page that was added twice?


Each week, Tom visits the supermarket and purchases: 4 apples at $2 each, 6 eggs at $0.50 each, 3 loaves of bread at $3 each, 2 packs of cheese at $6 each, and a $8 pack of chicken. This week, the store has a sale with apples priced at $1 each. Tom also has a coupon for $10 off any order of $40 or over. How much money does Tom spend on this shopping trip?


Square $EFGH$ has its center at $(10, -10)$ and an area of 16 square units. The top side of the square is horizontal. The square is then dilated with the center of dilation at $(1, -1)$ and a scale factor of 3. What are the coordinates of the vertex of the image of square $EFGH$ that is nearest to the origin? Give your answer as an ordered pair.

Solution: \boxed{(22, -22)}


What is the greatest number of Mondays that can occur in the first $51$ days of a year?


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


Let $b$ be a number such that $(3b+6)(b-2)=9b.$ What is the largest possible value of $b$? Express your answer as a common fraction.


Determine the value of $b$ such that the expression $\log_{y^8}{x^2}\cdot\log_{x^7}{y^4}\cdot\log_{y^6}{x^5}\cdot\log_{x^5}{y^6}\cdot\log_{y^4}{x^7}$ can be written as $b\log_y{x}$.


Consider the graph of the equation
\[
\sqrt{x^2 + y^2} + |y+2| = 5.
\]
It consists of portions of two parabolas. Compute the distance between the vertices of the parabolas.


Let $z = x + yi$, where $x$ and $y$ are positive integers, and suppose that
\[z^3 = -26 + di,\] for some integer $d$. Find $z$.

Solution: \boxed{1 + 3i}


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{2y} = \frac{1}{8}$, what is the least possible value of $xy$?


You own 8 shirts, 4 pairs of pants, and 8 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, plus white, yellow, red, and green. You refuse to wear an outfit where all three items are the same color or where any two items of the same type (e.g., two shirts or two hats) are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do you have?


The values of $f$, $g$, $h$, and $j$ are 9, 10, 11, and 12, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$, and $fj$?


A box contains 6 white balls and 6 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate in color?


$ABCD$ is a rectangle whose area is 20 square units. How many square units are contained in the area of trapezoid $EFBA$, where $E$ and $F$ are points on the segment $DC$ such that $DE = 1$ unit and $CF = 2$ units?

[asy]
size(4cm,4cm);
for(int i=0; i < 5; ++i){
for(int k=0; k < 6; ++k){
draw((0,i)--(5,i));
draw((k,0)--(k,4));
} }
draw((0,0)--(1,4));
draw((3,4)--(5,0));
label("$A$",(0,0),SW);
label("$B$",(5,0),SE);
label("$C$",(5,4),NE);
label("$D$",(0,4),NW);
label("$E$",(1,4),N);
label("$F$",(3,4),N);
[/asy]


My four friends and I have a weekly book club meeting. Each week, three of us are responsible for discussing a new book while the other two prepare snacks. How many different ways are there for us to choose who leads the discussion and who prepares the snacks?


How many four-digit positive integers have at least one digit that is a 1 or a 7?


Calculate the value of $$n \mod 9$$ where $$n = 2 + 333 + 5555 + 77777 + 999999 + 2222222 + 44444444 + 666666666$$ and determine the congruence of this sum modulo 9.


An equilateral triangle has two vertices at $(2,7)$ and $(10,7)$. If the third vertex is in the first quadrant, what is the y-coordinate?

Solution: \boxed{7 + 4\sqrt{3}}


From a group of ten students, how many different five-student committees can be formed? Furthermore, if two different five-student committees are chosen, in how many ways can they be chosen such that they share exactly three members?


Find $q(x)$ if the graph of $\frac{x^3 - 2x^2 - 5x + 6}{q(x)}$ has vertical asymptotes at $2$ and $-1$, no horizontal asymptote, and $q(3) = 18$.

Solution: \boxed{4.5x^2 - 4.5x - 9}


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

Solution: \boxed{\begin{pmatrix} -4 \\ 17 \end{pmatrix}}


If the six solutions of \( x^6=-729 \) are written in the form \( a+bi \), where \( a \) and \( b \) are real, find the product of those solutions with \( a>0 \).


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 40 dips?

Solution: \boxed{18.75 \text{ daps}}


How many integers $n$ satisfy the condition $200 < n < 300$ and the condition $n$ has the same remainder when divided by $7$ and by $9$?


