I want to create a license plate that is 4 characters long, where the first character is a digit, the second and third characters are letters, and the fourth character is a digit. I also require that the first and last digits are the same. How many ways can I choose such a license plate?


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

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


Determine the value of $n$ such that the quantity
\[\frac{\tan \frac{\pi}{7} + i}{\tan \frac{\pi}{7} - i}\] is a fourteenth root of unity, i.e., it equals $\cos \frac{2n \pi}{14} + i \sin \frac{2n \pi}{14}$ for some integer $n$ between 0 and 13 inclusive.


Consider a year 2015, which is a non-leap year like 2007. Determine the number of prime dates in 2015 where both the month and the day are prime numbers.


You have five shirts and six pairs of pants. Out of the six pairs of pants, three are formal and the rest are casual. Each formal pant can only be paired with one of the first three shirts. How many different outfits, consisting of one shirt and one pair of pants, can you make?


Find $x$ such that $\log_x 16 = \log_{64} 4$.


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


Find the number of seven-digit palindromes where the leading digit is either 1 or 2.


What is the greatest common factor of all three-digit palindromes? (Note: A palindrome is a number that reads the same forwards as backwards.)


Two lines $p$ and $q$ are parallel, and the measure of $\angle A$ is $\frac{1}{4}$ the measure of $\angle B$. Determine the degree measure of $\angle C$. Here, $\angle C$ is an alternate interior angle with respect to $\angle A$ when considering a transversal intersecting $p$ and $q$.


Assume a square ABCD. Points P and Q are located one third along AB from A, and two thirds along BC from B, respectively. What fraction of the interior of the square is shaded if the region inside the square but outside the triangle APQ is shaded?

[asy]
filldraw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,1)--(2,3)--(3,3)--(0,1)--cycle,white,linewidth(1));
label("P",(0,1),W);
label("Q",(2,3),N);
[/asy]


Given that \(x + \sin y = 2010\) and \(x + 2010 \cos y = 2009\), where \(0 \leq y \leq \frac{\pi}{2}\), find the value of \(x + y\).

Solution: \boxed{2009 + \frac{\pi}{2}}


A list of five positive integers has all of the following properties:
- The only integer in the list that occurs more than once is $7$.
- Its median is $10$.
- Its average (mean) is $12$.

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


The points $(0,0)$, $(a,15)$, and $(b,45)$ are the vertices of an equilateral triangle. Find the value of $ab$.


What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{137}$?


Three cards are drawn sequentially from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card is a 4 of any suit, the second card is a 5 of the same suit as the first card, and the third card is a 6 also of the same suit as the first and second cards?


Find the number of complex solutions to
\[\frac{z^4 - 1}{z^3 - 3z + 2} = 0.\]


Determine the positive difference between the two roots of the quadratic equation $5x^2 - 9x - 22 = 0$. Express your answer in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are integers and $p$ is not divisible by the square of any prime number. Find $p + q$.


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


In the coordinate plane, the curve $xy = 2$ intersects a circle at four points, three of which are $\left( 4, \frac{1}{2} \right),$ $\left( -2, -1 \right),$ and $\left( \frac{2}{3}, 3 \right).$ Find the fourth point of intersection.

Solution: \boxed{\left(-\frac{3}{4}, -\frac{8}{3}\right)}


The graph of the parabola $y = 3x^2 - 9x + 4$ has a $y$-intercept $(0,d)$ and two $x$-intercepts $(e,0)$ and $(f,0)$. Find $d + e + f$.


Of the five points (3, 11), (7, 19), (13, 32), (18, 38), and (21, 48), what is the sum of the $x$-coordinates of the points that lie in the region above the line $y = 2x + 5$ in the coordinate plane?


There are \(30\) different complex numbers \(z\) such that \(z^{30}=1\). For how many of these is \(z^{10}\) a real number?


Find the common ratio and the sum of the first three terms of the geometric series: $$\frac{4}{7}+\frac{16}{49}+\frac{64}{343}+\dots$$


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


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

\begin{tabular}{|c|c|c|c|c|c|}
\hline
& Bangkok & Cape Town & Honolulu & London & Tokyo \\ \hline
Bangkok & & 6300 & 6609 & 5944 & 5200 \\ \hline
Cape Town & 6300 & & 11,535 & 5989 & 9500 \\ \hline
Honolulu & 6609 & 11,535 & & 7240 & 4100 \\ \hline
London & 5944 & 5989 & 7240 & & 6000 \\ \hline
Tokyo & 5200 & 9500 & 4100 & 6000 & \\ \hline
\end{tabular}


Given a cylinder of fixed volume $V$, the total surface area (including the two circular ends) is minimized for a radius $R$ and height $H$. The surface area is given by $A = 2\pi R^2 + 2\pi RH$. If the volume of the cylinder is given by $\pi R^2 H = \frac{V}{2}$, find the ratio $\frac{H}{R}$.

Solution: \boxed{\frac{H}{R} = 2}


Let $x, y, z$ be positive real numbers such that $x + y + z = 2$. Find the minimum value of
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
\]


Solution: \boxed{x = \frac{27 + \sqrt{769}}{10} \text{ or } x = \frac{27 - \sqrt{769}}{10}}


Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane.  If $|p + q + r| = 48,$ find $|pq + pr + qr|.$


A $\textit{composite number}$ is a number that has two or more prime factors. The number 95 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?


A spinner is divided into 4 equal sections, each labeled with a different number: 1, 2, 5, and 6. You spin the spinner three times to form a three-digit number, where the outcomes from the first, second, and third spins represent the hundreds, tens, and units digits, respectively. What is the probability that the resultant three-digit number is divisible by 5? Express your answer as a common fraction.


Let $g(x) : \mathbb{R} \to \mathbb{R}$ be a function such that
\[\frac{g(x) g(y) - g(xy)}{4} = x + y + 3\] for all $x,$ $y \in \mathbb{R}.$ Find $g(x).$

Solution: \boxed{x + 4}


A sector is removed from a circular piece of paper with a radius of 6 inches to create a right circular cone. The sector removed constitutes \(300^\circ\) of the original circle. Determine the circumference of the base of the resulting cone, and express your answer in terms of \(\pi\).


Determine the sum of the squares of the medians of a triangle with side lengths 13, 14, and 15.


In Mr. Johnson's class, 12 out of 20 students received an 'A' grade and the rest received a 'B' grade. Mrs. Smith, teaching a different class, observed that the proportion of students getting 'A' was the same. If Mrs. Smith has 30 students total, how many students received an 'A' grade? Moreover, if the same proportion received 'B' as in Mr. Johnson’s class, how many students in Mrs. Smith’s class received 'B'?


The third and twenty-third terms of an arithmetic sequence are $\frac{2}{11}$ and $\frac{3}{7}$, respectively. What is the thirteenth term?


Solve the equation $x^2 + 14x = 84$ and find the positive solution in the form $\sqrt{c} - d$ for positive natural numbers $c$ and $d$. What is $c+d$?


Simplify $\tan \frac{\pi}{12} + \tan \frac{5\pi}{12}$.


In the diagram, $QRS$ is a straight line. What is the measure of $\angle RPS,$ in degrees? [asy]
pair Q=(0,0);
pair R=(1,0);
pair SS=(2,0);
pair P=(.6,0.9);

draw(P--Q--R--SS--P--R);

label("$Q$",Q,S);
label("$R$",R,S);
label("$S$",SS,S);
label("$P$",P,N);

label("$35^\circ$",Q+(.08,.04),NE);
label("$80^\circ$",P-(.02,.13),S);
label("$40^\circ$",SS+(-.28,.04),NW);

[/asy]


Consider a function $g$ defined on the complex numbers such that $g(z) = (c+di)z$, where $c$ and $d$ are real numbers. It is known that for any complex number $z$, $g(z)$ is equidistant from both $z$ and the origin. If $|c+di| = 10$, find $d^2$.


If \(\tan \alpha = 5\) and \(\tan \beta = 3,\) then find \(\tan (\alpha - \beta)\) and \(\tan (\alpha + \beta)\).


What is the shortest distance between the circles defined by \(x^2 - 6x + y^2 + 2y - 11 = 0\) and \(x^2 + 10x + y^2 - 8y + 25 = 0\)?

Solution: \boxed{\sqrt{89} - \sqrt{21} - 4}


Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 2,$ $\|\mathbf{c}\| = 3,$ and
\[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + 2\mathbf{b} = \mathbf{0}.\] Find the largest possible angle between $\mathbf{a}$ and $\mathbf{c},$ in degrees.


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


What is the coefficient of $x^3$ in the product of the polynomials $$x^4 - 2x^3 + 3x^2 - 4x + 5$$ and $$3x^3 - 4x^2 + x + 6$$ after combining like terms?


A "super ball" is dropped from a window 20 meters above the ground. On each bounce, it rises to 80% of the height reached in the previous bounce. How far has the ball traveled when it is caught as it reaches the high point after hitting the ground for the fourth time?


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

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


Suppose the function \( g(x) \) is defined for \( x \in \{a, b, c\} \) such that the points \( (a, g(a)), (b, g(b)), (c, g(c)) \) form a triangle with an area of \( 24 \). The graph of \( y = 3g(3x) \) likewise contains just three points. Determine the area of this triangle.


Evaluate $\lfloor -5 -.5 \rfloor \cdot \lceil 5 +.5 \rceil \cdot \lfloor -4 -.5 \rfloor \cdot \lceil 4 +.5 \rceil \cdot \lfloor -3 -.5 \rfloor \cdot \lceil 3 +.5 \rceil \cdot \lfloor -2 -.5 \rfloor \cdot \lceil 2 +.5 \rceil \cdot \lfloor -1 -.5 \rfloor \cdot \lceil 1 +.5 \rceil$.


A polygon $P_1$ with $n$ sides is drawn in the Cartesian plane. The sum of the x-coordinates of the $n$ vertices equals $3n$. The midpoints of the sides of $P_1$ form a second polygon $P_2$. Finally, the midpoints of the sides of $P_2$ form a third polygon $P_3$. Find the sum of the x-coordinates of the vertices of $P_3$.


Give an example of a quadratic function that has zeroes at $x=1$ and $x=5$, and that takes the value $24$ when $x=3$.

Solution: \boxed{-6x^2 + 36x - 30}


Calculate the value of $\frac{15!}{6!9!}$.


What is the matrix $\mathbf{N}$ that performs the transformation which sends square $EFGH$ to square $E'F'G'H'$? Assume $E$ goes to $E'$, and so on. The square $EFGH$ is rotated by $30^\circ$, scaled by 2, and then translated by the vector $(1, 2)$.

Solution: \boxed{\begin{pmatrix} \sqrt{3} & -1 & 1 \\ 1 & \sqrt{3} & 2 \\ 0 & 0 & 1 \end{pmatrix}}


Suppose $x$ and $y$ are positive real numbers such that $x^2 - 3xy + 4y^2 = 12$. Find the maximum possible value of $x^2 + 3xy + 4y^2$.


The graphs of a function $f(x) = 2x + b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-4, a)$. Given that $b$ and $a$ are both integers, what is the value of $a$?


The shaded region consists of 24 congruent squares arranged in a rectangular grid. If $PQ = 10$ cm, where $PQ$ represents the diagonal of the rectangle formed by a 6x4 arrangement of these squares, what is the area of the entire shaded region?

Solution: \boxed{\frac{600}{13} \text{ square cm}}


How many positive integers smaller than \(500,000\) are powers of \(2\), but are not powers of \(4\)? You may find it useful to know that \(2^9 = 512\).


What is the sum of the 2010 fractions of the form $\frac{3}{n(n+3)}$ for $n$ being the positive integers from 1 through 2010? Express your answer as a decimal to the nearest thousandth.


Evaluate the value of the expression $(45 + 15)^2 - (45^2 + 15^2 + 2 \times 45 \times 5)$.


Find the units digit of \(7^{7(13^{13})}\).


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


There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4,$ $b_5,$ $b_6,$ $b_7$ such that
\[\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta\]
for all angles $\theta.$ Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2.$


Find the product of all the divisors of $36$.


For some integers that are not palindromes, like 73, a person can create a palindrome by repeatedly reversing the number and adding the original number to its reverse. For example, $73 + 37 = 110$. Then $110+011 = 121$, which is a palindrome. Of all positive integers between 10 and 100, what is the sum of the non-palindrome integers that take exactly five steps to become palindromes?


Let $z$ and $w$ be complex numbers such that $|z| = 2$ and $|w| = 4$. If $|z+w| = 3$, what is $ \left | \frac{1}{z} + \frac{1}{w} \right| $?


Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be vectors with $\|\mathbf{a}\| = 5$, $\|\mathbf{b}\| = 8$, and $\|\mathbf{c}\| = 6$. If $\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} = 0$, find all possible values of $\mathbf{a} \cdot \mathbf{b}$.

Solution: \boxed{[-40, 40]}


Evaluate the number of zeros located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of the fraction $\frac{1}{2^4 \cdot 5^7}$.


For how many values of \( c \) in the interval [0, 2000] does the equation \[
10 \lfloor x \rfloor + 3 \lceil x \rceil = c
\] have a solution for \( x \)?


Given three vertices of parallelogram $EFGH$ as $E = (2, 3, -1)$, $F = (0, 5, 3)$, and $G = (4, 2, 5)$. Find the coordinates of $H$.

Solution: \boxed{(6, 0, 1)}


Noelle needs to follow specific guidelines to earn homework points: For each of the first ten homework points she wants to earn, she needs to do one homework assignment per point. For each homework point from 11 to 15, she needs two assignments; for each point from 16 to 20, she needs three assignments and so on. How many homework assignments are necessary for her to earn a total of 30 homework points?


A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\sqrt{d} - b$ cm. What is the value of $b + d$?


What is the smallest positive integer $n$ such that $9450$ is a factor of $n!$?


Lark has forgotten her locker combination. It is a sequence of three numbers, each in the range from 1 to 50, inclusive. She remembers that the first number is a multiple of 5, the second number is a multiple of 4, and the third number is a multiple of 6. How many combinations could possibly be Lark's?


Find the area of the triangle with vertices $(1,8,11),$ $(0,7,9),$ and $(-3,10,9).$


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


A fair 8-sided die is rolled. If the roll is even and less than 8, then you win that amount of dollars. If you roll an 8, your winnings are doubled from the sum of the previous even numbers you could have won. If the roll is odd, you win nothing. What is the expected value of your winnings?


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


What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side or overlapping vertically are the same color? The tessellation consists of a base layer of rectangles uniformly placed and a top layer consisting of circles, where each circle overlaps four rectangles (one on each side, without side sharing).


Two lines each have a different non-zero $y$-intercept. The first line has a slope of 8 and an $x$-intercept of $(r, 0)$. The second line has a slope of 4 and an $x$-intercept of $(q, 0)$. If the first line's $y$-intercept is double that of the second line, what is the ratio of $r$ to $q$?


In the expansion of \( (x+2)^{50} \), what is the coefficient of the \( x^3 \) term?

Solution: \boxed{19600 \cdot 2^{47}}


Let $n$ equal the number of sides in a regular polygon. For $3 \leq n < 12$, how many values of $n$ result in a regular polygon where the common degree measure of the interior angles is not an integer?


Compute the largest integer $k$ such that $2016^k$ divides $2016!$.


A wizard is preparing an invisibility elixir. He must choose one of four magical herbs and one of six mystical stones. However, one particular stone is incompatible with three of the herbs, but the rest can be combined freely. How many valid combinations can the wizard use to prepare his elixir?


What is the smallest positive integer that is a multiple of both 42 and 56 but not a multiple of 18?


The mean of $7, 10$, and $23$ is equal to the mean of $18$ and $z$. What is the value of $z$?


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


An infinite geometric series has a first term of $24$ and a second term of $8$. A second infinite geometric series has the same first term of $24$, a second term of $8 + m$, and a sum of three times that of the first series. Find the value of $m$.


Two standard 6-sided dice are rolled. What is the probability that the sum of the numbers rolled is 4 or 16?


Determine the tens digit of $13^{2023}$.


In how many ways can the digits of $33,\!520$ be arranged to form a 5-digit number, ensuring that the number does not start with 0?


In rectangle $ABCD$, $AB = 8$ cm, $BC = 10$ cm, and $DG = DF$. The area of triangle $DFG$ is one-fifth the area of rectangle $ABCD$. What is the length in centimeters of segment $DG$? Express your answer in simplest radical form.

Solution: \boxed{4\sqrt{2} \text{ cm}}


Let
\[
f(x) =
\begin{cases}
5x^2 + 4 & \text{if } x \le 4, \\
bx + 2 & \text{if } x > 4.
\end{cases}
\]
Find \( b \) if the graph of \( y = f(x) \) is continuous.


In how many distinct ways can I arrange my seven keys on a keychain, if I want to place my house key, car key, and office key next to each other in any order? Two arrangements are not considered different if the keys can be made to be in the same order by reflection or rotation.


A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure $1/\pi$, as shown. What is the perimeter of this region?


How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^3\cdot5^{10}}$?


A large sphere has a volume of $450\pi$ cubic units. A smaller sphere has a volume which is $27.04\%$ of the volume of the large sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a decimal.


If
\[(1 + \tan 2^\circ)(1 + \tan 4^\circ)(1 + \tan 6^\circ) \dotsm (1 + \tan 88^\circ) = 2^m,\]
then find $m.$


Let $\theta$ be the angle between the planes $3x - y + z - 4 = 0$ and $4x + 2y - z + 2 = 0.$ Find $\cos \theta.$


Calculate $\left(\frac{2}{3}\right)^{6} \cdot \left(\frac{4}{5}\right)^{-4}$.


Ella, Finn, Grace, and Henry weigh themselves in pairs. Together Ella and Finn weigh 280 pounds, Finn and Grace weigh 230 pounds, Ella and Henry weigh 300 pounds. Determine how many pounds do Grace and Henry weigh together.


Express $342_{5} + 2C9_{13}$ as a base 10 integer. Here $C$ denotes the digit whose value is 12 in base 13.


Find the radius of the circle with equation $x^2 - 4x + y^2 - 6y + 13 = 0$.


What is the remainder when $(x + 1)^{2016}$ is divided by $x^2 + x + 1$?


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


For **v** = \( \begin{pmatrix} 8 \\ -12 \end{pmatrix} \) and **w** = \( \begin{pmatrix} -6 \\ 9 \end{pmatrix} \), compute \( \text{proj}_{\mathbf{w}} \mathbf{v} \).

Solution: \boxed{\begin{pmatrix} 8 \\ -12 \end{pmatrix}}


In $\triangle ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM= 15$ and $BN = 20$, find the length of $AB$.