If triangle $DEF$ has sides of length $DE = 7,$ $DF = 8,$ and $EF = 5,$ then calculate
\[
\frac{\cos \frac{D - E}{2}}{\sin \frac{F}{2}} - \frac{\sin \frac{D - E}{2}}{\cos \frac{F}{2}}.
\]


Calculate the total number of digits used when the first 1500 positive even integers are written.


Let $a,$ $b,$ $c$ be integers such that
\[\mathbf{A} = \frac{1}{7} \begin{pmatrix} -5 & a \\ b & c \end{pmatrix}\]
and $\mathbf{A}^2 = \mathbf{I}.$  Find the largest possible value of $a - b + c.$


Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(10, 0)$.  A line through $B$ cuts the area of $\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line.


If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{z}$ where $z$ is an integer such that $0<z\le12$, then what is the largest possible value of $a+b+c$?


Ellie Elephant begins walking on a number line at 0, aiming to get to 5 meters away. She can walk $\frac{1}{4}$ of the remaining distance to the target with every step. Each step reduces her stamina so she can only walk $\frac{1}{4}$ of the distance left for the next step. How far has Ellie walked after six steps?


Rosie can make two pies from nine apples. She has thirty-six apples now. How many pies can she make, and will there be any apples left?


A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. Find the solution to the equation $2024 \,\diamondsuit\, (8\,\diamondsuit\, x)=200.$


How many different counting numbers will each leave a remainder of 7 when divided into 53?


What is the remainder when \(2,468,135,790\) is divided by \(99\)?


An alien creature tells me that it has laid $215_7$ eggs. In base 10, how many eggs did it lay?


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


What is the value of $y$ in the equation $\frac{3}{y} - \frac{5}{y} \div \frac{7}{y} = 1.2$?


Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and $ab + ac + ad + bc + bd + cd = 20$. Find the largest possible value of $a$.

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


On the first day, Barry Potter used his magic wand to increase the length of an object by $\frac{1}{2}$. If the initial length was \( x \), it became \( x + \frac{1}{2}x \). On each subsequent day, Barry increased the object's most recent length by $\frac{1}{k+1}$ where \( k \) is the number of the day starting from \( k = 1 \) on day one. Barry continued this pattern, and on the \( n^\text{th} \) day, the object's length became exactly 50 times its original length. Determine \( n \).


When three standard dice are rolled and the numbers $a, b, c$ appear on their top faces, find the probability that the product $abc = 72$ and the sum $a+b+c=13$.


What is the 415th digit to the right of the decimal point when $\frac{17}{39}$ is expressed as a decimal?


Two chords, $EF$ and $GH$, intersect inside a circle at point $Q$. If $EQ = 5$ and $GQ = 12$, find the ratio $\frac{FQ}{HQ}$ if $HQ = 3$.


Given two similar triangles, \(\triangle ABC\) and \(\triangle DBE\), where \(BC = 25\) cm and \(BE = 15\) cm. The length \(AC\) is 18 cm. How many centimeters long is \(DE\)? Express your answer as a decimal to the nearest tenth.

Solution: \boxed{10.8\text{ cm}}


What is the probability that Sarah gets fewer tails than heads if she flips 8 coins?


Given that $\log_{10} \sin x + \log_{10} \cos x = -2$ and that $\log_{10} (\sin x + \cos x)^2 = \log_{10} n - 1,$ find $n.$


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


Determine the area, in square units, of triangle $PQR$, where the coordinates of the vertices are $P(-3, 4)$, $Q(4, 9)$, and $R(5, -3)$.


Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - n \]for all \( n > 1 \). Find the smallest possible value of \( a_1 \).


To make lemonade, the recipe requires a ratio of $5$ parts water to $2$ parts lemon juice. If I want to make 1.5 gallons of lemonade, and there are 4 quarts in a gallon, how many cups of water does the recipe require?

Solution: \boxed{\frac{120}{7} \text{ cups}}


A natural number is abundant if it is less than the sum of its proper divisors. What is the smallest even abundant number greater than 12?


Maria measures the radius of a circular field as part of her study on crop circles. The actual radius of the field is 15 cm, but Maria's measurement has an error of up to 10%. What is the largest possible percent error in Maria's computed area of the circle?


Find the modular inverse of $28$ modulo $29$.


If $(5, 20)$ and $(15, -6)$ are the coordinates of two opposite vertices of a parallelogram, what is the sum of the $y$-coordinates of the other two vertices?


Mrs. Thompson decides to award extra credit in her class to students whose scores exceed the average score of the class on a certain test. If there are 150 students in her class, what is the maximum number of students who can receive extra credit?


A musical has three different male roles, three different female roles, and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If seven men and eight women audition, in how many ways can the eight roles be assigned?


Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^4 - 10x^3 + px^2 - qx + 24 = 0\] are four distinct positive integers. Compute $p + q.$


Suppose the function $g(x)$ is defined on the domain $\{x_1, x_2, x_3\}$, such that the graph of $y=g(x)$ consists of three points which form a triangle of area $50$. What is the area of the triangle formed by the points in the graph of $y=3g(3x)$?


Find $x$ if
\[2 \arctan \frac{1}{3} + \arctan \frac{1}{15} + \arctan \frac{1}{x} = \frac{\pi}{4}.\]


Let $\omega$ be a nonreal root of $z^4 = 1$ that is not a root of $z^2 = 1$. Find the number of ordered pairs $(a,b)$ of integers such that $|a \omega + b| = 1$.


A point is selected at random from the portion of the number line from 0 to 7. What is the probability that the point is closer to 6 than to 1? Express your answer as a decimal to the nearest tenth.


What is the value of $\frac12\cdot\frac91\cdot\frac18\cdot\frac{64}{1}\cdot\frac{1}{128}\cdot\frac{729}{1}\cdot\frac{1}{2187}\cdot\frac{19683}{1}$?


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


Let $p, q, r,$ and $s$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 2500, \\
pq &=& rs &=& 1152.
\end{array}
\]
If $T=p+q+r+s$, compute the value of $\lfloor T \rfloor$.


How many integers are between $(10.4)^3$ and $(10.5)^3$?


How many pairs of positive integers \((x, y)\) satisfy \(x^2 - y^2 = 45\)?


City School UK has 180 students, with 100 boys and 80 girls. If $\frac{1}{5}$ of the boys and $\frac{1}{4}$ of the girls are absent one day, what percent of the total student population is absent?


A Senate committee has 6 Democrats and 6 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table if two specific politicians, one Democrat and one Republican, must not sit next to each other?


Points $X$, $Y$, $Z$, and $W$ lie on a straight line in that order. If $XY=3$ units, $YZ=4$ units, and $XW=20$ units, what is the ratio of $XZ$ to $YW$? Additionally, find the ratio of $YZ$ to $XW$.


The area of a square equals the square of the length of one of its sides. The perimeter of a square equals four times the length of one side. The sum of the areas of two squares is 145, while the difference in their areas is 85. Find the sum of the diagonals of both squares.

Solution: \boxed{\sqrt{230} + \sqrt{60}}


The graph of the rational function $\frac{1}{q(x)}$ has vertical asymptotes at $x = -2$ and $x = 3$. If $q(x)$ is a quadratic and $q(1) = 8$, find $q(x)$.

Solution: \boxed{q(x) = -\frac{4}{3}x^2 + \frac{4}{3}x + 8}


A root of unity is a complex number that satisfies $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that also satisfy $z^3 + az^2 + bz + c = 0$ for integers $a$, $b$, and $c$.


The measure of each exterior angle of a regular polygon is $24$ degrees. What is the sum of the measures of the interior angles of the polygon, in degrees? Also, calculate the total number of diagonals in the polygon.


What is the largest four-digit integer $n$ that satisfies $$45n \equiv 180 \pmod{315}~?$$


How many values of $x$, $-50<x<120$, satisfy the equation $\cos^2 x + 2\sin^2 x = 1?$ (Note: $x$ is measured in radians.)


Find the smallest digit $d$ so that $457{,}d01$ is divisible by 11.


In a set of six consecutive integers, the second largest integer is less than three times the average of the six integers. What is the smallest integer that could be in the set?


The Gnollish language consists of 3 words, "splargh", "glumph", and "amr". In a sentence, "splargh" cannot come directly before "glumph", and "amr" cannot come directly after "glumph". How many valid 3-word sentences are there in Gnollish?


A regular hexagon is inscribed in a circle of radius 3 units. In square units, what is the area of the hexagon, and what is its perimeter?


How many unordered pairs of prime numbers have a sum of 60?


What is the largest three-digit multiple of 6 whose digits' sum is 15?


How many positive real solutions are there to $x^8 + 5x^7 + 10x^6 + 1728x^5 - 1380x^4 = 0$?


What is the domain of the function \( f(x) = \log_5(\log_7(\log_4x - 1)) \)?

Solution: \boxed{(16, \infty)}


A sphere is inscribed in a right cone with base radius \(9\) cm and height \(27\) cm. Determine the radius of the sphere, expressed in the form \(a\sqrt{c} - b\). What is the value of \(a + b + c\)?