In a right triangle $DEF$ with $\angle E = 90^\circ$, it is given that $$5\sin D = 12\cos D.$$ What is $\sin D$?

Solution: \boxed{\sin D = \frac{12}{13}}


Determine how many positive divisors does \(8!\) have?


Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 45^\circ$, and $DF=8$. Find the radius of the incircle of $\triangle DEF$.

Solution: \boxed{8 - 4\sqrt{2}}


Convert the point \((7, \frac{\pi}{3})\) in polar coordinates to rectangular coordinates.

Solution: \boxed{\left(3.5, \frac{7\sqrt{3}}{2}\right)}


Each of two spinners is divided into different sections. The first spinner has 5 sections labeled 2, 3, 5, 7, and 10. The second spinner has 4 sections labeled 6, 9, 11, and 14. If each spinner is spun once, what is the probability that the product of the results of the two spins will be an even number? Express your answer as a common fraction.


What is the greatest integer less than 150 for which the greatest common factor of that integer and 30 is 5?


The equation $x^2 + bx = -21$ has only integer solutions for $x$. If $b$ is a positive integer, what is the greatest possible value of $b$?


I had $50$ in allowance money and spent it according to a new distribution shown in the pie graph. How many dollars did I spend on burgers if the fractions spent on movies, ice cream, and music are respectively $\frac{1}{4}$, $\frac{1}{6}$, and $\frac{1}{3}$?


How many integers (positive and negative) greater than 1 are divisors of 30?


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


Determine how many integers $n$ between 1 and 15 (inclusive) result in a fraction $\frac{n}{30}$ that has a repeating decimal.


The sequence starts with 800,000; each subsequent term is obtained by dividing the previous term by 3. What is the last integer in this sequence?


Let $a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p$ be real numbers such that $abcd = 16$, $efgh = 16$, $ijkl = 16$, and $mnop = 16$. Find the minimum value of
\[
(aeim)^2 + (bfjn)^2 + (cgko)^2 + (dhlp)^2.
\]


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


Two identical isosceles triangles are positioned such that their bases are on the same line but they do not touch each other. The vertex angles of the triangles are each $40^\circ$. The triangles are rotated such that their equal sides are facing each other and the vertices of these angles are $4$ units apart. What is the sum of the angles at the vertices facing each other?


The graph of the parabola $x = 3y^2 - 9y + 4$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$.


In a modified walking game, I start at a point and follow these rules: On move 1, I do nothing. For moves from 2 to 40, I take one step forward if the number is prime, two steps backwards if the number is composite, and stand still if the number is a perfect square. After completing 40 moves, I walk back to my original starting point. How many steps do I need to walk back?


Let $b_n = 5^n + 9^n$. Determine the remainder when $b_{77}$ is divided by $40$.


Let $x$, $y$, and $z$ be positive real numbers. Find the maximum value of
\[
\frac{(x + y + z)^3}{x^3 + y^3 + z^3}.
\]


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


One of the six faces of a rectangular prism will be used as the base of a new pyramid. Calculate the maximum value of the sum formed by adding the number of exterior faces, vertices, and edges of this combined shape (the fusion of the prism and the pyramid), depicted below:

[asy]
draw((0,0)--(20,0)--(22,5)--(2,5)--cycle);
draw((0,0)--(0,12));
draw((20,0)--(20,12)--(22,17)--(2,17)--(0,12));
draw((22,5)--(22,17));
[/asy]


How many numbers are in the list $$1, 5, 9, \ldots, 2021, 2025?$$


The product of 3 inches and twice the circumference of a circle, in inches, equals 3 times the circle's area. What is the length of the radius of the circle, in inches?


Let $a$ and $b$ be the real values of $x$ for which \[\sqrt[3]{x} + \sqrt[3]{30 - x} = 3.\] Determine $x$ when $a > b$.

Solution: \boxed{\left(\frac{9 + \sqrt{93}}{6}\right)^3}


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


The quadratic $ax^2 + bx + c$ is given in the form $3(x - 3)^2 + 6$. When this quadratic $4ax^2 + 4bx + 4c$ is rewritten, what will be the constant $h$ in the vertex form $n(x - h)^2 + k$?


Determine the value of \( b \) such that the lines given by the equations below are perpendicular:
\begin{align*}
y &= 3x - 4, \\
2y + bx &= 14.
\end{align*}


In trapezoid $PQRS$ the lengths of the bases $PQ$ and $RS$ are 10 and 21 respectively. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a common fraction.


Find the smallest prime number that appears after a sequence of seven consecutive positive integers, all of which are nonprime, in ascending order of natural numbers.


The courier company charges an additional $\$0.15$ for envelopes if the length, in inches, divided by its height, in inches, is less than $1.5$ or greater than $3.0$. Determine how many of these four envelopes need this extra charge:
\begin{tabular}[t]{ccc}
Envelope & Length in inches & Height in inches\\\hline
E &7 &5\\
F &10 &2\\
G &8 &8\\
H &12 &3
\end{tabular}


Jessica runs a bakery where she typically makes 40 pie crusts every day, using up a total of 5 cups of flour. On a particular day, she decides to make only 25 pie crusts but wants to use the same total amount of flour. How much flour will each of these new pie crusts require? Express your answer in simplest fractional form.


For any number $y$, define two operations: $y\star = 9 - y$ and $\star y = y - 9$. What is the value of $\star(15\star)$?


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


The orchestra has more than 150 members but fewer than 250 members. When they line up in rows of 4, there are two extra persons; when they line up in rows of 5, there are three extra persons; and when they line up in rows of 7, there are four extra persons. How many members are in the orchestra?


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


Find the product of all positive integral values of \( n \) such that \( n^2 - 40n + 399 = p \) for some prime number \( p \). Note that there is at least one such \( n \).


Suppose the numbers \[2 - 3\sqrt{3}, \; -2-3\sqrt{3}, \; 2+\sqrt{11}, \; 2-\sqrt{11}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


How many positive integers $N$ less than $500$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?


At the PetPals Training Center, cats can learn to do three tricks: jump, fetch, and spin. The current training records show:
\begin{tabular}{l@{\qquad}l}
60 cats can jump & 25 cats can jump and fetch \\
40 cats can fetch & 20 cats can fetch and spin \\
50 cats can spin & 30 cats can jump and spin \\
15 cats can do all three & 5 cats can do none
\end{tabular}
How many cats are in the center?


Points \( A \) and \( B \) are selected on the graph of \( y = -\frac{1}{4}x^2 \) so that triangle \( ABO \) is equilateral. Find the length of one side of triangle \( ABO \).


Determine the domain of the real-valued function $$f(x)=\sqrt{4-\sqrt{7-\sqrt{x^2}}}.$$

Solution: \boxed{[-7, 7]}


Round $67.\overline{836}$ to the nearest thousandth.


If $a$, $b$, and $c$ are positive numbers where $ab = 15\sqrt{3}$, $bc = 21\sqrt{3}$, and $ac = 10\sqrt{3}$, what is the value of $abc$?


What is the sum of the first seven distinct positive integer multiples of 5 that are squares?


The angles of quadrilateral $EFGH$ satisfy $\angle E = 3\angle F = 4\angle G = 6\angle H$. What is the degree measure of $\angle E$, rounded to the nearest whole number?


What is the total number of digits used when the first 3002 positive even integers are written?


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


In Vermont, 780 people were surveyed about their preferred term for soft drinks. The results of the survey are shown in a pie chart. The central angle for the "Soda" sector is $195^\circ$. How many individuals selected "Soda"? Provide your answer as a whole number.


A circle's area is denoted as $x$ square units, and its circumference is denoted as $y$ units. If $x + y = 150\pi$, what is the radius of this circle?

Solution: \boxed{\sqrt{151} - 1}


I run at a constant pace, and it takes me 30 minutes to run to the park from my home. If the distance to the park is 5 miles, and a nearby coffee shop is 2 miles away from my home, how many minutes will it take me to run from my home to the coffee shop?

Solution: \boxed{12 \text{ min}}


For how many positive integers \( n \) less than or equal to 500 is
$$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$
true for all real \( t \)?


The product of two consecutive page numbers is $2,156.$ What are the sum of these two page numbers?


Let $A,B$ be the points on the coordinate plane with coordinates $(2t-3, t)$ and $(t-1, 2t+4)$, respectively. The square of the distance between the midpoint of $\overline{AB}$ and an endpoint of $\overline{AB}$ is equal to $(t^2 + t)/2$. Find the value of $t$.


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

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


What is the greatest 3-digit base 8 positive integer that is divisible by 7? (Express your answer in base 8.)


Add $-45.367$, $108.2$, and $23.7654$, then round your answer to the nearest tenth.


Solution: \boxed{\csc 20}


On the complex plane, consider a parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$. The area of this parallelogram is $\frac{12}{13}$. If the real part of $z$ is positive, find the smallest possible value of $\left| z + \frac{1}{z} \right|$ and compute the square of this minimum value, $d^2$.


A line segment with endpoints at $A(1, 3)$ and $B(13, 9)$ is extended from $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates of point $C$?

Solution: \boxed{(19, 12)}


Find the common ratio of the infinite geometric series: $$\frac{8}{10} - \frac{12}{25} + \frac{36}{125} - \dots$$


There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 6 \le 0.\]
How many integer values of $b$ are possible?


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

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


A point has rectangular coordinates $(2, -3, 6)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, \pi + \theta, \phi)$.

Solution: \boxed{(-2, 3, 6)}


How many numbers less than 35 are abundant numbers?


Consider a polynomial $p(x) = x^2 + ax + b$. Define its munificence as the maximum value of $|p(x)|$ on the interval $-1 \le x \le 1$. Find the smallest possible munificence of such a polynomial.


What is the average of all integer values of \( N \) such that \( \frac{N}{84} \) is strictly between \( \frac{2}{7} \) and \( \frac{1}{6} \)?


The center of a circle has coordinates \((4, -3)\). The circle is reflected about the line \(y=-x\). What are the \(x,y\) coordinates of the center of the reflected circle? State the \(x\) coordinate first.

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


Suppose \(a, b,\) and \(c\) are nonnegative real numbers such that the sum of their squares is 52 and the sum of their pairwise products is 28. What is the maximum possible value of \(a+b+c\)?


A math class has fewer than $50$ students. When the students try to sit in rows of $8$, $6$ students are left in the last row. When the students try to sit in rows of $5$, $1$ student is left in the last row. How many students are in this class?


Suppose a parabola has vertex $\left(\frac{3}{4},-\frac{25}{16}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer.  Find the smallest possible value of $a.$


Let $ABCDEFGH$ denote a cube with a side length of 2. Find the volume of pyramid $ABCG$.


There are 72 chips in a box. Each chip is either small or large. If the number of small chips is greater than the number of large chips by a composite number of chips, what is the greatest possible number of large chips?


Calculate $\sqrt{15p^2} \cdot \sqrt{8p} \cdot \sqrt{27p^5}$ and express your answer in simplest radical form in terms of $p$.


April has four different basil plants, four different tomato plants, and two different pepper plants. In how many ways can she arrange the plants in a row if all tomato plants are grouped together, and all pepper plants are also grouped together?


If $x = \frac23$ and $y = \frac32$, find the value of $\frac34 x^4 y^5$.


Suppose the function $G(x) = (x-2)^2 - 1$ represents a portion of a graph shown in a specific color below. The distance between grid lines is $1$ unit. Compute $G(G(G(G(G(2)))))$.


Let $p,$ $q,$ $r,$ $s$ be nonzero integers such that
\[\begin{pmatrix} p & q \\ r & s \end{pmatrix}^2 = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}.\] Find the smallest possible value of $|p| + |q| + |r| + |s|$.


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


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


Determine the range of the function $$F(x) = |2x + 4| - |x - 2|.$$
Express your answer in interval notation.

Solution: \boxed{[-4, \infty)}


Let $D$ be the circle with the equation $x^2 - 16y + 81 = -y^2 + 14x$. Find the sum $a + b + r$ where $(a, b)$ is the center of $D$ and $r$ is its radius.

Solution: \boxed{15 + 4\sqrt{2}}


Tamika selects two different numbers at random from the set $\{7,8,9\}$ and subtracts the smaller number from the larger one. Carlos takes two different numbers at random from the set $\{2,4,5\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.


If $\sec x + \tan x = 5,$ then find $\sec x - \tan x.$


In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 5 and 20 respectively. The legs of the trapezoid $PR$ and $QS$ are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a common fraction.


In a local frisbee league, teams have 7 members including at least one woman per team, and each of the 5 teams takes turns hosting tournaments. At each tournament, each team selects two members of that team to be on the tournament committee, except the host team, which selects three members, ensuring at least one woman from each team is included. How many possible 11-member tournament committees are there?


For how many positive integral values of \( a \) is it true that \( x = 2 \) is the only positive integer solution of the system of inequalities
$$
\begin{cases}
2x \geq 4x - 4\\
3x - a > -9
\end{cases}
$$


In triangle $PQR,$ the midpoint of $\overline{QR}$ is $(2,7,2),$ the midpoint of $\overline{PR}$ is $(3,5,-3),$ and the midpoint of $\overline{PQ}$ is $(1,8,5).$ Find the coordinates of vertex $P.$

Solution: \boxed{(2, 6, 0)}


In triangle $ABC$, point $E$ is on side $BC$, $\angle BAE = 20^\circ$ and $\angle EAC = 40^\circ$. Determine the measure of angle $BEA$, in degrees.


How many integers between 0 and 20 inclusive have an inverse modulo 21?


If $\displaystyle\frac{a}{b} = 5$, $\displaystyle\frac{b}{c} = \frac{1}{2}$, $\displaystyle \frac{c}{d} = 3$, and $\displaystyle \frac{d}{e} = \frac{1}{4}$, then what is $\displaystyle\frac{e}{a}$?

Solution: \boxed{\frac{e}{a} = \frac{8}{15}}


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, such that the first outcome is the thousands digit, 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 5? 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]


Henry's little brother now has $10$ identical stickers and $5$ sheets of paper, each a different color. He distributes all the stickers on the sheets of paper. How many ways can he do this, if only the number of stickers on each sheet of paper matters?


What is the sum of the different prime factors of $504210$?


Consider a function $y = g(x)$ with the property that if it is shifted $24$ units to the right, then the resulting graph is identical to the original graph of $y = g(x)$. Determine the smallest positive $a$ such that if the graph of $y = g\left(\frac{x}{4}\right)$ is shifted $a$ units to the right, then the resulting graph is identical to the original graph of $y = g\left(\frac{x}{4}\right)$.


An equilateral triangle has an area of $36\sqrt{3}$ $\text{cm}^2$. If each side of the triangle is increased by 3 cm, by how many square centimeters is the area increased?


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


An equilateral triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of the side of the triangle to the width of the rectangle? Express your answer as a common fraction.


How many different counting numbers will each leave a remainder of 7 when divided into 55?


Let \( x, y, z \) be positive real numbers such that
\[
\frac{2}{x} + \frac{3}{y} + \frac{1}{z} = 12.
\]
Find the minimum value of \( x^2 y^3 z \).


A school library now possesses eight identical copies of a certain book. Each day, books are either in the library or checked out. Determine the number of different configurations possible if at least two books must remain in the library and at least two must be checked out.


Let $\mathbf{N}$ be a matrix such that
\[\mathbf{N} \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 5 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{N} \begin{pmatrix} -2 \\ 4 \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}.\]Compute $\mathbf{N} \begin{pmatrix} 7 \\ 0 \end{pmatrix}.$

Solution: \boxed{\begin{pmatrix} 17.5 \\ 7 \end{pmatrix}}


A point \((x, y)\) is randomly selected such that \(0 \leq x \leq 4\) and \(0 \leq y \leq 5\). What is the probability that \(x + 2y \leq 6\)? Express your answer as a common fraction.


In the triangle $ABC$, $BP$ and $BQ$ trisect $\angle ABC$, which measures $120^\circ$. The line $BM$ bisects $\angle PBQ$. Calculate the ratio of the measure of $\angle MBQ$ to the measure of $\angle ABQ$.


What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!$?


Solution: \boxed{10 \text{ mph}}


Consider the arithmetic sequence: $ -25, -18, -11, \ldots, 89, 96.$ How many numbers are in the list?


Find the area of the parallelogram generated by vectors $\begin{pmatrix} 4 \\ 2 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 5 \\ 3 \end{pmatrix}.$


If Ben rolls 12 dice, what is the probability that he gets more numbers greater than 3 than numbers less than or equal to 3?


If the consecutive integers from $40$ to $1$ were written as $$4039383736...,$$ what would be the $57^{\text{th}}$ digit to be written?


Solution: \boxed{x_1 = \frac{2}{3}}


When Alex divides his stickers into groups of 6, he has 4 remaining, and when he divides them into groups of 8, he has 6 remaining. If Alex has fewer than 100 stickers, what is the sum of all possible quantities of stickers that he could have?


A square is divided into five congruent rectangles. If the perimeter of each of these rectangles is 48 inches, what is the perimeter of the square, in inches?


In how many ways is it possible to arrange the digits of 11250 to get a five-digit multiple of 5?


Alexio now has 120 cards numbered from 1 to 120. He randomly selects one card. What is the probability that the number on the card is a multiple of 2, 4, or 6? Express your answer as a common fraction.


Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $5000$?


Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 24, what is the smallest value of $n$ such that $f(n) > 15$?


What is the volume in cubic meters of a sphere with a diameter of 10 meters? Express your answer in terms of $\pi$.

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


Find the minimum value of
\[ \cos \frac{\theta}{2} \cdot (2 - \sin \theta) \]
for \( 0 < \theta < \pi \).


For how many integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 6?


In trapezoid \(PQRS\), the lengths of the bases \(PQ\) and \(RS\) are 10 and 23 respectively. The legs of the trapezoid are extended beyond \(P\) and \(Q\) to meet at point \(T\). What is the ratio of the area of triangle \(TPQ\) to the area of trapezoid \(PQRS\)? Express your answer as a common fraction.


Jenna rolls a fair icosahedral die with numbers $1,2,3,...,20$ on its faces. What is the expected number of digits in the number she obtains?


What is the smallest integer value of $x$ for which $7 - 5x < 25$ and $y - 3x > 6$, where $y = 10$?


Define $h(x) = 7x - 6$. If $h(x) = f^{-1}(x) - 5$ and $f^{-1}(x)$ is the inverse of the function $f(x) = ax + c$, find $3a + 4c$.


Determine the domain of the function $f(x) = \frac{\sqrt{x - 3}}{\sqrt{8 - x} + \sqrt{x - 1}}$.

Solution: \boxed{[3, 8]}


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


For how many integers $n$ is it true that $\sqrt{n+1} \le \sqrt{5n - 7} < \sqrt{3n + 4}$?


In a trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, what is the area of trapezoid $PQRS$?


How many times does the digit 9 appear in the list of all integers from 1 to 800?


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


Ed and Sue cycle, jog, and swim at constant rates. Ed covers 66 kilometers biking for 3 hours, jogging for 2 hours, and swimming for 4 hours, while Sue covers 96 kilometers after jogging for 3 hours, swimming for 2 hours, and biking for 4 hours. Their cycling, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's cycling, jogging, and swimming rates.


A regular decagon \(Q_1 Q_2 \dotsb Q_{10}\) is drawn in the coordinate plane with \(Q_1\) at \((2,0)\) and \(Q_6\) at \((4,0)\). If \(Q_n\) is the point \((x_n,y_n)\), compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).\]

Solution: \boxed{59049 - 1 = 59048}


Our school's boys basketball team has 16 players, including a set of twins, Ben and Jerry, and a set of triplets, Tom, Tim, and Ted. In how many ways can we choose 5 players if at most one of the triplets and one of the twins can be in the starting lineup?


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


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


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


The planet Zircon orbits its star in an elliptical path with the star located at one of the focusses. The closest Zircon gets to its star (perigee) is 3 astronomical units (AU) and its furthest distance (apogee) is 15 AU. Determine the distance from Zircon to its star when Zircon is halfway through its orbital path.

Solution: \boxed{9 \text{ AU}}


Let $x = (3 + \sqrt{8})^{500},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[x^2(1 - f).\]

Solution: \boxed{(3 + \sqrt{8})^{500}}


Triangle $PQR$ has side lengths $PQ=8$, $QR=10$, and $PR=12$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle. One bug crawls clockwise at 2 units per minute, and the other bug crawls counterclockwise at 3 units per minute. They meet at point $S$ along $QR$. Determine the length $RS$.

Solution: \boxed{6 \text{ units}}


Given positive integers $a$ and $b$ such that $\frac{1}{a} + \frac{1}{3b} = \frac{1}{9}$, what is the least possible value of $ab$?


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


Assume that $x$ is a positive multiple of $4$. If $x^3$ is less than $2000$, what is the greatest possible value of $x$?


Find all values of $r$ such that $\lfloor r \rfloor + r = 18.75$.


How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16 \}$?


Find the minimum value of
\[(15 - x)(8 - x)(15 + x)(8 + x).\]


What fraction of $\frac{3}{4}$ is $\frac{1}{5}$?


If the consecutive integers from $60$ to $1$ are written as $$6060595857...,$$ what would be the $80^{\text{th}}$ digit to be written?


A sequence $b_1, b_2, b_3, \dots,$ is defined recursively by $b_1 = 2, b_2 = 3,$ and for $k \geq 3,$
\[ b_k = \frac{1}{2}b_{k-1} + \frac{1}{6}b_{k-2}. \]
Evaluate $b_1 + b_2 + b_3 + \dotsb.$


Find the number of odd digits in the base-5 representation of $365_{10}$.


Determine how many integers $i$ exist such that $1 \le i \le 5000$ and $f(i) = 1 + \sqrt{i} + i + \sqrt[3]{i}$?


In triangle $XYZ,$ the midpoint of $\overline{YZ}$ is $(2,6,-2),$ the midpoint of $\overline{XZ}$ is $(1,5,-3),$ and the midpoint of $\overline{XY}$ is $(3,4,5).$  Find the coordinates of vertex $X.$

Solution: \boxed{(2, 3, 4)}


What value of $x$ will give the maximum value for the function $-2x^2 + 8x -10$? Additionally, find the values of $x$ where the function equals zero.

Solution: \boxed{\text{Maximum value is -2 at } x = 2}


Let $C$ equal the number of four-digit numbers that are both odd and divisible by 5. Let $D$ equal the number of four-digit numbers that are divisible by 25. Find $C+D$.


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


Six congruent equilateral triangular fields each with side length \(s\) are enclosed using fencing. If the exact amount of fencing that enclosed the six small triangles is reused to form one large equilateral triangular field, what is the ratio of the total area of the six small triangles to the area of the new large triangle? Express your answer as a common fraction.


How many obtuse angles can an isosceles triangle have?


Two wheels, each with a radius of $1 \text{ m}$, are attached side-by-side and are rolled in a straight line on a flat horizontal surface. The first wheel (Wheel A) is in contact with the second wheel (Wheel B), and both complete one full revolution without slipping. How many meters did the center of Wheel A travel horizontally from its starting point?


Below is the graph of \( y = A \sin (Bx + C) + D \) where \( A \), \( B \), \( C \), and \( D \) are constants, with \( A = 1 \) and \( D = 1 \). The graph completes two full cycles between \( x = 0 \) and \( x = 2\pi \). Find \( B \).


The Rotokas alphabet has twelve letters: A, E, G, I, K, O, P, R, S, T, U, and V. Design license plates of five letters using only these letters where the license plate ends with either G or K, starts with S, cannot contain T, and where no letters repeat. How many such license plates are possible?


Suppose that $x$ is a positive multiple of $4$. If $x$ cubed is less than $5000$, what is the greatest possible value of $x$?


For some positive integer $k$, when 80 is divided by $k^2$, the remainder is 8. What is the remainder when 150 is divided by $k$?


Let $a, b, c$ be constants, and suppose that the inequality \[\frac{(x-a)(x-b)}{x-c} \le 0\] is true if and only if either $x < -6$ or $|x-30| \le 2$. Given that $a < b$, find the value of $a + 2b + 3c.$


A designer has 5 fabric colors to choose from for a dress: red, green, blue, yellow, and purple. There are six different patterns available for the dresses. However, the sixth pattern can only be used with blue or purple fabrics. If each dress design requires exactly one color and one pattern, how many different dress designs are possible?


Each of the symbols $\star$ and $*$ represents a unique operation in the set $\{+, - , \times, \div\}$. Assume $\frac{16 \star 4}{10*2} = 4$. What is the value of $\frac{5 \star 15}{8 * 12}$? Express your answer as a common fraction.


Find the quadratic function $f(x) = x^2 + cx + d$ such that
\[\frac{f(f(x) + x)}{f(x)} = x^2 + 1779x + 2013.\]

Solution: \boxed{x^2 + 1777x + 235}


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


What is the least positive integer value of $x$ such that $(2x + 51)^2$ is a multiple of 59?


Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20?


The line \( y = -\frac{5}{3}x + 15 \) crosses the \( x \)-axis at \( P \) and the \( y \)-axis at \( Q \). Point \( T(r,s) \) is on line segment \( PQ \). If the area of \( \triangle POQ \) is twice the area of \( \triangle TOP \), then what is the value of \( r+s \)?


A rectangular prism has dimensions 12 inches, 18 inches, and 15 inches. What is the length, in inches, of the diagonal connecting the opposite corners at points A and B of the prism?

Solution: \boxed{3\sqrt{77} \text{ inches}}


For how many integer values of $n$ between 1 and 493 inclusive does the decimal representation of $\frac{n}{425}$ terminate?


Each triangle in a sequence of four connected triangles is a 30-60-90 triangle, and the hypotenuse of each triangle is the longer leg of the subsequent triangle. The hypotenuse of the largest triangle is 16 centimeters. Determine the length of the longer leg of the smallest triangle.

Solution: \boxed{9 \text{ cm}}


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


A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$


How many integers between $200$ and $300$ have three different digits in increasing order and are divisible by 5?


For how many different digits $n$ is the three-digit number $25n$ divisible by $n$?


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


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


Determine the number of $x$-intercepts on the graph of the function $y = \cos \frac{1}{x}$ in the interval $(0.00005, 0.0005)$ in radians.


In a modified cross country meet between 2 teams of 6 runners each, a runner who finishes in the $n$th position contributes $n$ points to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different possible winning scores are there?


Consider the set of all points \((x, y)\) in the coordinate plane for which one of the coordinates is either three times the other or one third of the other. If these points were plotted, into how many regions would the resulting graph split the plane?


What integer $n$ satisfies $0 \leq n < 103$ and $$100n \equiv 85 \pmod{103}~?$$


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\overline{13}$?


An arithmetic sequence with the first term $2$ has a common difference of $5$. A second sequence begins with $3$ and has a common difference of $8$. In the range of $1$ to $150$, what is the largest number common to both sequences?


How many rows of Pascal's Triangle contain the number $12$?


Determine the integers $a$ and $b$ such that $a < \log_3 200 < b$. Find the sum of integers $a$ and $b$.


A roll of quarters now contains $30$ quarters, and a roll of dimes contains $60$ dimes. Michael has a jar with $94$ quarters and $184$ dimes. Sara has a jar with $137$ quarters and $312$ dimes. Michael and Sara combine their quarters and dimes to make complete rolls. Calculate in dollars the value of the leftover quarters and dimes after making as many complete rolls as possible.


Solve for $x$ in the equation $3 - 2ix = 5 + 4ix$ (where $i^2 = -1$).

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


The square of an integer is 256 greater than the integer itself. What is the sum of all integers for which this is true?


The points $(1,3)$ and $(4,6)$ are adjacent vertices of a square. What is the area of the square?


When the base-10 integers 150 and 950 are expressed in base 2, how many more digits does 950 have than 150 (after being converted)?


Compute $\arccos (\cos 8).$ All functions are in radians.

Solution: \boxed{8 - 2\pi}


The equation $y = -4.9t^2 + 30t$ describes the height (in meters) of a projectile launched from the ground at 30 meters per second. In how many seconds will the projectile first reach 35 meters in height?

Solution: \boxed{t = 1.57 \text{ seconds}}


Let $a$ and $b$ be nonnegative real numbers. Consider the following five statements:

1. $\frac{1}{a} < \frac{1}{b}$
2. $a^2 > b^2$
3. $a < b$
4. $a > 0$
5. $b > 0$

What is the maximum number of these statements that can be true for any values of $a$ and $b$?


Suppose $x$ and $y$ are positive real numbers such that $x^2 - xy + y^2 = 9$. Find the maximum possible value of $x^2 + xy + y^2$.


Find the greatest common divisor of 1821 and 2993.


In $\triangle ABC, AB = 10, BC = 8, CA = 7$, and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$.


Given the equations $3a + 2b = 18$ and $5a + 4b = 31$, find the value of $2a+b$.


Compute the distance between the vertices of the parabolas that are part of the graph of the equation \(\sqrt{x^2+y^2} + |y-1| = 5\).


An airplane has three sections: First Class (30 seats), Business Class ($20\%$ of the total number of seats) and Economy ($\frac{3}{4}$ of the total number of seats). How many seats does the plane have?


In a gym class, students line up for a group activity. They notice that when they form rows of six, there are two extra students left, when they form rows of seven, there are three extras, and when they line up in groups of eight, four students are left over. What is the smallest number of students possible in the gym class?


I was planning to work 25 hours a week for 15 weeks this summer to earn $\$4500$ to fund my education. Unfortunately, I was unable to work for the first three weeks of summer. How many hours per week will I need to work for the rest of the summer to still meet my financial goal?


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


How many integers between 1 and 300 are multiples of both 6 and 10 but not of either 5 or 8?


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

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


Let $z$ be a complex number such that
\[z^2 + |z|^2 = 6 - 9i.\]
Find $|z|^2.$


Points $P$, $Q$, $R$, $S$, and $T$ are located in 2-dimensional space with $PQ = QR = RS = ST = TP = 3$ units and $\angle PQR = \angle RST = \angle STP = 120^\circ$. The line through $PQ$ is parallel to the segment $ST$. What is the perimeter of triangle $QRS$?

Solution: \boxed{9 \text{ units}}


Let $ABCD$ be a square with side length 10. A circle is inscribed such that it is tangent to two adjacent sides, $AB$ and $AD$, with tangent points $X$ and $Y$ respectively. The point on the circle diametrically opposite $X$ lies on the side $CD$. Find the area of the portion of the circle that lies outside the square.


Find \[\left|\left(\frac 23+\frac 56i\right)^8\right|\]


The ratio of irises to roses in Tim’s garden is 3:7. He initially has 49 roses. He plans to add 35 more roses and wants to maintain the same ratio by adding more irises. How many irises must he add, and what will be the total number of flowers in the garden after this addition?


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


$\textbf{Catherine's Vintage Postcards}$

Catherine organizes her postcard collection by country and by the decade in which they were acquired. The prices she paid for them at an antique store were: Italy and Germany, $10$ cents each, Canada $7$ cents each, and Mexico $8$ cents each. (Italy and Germany are European countries and Canada and Mexico are in North America.) Catherine has the following number of postcards from each decade and country:

| Decade  | Italy | Germany | Canada | Mexico |
|---------|-------|---------|--------|--------|
| 1950s   | 10    | 5       | 8      | 12     |
| 1960s   | 16    | 12      | 10     | 15     |
| 1970s   | 12    | 18      | 13     | 9      |

In dollars and cents, how much did her European postcards issued before the $80s$ cost her?


The graph of the function $y = g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers, find $A+B+C$. Assume the graph includes vertical asymptotes at $x = -3$ and $x = 4$ with a horizontal line approaching $y = \frac{1}{A} < 1$ for large $x$.


Nine positive integers form a list, of which six are given as $6, 7, 2, 4, 8, 5$. What is the largest possible value of the median of this list if three of the remaining integers are less than or equal to 2?


What is the remainder when \(2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005 \cdot 2006 \cdot 2007\) is divided by 17?


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 $d$.

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


Find the number of integers $n$ that satisfy:
\[50 < n^2 < 200.\]


George flips an unfair coin 5 times. The coin has a $\frac{1}{4}$ probability of coming up heads and a $\frac{3}{4}$ probability of coming up tails. What is the probability that he flips at least 3 tails, and the last two flips are tails?


An entrepreneur invested $\$20,\!000$ in a nine-month term deposit that paid a simple annual interest rate of $8\%$. After the term ended, she reinvested all the proceeds into another nine-month term deposit. At the end of the second term, her total investment had grown to $\$22,\!446.40$. If the annual interest rate of the second term deposit is $s\%$, what is $s?$


Henry’s Hamburger Heaven has expanded its menu to include an additional option - bacon. Now, the condiments available are ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, onions, and bacon. Customers can choose to have 1, 2, or 3 meat patties like before, but they can also opt for a meatless, veggie patty. How many different kinds of hamburgers can a customer now order?


Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}
\]
over all real numbers $x$.


Let \( A = (2,0) \) and \( B = (7,6) \). Let \( P \) be a point on the parabola \( y^2 = 8x \). Find the smallest possible value of \( AP + BP \).


A large academy has an incoming class of fewer than 1000 students. When these students are lined up in columns of 25, there are 18 students in the last column. When lined up in columns of 28, there are 26 students in the last column. How many students are there in the incoming class?


For a given positive integer \( n > 3^2 \), what is the greatest common divisor of \( n^4 + 4^2 \) and \( n + 3 \)?


An integer $B$ is considered lucky if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest lucky integer?


Two circles are drawn in a 15-inch by 18-inch rectangle. Each circle has a diameter of 7 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?

Solution: \boxed{\sqrt{185}\text{ inches}}


For how many positive integers \( x \) is \( x^2 - 4x + 4 \) between 30 and 60?


Calculate $\sqrt{42p} \cdot \sqrt{14p} \cdot \sqrt{7p}$ and express your answer in simplest radical form in terms of $p$.

Solution: \boxed{14p \sqrt{21p}}


Two cylindrical cans have the same volume. The height of one can is quadruple the height of the other. If the radius of the narrower can is 10 units, what is the length of the radius of the wider can? Express your answer in simplest radical form.


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


For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n+1$ digits after the decimal point. Calculate the total number of positive integer divisors of $f(1005)$.


At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink, and one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals can she buy from these options?

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


For what values of $x$ is $x^2 - 50x + 625 \le 25$?

Solution: \boxed{[20, 30]}


What is the remainder when 3123 is divided by 28?


The graphs of $y = x^3 - 3x + 2$ and $x + 6y = 6$ intersect in points $(x_1,y_1),$ $(x_2,y_2),$ and $(x_3,y_3).$ If $x_1 + x_2 + x_3 = A$ and $y_1 + y_2 + y_3 = B,$ compute the ordered pair $(A,B).$

Solution: \boxed{(0, 3)}


The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The vertex of the parabola is at $(2,7)$. (2) The graph passes through the point $(0,-7)$. Determine the value of $y$ when the graph passes through the point $(5, n)$.


Seventy percent of a train's passengers are women and fifteen percent of those women are in business class. What is the number of women in business class if the train is carrying 300 passengers?


What is the coefficient of the $m^6n^6$ term in the expansion of $(m+n)^{12}$?


Consider the function $\zeta(x)$ for a positive real number $x > 2$ defined by
\[
\zeta(x) = \sum_{n = 1}^\infty \frac{1}{n^x}.
\]
Compute the sum
\[
\sum_{k = 1}^\infty \{\zeta(2k)\}.
\]
Note: For a real number $x,$ $\{x\}$ denotes the fractional part of $x.$


What is the sum of the digits in the base 8 representation of $4321_{10}$?


A notebook and its cover together cost $\$3.30$. The notebook costs \$2 more than its cover. If the displayed price includes a 10% sales tax, what is the pre-tax cost of the notebook?


The graph of $y = f(x)$ is shown below.

[asy]
unitsize(0.3 cm);

real func(real x) {
  real y;
  if (x >= -3 && x <= 0) {y = -2 - x;}
  if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
  if (x >= 2 && x <= 3) {y = 2*(x - 2);}
  return(y);
}

int i, n;

for (i = -8; i <= 8; ++i) {
  draw((i,-8)--(i,8),gray(0.7));
  draw((-8,i)--(8,i),gray(0.7));
}

draw((-8,0)--(8,0),Arrows(6));
draw((0,-8)--(0,8),Arrows(6));

label("$x$", (8,0), E);
label("$y$", (0,8), N);

draw(graph(func,-3,3),red);

label("$y = f(x)$", (4,-3), UnFill);
[/asy]

The graph of $y = h(x)$ is obtained by applying a vertical reflection to $f(x)$, then reflecting it on the $y$-axis, and finally shifting 5 units to the right. What is $h(x)$ in terms of $f(x)$?

Solution: \boxed{h(x) = -f(5 - x)}


Compute the number of increasing sequences of positive integers $b_1 \le b_2 \le b_3 \le \cdots \le b_{15} \le 3005$ such that $b_i-i$ is odd for $1\le i \le 15$. Express your answer as ${p \choose q}$ for some integers $p > q$ and find the remainder when $p$ is divided by 1000.


Given vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ such that $\|\mathbf{u}\| = 4,$ $\|\mathbf{v}\| = 5,$ $\|\mathbf{w}\| = 6,$ $\mathbf{u} \cdot \mathbf{v} = 12,$ and $\mathbf{v} \cdot \mathbf{w} = 15,$ find $\|\operatorname{proj}_{\mathbf{v}} \mathbf{u}\|$ and $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|$.

Solution: \boxed{\|\operatorname{proj}_{\mathbf{v}} \mathbf{u}\| = \frac{12}{5}, \|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\| = \frac{5}{2}}


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 $b$?


Ben now throws five identical darts. Each hits one of four identical dartboards on the wall. After throwing the five darts, he lists the number of darts that hit each board, from greatest to least. How many different lists of results are possible?


The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers with $1 < c < b < a$. What is the largest possible value of $b$?


Carolyn and Paul are playing a game starting with a list of integers from $1$ to $n$. In this variation:
- Paul always has the first turn.
- Carolyn and Paul alternate turns.
- On each of his turns, Paul must remove one number from the list such that this number has at least one positive divisor (other than itself) remaining in the list.
- On each of her turns, Carolyn must remove all positive divisors of the number that Paul has just removed.
- If Paul cannot remove any more numbers, then Carolyn removes the rest of the numbers.

Suppose $n=5$ and Paul removes the integer $3$ on his first turn. Determine the sum of the numbers that Carolyn removes.


Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\]


Given $n = 2^{10} \cdot 3^{14} \cdot 5^{8}$, how many of the natural-number factors of $n$ are multiples of 360?


Jo adds up all the positive integers from 1 to 100. Anne does a similar thing with the first 100 positive integers; however, she first rounds every integer to its nearest multiple of 5 (rounding up on .5) and then adds the 100 values. What is the positive difference between Jo's sum and Anne's sum?


The graphs of $y=|x|$ and $y=-x^2-4x-3$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments.


A school has eight identical copies of a specific textbook. During the school day, some of these textbooks can be in the classroom, and others in the library. How many different ways are there for some of the books to be in the classroom and the rest in the library if at least one book must be in both locations?


Roberto has 5 pairs of trousers, 7 shirts, and 4 jackets. However, one particular pair of trousers cannot be worn with one specific jacket. How many different outfits can Roberto put together if an outfit consists of a pair of trousers, a shirt, and a jacket?


Determine the value of \( k \) such that the line \( x = k \) intersects the graph of the parabola \( x = -y^2 - 4y + 2 \) at exactly one point.


The Cookie Monster now finds a new cookie with a boundary described by the equation $x^2 + y^2 - 6x + 10y + 12 = 0$. He needs to know the size of this cookie to decide if it's suitable for his dessert. What is the radius of this cookie?


Find the remainder when $2^{2^{2^2}}$ is divided by 500.


Solution: \boxed{\sin (1^\circ)}


The quadratic $6x^2 - 13x + 5$ has two real roots. Find the sum of the squares of these roots. Express the answer as a common fraction in its lowest terms.


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 314 hours, 21 minutes, and 56 seconds the time will be $A:B:C$. What is the value of $A + B + C$?


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


Melissa spilled some ink on her mathematics homework. She cannot determine the coefficient of \(x\), but she notes that the equation whiel spill still maintains two distinct negative integer solutions. The equation looks like:
\[
x^2 + ?x + 48 = 0
\]
What is the sum of all the distinct possible integers that could have been under the ink stain?


Determine the value of $k$ such that the lines represented by the equations
\[\begin{pmatrix} 1 \\ -3 \end{pmatrix} + t \begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} -4 \\ 2 \end{pmatrix} + s \begin{pmatrix} 3 \\ k \end{pmatrix}\]
do not intersect for any real values of $t$ and $s$.


How many three-digit numbers whose digits add up to $9$ are divisible by $3$?


What is the units digit of the sum of the squares of the first 2035 odd, positive integers?


Square EFGH has its center at $(5, 3)$ and has an area of 16 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at $(0,0)$ and a scale factor of $\frac{1}{2}.$ What are the coordinates of the vertex of the image of square EFGH that is closest to the origin? Give your answer as an ordered pair.

Solution: \boxed{(1.5, 0.5)}


The Sunnydale Farm has 5 chickens, 3 dogs, 4 cats, and 3 rabbits. How many ways can these 15 animals be placed in a row of 15 cages, such that all of the animals of each type are next to each other? Each animal within a species remains distinguishable from others.


In a modified version of a walking game, I make moves according to the following rules: On move 1, I do nothing, but on move $n$ where $2 \le n \le 20$, I take two steps forward if $n$ is a prime number and three steps backward if $n$ is composite. After finishing all 20 moves, I measure the distance I need to walk back to return to my starting point. How many steps long is my walk back?


Find the smallest positive four-digit number that is divisible by each of the five smallest prime numbers.


Determine the 58th card in the sequence if Emily orders her playing cards continuously in the order $$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, 2, 3, \dots$$.


In the eight-term sequence $P, Q, R, S, T, U, V, W$, the value of $R$ is $8$ and the sum of any three consecutive terms is $35$. What is $P+W$?


Let $m$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers such that $\sum_{i = 1}^4 x_i = 66.$ Find $\frac m{100}.$


Find the sum of all possible positive integer values of $c$ such that the quadratic equation $3x^2 - 7x + c = 0$ has rational roots.


There are \(30\) different complex numbers \(z\) such that \(z^{30} = 1\). For how many of these is \(z^{10}\) a real number?


What is the largest positive integer $n$ for which $n^4 + 119$ is divisible by $n+11$?


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

Solution: \boxed{144 + 108\pi}


Ryosuke picks up his friend from the library and the odometer reads 75,200 when he picks up his friend. The odometer reads 75,238 when he drops his friend off at their house. Ryosuke's car now gets 32 miles per gallon and the price of one gallon of gas remains $\$4.05$. How much did the gas cost for the trip from the library to the friend's house?


Find the largest integer $n$ for which $18^n$ evenly divides $30!$.


Find the numerical value of $k$ such that
\[\frac{5}{x + y} = \frac{k}{x + z} = \frac{9}{z - y}.\]


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


A triangle has vertices at coordinates (3, 3), (5, 7), and (7, 3). What is the number of units in the length of the longest side of the triangle?


How many rows of Pascal's Triangle contain the number $101$?


Given vectors \(\mathbf{c}\) and \(\mathbf{d}\) such that \(\|\mathbf{c}\| = 5\), \(\|\mathbf{d}\| = 7\), and \(\|\mathbf{c} + \mathbf{d}\| = 10\). Find \(\cos \phi\), where \(\phi\) is the angle between \(\mathbf{c}\) and \(\mathbf{d}\).


In the strategy game "CodeMaster", secret codes are formed by placing pegs of any of eight different colors into five slots. Colors may be repeated, but no slot may remain empty. However, one color, black, is only allowed in the first two slots. How many different secret codes are possible?


Determine the integer when the expression $y^2 + 16y + 60$ is expressed as a combination of the square of a binomial and an integer.


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


Let \( s \) be the positive real solution to \( x^3 + \frac{1}{4} x - 1 = 0 \). Find the exact numerical value of
\[ s^2 + 2s^5 + 3s^8 + 4s^{11} + \dotsb. \]


For the function \( y = \frac{2-x}{3x+4} \) where \( x \neq -\frac{4}{3} \), determine the value of \( y \) that cannot be reached by any real value of \( x \). Express your answer as a common fraction.


A shopkeeper has a $\$20$ hat and a $\$15$ tie. He decides to discount the hat by $25\%$ and later by $20\%$ of the reduced price. For the tie, he gives an initial discount of $10\%$ followed by an additional $30\%$ of the new price. What are the final prices of the hat and the tie?


Convert the decimal number $256_{10}$ to its equivalent in base five.


Our football team has 12 members, of which only 4 are strong enough to play offensive lineman, while all other positions can be played by anyone. In how many ways can we choose a starting lineup consisting of a quarterback, a running back, an offensive lineman, a wide receiver, and a tight end?


The spinner used by Jane and her brother now has six congruent sectors, numbered 1 through 6. They each spin the spinner once. If the non-negative difference of their numbers is less than 2, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.


What is the base ten equivalent of $37564_7$?


Each symbol $\diamond$ and $\circ$ represents an operation from $\{+, -, \times, \div\}$. Find the value of $\frac{8 \diamond 3}{10 \circ 5}$ if it is given that $\frac{10 \diamond 4}{6 \circ 2} = 5$.


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


Determine the domain of the function $$f(x) = \frac{1}{x-5} + \frac{1}{x^2-4} + \frac{1}{x^3-27}.$$

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


Express $3.\overline{45}$ as a common fraction in the lowest terms.


Completely factor the following expression: $16x^2 + 8x - 24$.

Solution: \boxed{8(2x + 3)(x - 1)}


Given the equation:
\[
\frac{\sin^6 \alpha}{\sin^3 \beta} + \frac{\cos^6 \alpha}{\cos^3 \beta} = 1,
\]
find the sum of all possible values of:
\[
\frac{\cos^6 \beta}{\cos^3 \alpha} + \frac{\sin^6 \beta}{\sin^3 \alpha}.
\]


Calculate \[
\frac{1}{4} \cdot \frac{2}{5} \cdot \frac{3}{6} \cdot \frac{4}{7} \cdots \frac{59}{62} \cdot \frac{60}{63}
\]
Express your answer as a common fraction.


A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the thousands digit and the hundreds digit of the number are both even?


Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $2310$, then what is the sum of the numbers written on the faces?


What number is thirty times a multiple of every integer?


Elisa has 8 daughters and no sons. Some of her daughters have 7 daughters each, and the rest have none. Elisa has a total of 36 daughters and granddaughters, all of whom have no great-granddaughters. How many of Elisa's daughters and granddaughters have no daughters?


A truncated cone has horizontal bases with radii 24 and 6. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?


In triangle \(DEF,\) \(DE = 25\) and \(EF = 20.\) Find the largest possible value of \(\sin D.\)


If $11 = x^4 + \frac{1}{x^4}$, what is the greatest possible value of $x + \frac{1}{x}$?

Solution: \boxed{\sqrt{\sqrt{13} + 2}}


Maria rewrites the quadratic $4x^2 - 24x + 35$ in the form of $(dx + e)^2 + f,$ where $d,$ $e,$ and $f$ are all integers. What is $de$?


For how many different negative values of $x$ is $\sqrt{x + 196}$ a positive integer?


What is the smallest integer \( n \) such that \( n^2 + n \) is divisible by some but not all integer values of \( k \) when \( 1 \leq k \leq n \)?

Solution: \boxed{n = 4}


In a right-angled triangle $LMN$, suppose $\sin N = \frac{5}{13}$ with $LM = 10$. Calculate the length of $LN$.


Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers?


In $\triangle XYZ$, $\angle X = 90^\circ$ and $\tan Z = \frac{5}{12}$. If $YZ = 60$, what is the length of $XY$?


A park features a central circular fountain, surrounded by a flower garden and an outer circular walking path. The fountain has a diameter of 12 feet, the garden ring is 10 feet wide, and the walking path is 6 feet wide in all places. Determine the diameter, in feet, of the circle that forms the outer boundary of the walking path.


Emily shuffled two standard decks of 52 playing cards together and arranged them in the following sequence repeated: $$A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, Joker, A, 2, 3, ...$$ What is the $52^\text{nd}$ card in this sequence?


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $100$, and $BC=20$. Find the volume of the tetrahedron.


What is the degree measure of the smaller angle formed by the hands of a clock at 9:15?


A ball is dropped from a height of 80 feet and rebounds to three-fourths of the distance it falls each time it bounces. Calculate the total distance the ball travels by the time it hits the ground for the fourth time.


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


Find the product of all $x$ such that the expression $\frac{x^2 - 4x + 4}{x^2 - 5x + 6}$ is undefined.


Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{6 digits}.\] Calculate the sum of the digits of $N$.


Let \( p, q, r, \) and \( s \) be points on a line such that \( |p-q| = 1 \), \( |q-r| = 4 \), and \( |r-s| = 5 \). What is the sum of all possible values of \( |p-s| \)?


Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$.


A triangle has three different integer side lengths and a perimeter of 24 units. What is the maximum length of any one side if the shortest side is at least 4 units?


Chandra has expanded her collection and now has six bowls and six glasses, each set containing items in red, blue, yellow, green, orange, and purple. Each item in a set is uniquely colored. How many different pairings of a bowl and a glass can Chandra make?


In a circle with center $O$, angle $MAN$ measures 60 degrees. Point $M$ is on the circle, point $A$ is at the center, and point $N$ is also on the circle. What is the measure of minor arc $MN$ in degrees?


How many integer palindromes are there between 300 and 800?


**What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 10:30?**


What is the smallest positive integer that has a remainder of 1 when divided by 4, a remainder of 2 when divided by 5, and a remainder of 3 when divided by 6?


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


The sum of four positive integers that form an arithmetic sequence is 50. Of all such possible sequences, what is the greatest possible third term?


What is the minimum number of colors required to color a tessellation composed of hexagons, such that no two adjacent hexagons share the same color?


What is the greatest possible number of digits in the product of a 5-digit whole number and a 3-digit whole number?


Let \(b_n = \frac{2^n-1}{3}\). Define \(c_n\) to be the greatest common divisor of \(b_n\) and \(b_{n+1}\). What is the maximum possible value that \(c_n\) can take on?


A circular pizza with a diameter of $16\text{ cm}$ is cut into four equal-sized sector-shaped slices. Let $m$ be the number of centimeters in the length of the longest line segment that can be drawn in one of these slices. What is $m^2$?


How many 12-digit positive integers have all digits either 1 or 2, and have numbers ending in 12, but have no two consecutive 1's?


Consider a polynomial $f(x)$ of degree $15$ that, when divided by $d(x)$, results in a quotient of degree $7$ and a remainder of $4x^4 + 5x^3 - 2x^2 + 3x + 20$. What is the degree of $d(x)$?


For how many different negative values of $x$ is $\sqrt{x + 200}$ a positive integer?


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


A square has a side length of $70\sqrt{2}$ cm. A circle is drawn such that its diameter is the same as the diagonal of this square. What is the length of the diagonal of the square, and what is the circumference of the circle?

Solution: \boxed{140\pi \text{ cm}}


You are given that \(7^{500} \equiv 1 \pmod{1250}\). What are the last three digits of \(7^{10000}\)?


A roll of quarters now contains 45 quarters and a roll of dimes contains 60 dimes. Emily has a jar that contains 105 quarters and 215 dimes. Jack has a jar that contains 140 quarters and 340 dimes. Emily and Jack combine all their quarters and dimes and roll them as completely as possible. How much, in dollars, are the leftover quarters and dimes worth after making all possible complete rolls?


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 two books are in the library and at least one is checked out?


As $m$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $15m+4$ and $14m+3$?


In triangle $DEF$, where $DE = 10$, $EF = 8$, and $FD = 6$. Let $J$ be the incenter of triangle $DEF$. Represent the position vector of $J$, denoted as $\overrightarrow{J}$, in terms of the position vectors of the vertices of the triangle, i.e., $\overrightarrow{D}, \overrightarrow{E}, \overrightarrow{F}$ like so:
\[
\overrightarrow{J} = p \overrightarrow{D} + q \overrightarrow{E} + r\overrightarrow{F}
\]
where $p$, $q$, and $r$ are constants such that $p + q + r = 1$. Find the ordered triple $(p, q, r)$.

Solution: \boxed{\left(\frac{1}{3}, \frac{1}{4}, \frac{5}{12}\right)}


Let $A$ be the product of the divisors of $60$. How many distinct prime factors does $A$ have, and how many divisors does $A$ itself have?


The set of vectors $\left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ k \\ 6 \end{pmatrix} \right\}$ is linearly dependent. Find all possible values of $k.$ Enter all the possible values, separated by commas.


An equilateral triangle has a perimeter of 30 inches, a square has a perimeter of 40 inches, and a regular pentagon has a perimeter of 50 inches. What is the ratio of the length of the side of the triangle to the length of the side of the square to the length of the side of the pentagon? Express your answer as a common fraction.


For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 20\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 200x + c.\] What is $f(1)$?


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


The points $(12, -7)$ and $(-6, 4)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle?


A box contains tiles, numbered 1, 2, 3, ..., 49, 50. Only tiles which are marked with a number congruent to $1 \pmod{4}$ are blue. One tile is chosen randomly from the box. What is the probability that the tile is blue?


Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=\frac{20}{3}-x$.


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


In how many ways can 7 people be seated in a row of chairs if three of them, Wilma, Paul, and Natasha, each refuse to sit next to the other two?


What is the probability that the square root of a randomly selected two-digit whole number is less than seven? Express your answer as a common fraction.


Evaluate the expression $8^3 + 4a(8^2) + 6a^2(8) + a^3$ for $a = 2$.


Using the digits 1, 2, 3, 4, 5, how many even three-digit numbers less than 300 can be formed if each digit can be used more than once but the tens digit must be odd?


A triangle is constructed with wood sticks of lengths 9, 14, and 16 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. What is the length of the smallest piece that can be cut from each of the three sticks to achieve this?


A play has three different male roles, three different female roles, and three 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 nine roles be assigned?


Compute the sum
\[
\sum_{n=1}^{25} \lfloor \sqrt{n} \rfloor.
\]


Let $a, b, c, d$ be real numbers such that
\[a^2 + b^2 + c^2 + d^2 = 10.\]
Find the maximum value of $a^4 + b^2 + c^2 + d^2.$


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


The lengths of the sides of a triangle with positive area are \( \log_{2}8 \), \( \log_{2}48 \), and \( \log_{2}n \), where \( n \) is a positive integer. Find the number of possible values for \( n \).


If 520 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?


How many three-digit positive integers $x$ satisfy $2597x + 763 \equiv 1459 \pmod{17}$?


Given the square of an integer $y$ is 9801, what is the value of $(y+2)(y-2)$?


What is the sum of the digits of the base-2 expression for $310_{10}$?


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


How many different primes are in the prime factorization of $97 \cdot 101 \cdot 104 \cdot 107 \cdot 109$?


Simplify the expression: $[(3+4+5+6)\div3] + [(3\cdot4+9)\div4]$.


Determine values of $a$, $b$, and $c$ such that $a+b+c$ is minimal in the equation:
\[ \sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2 \]
can be reduced to the equivalent equation
\[ \cos ax \cos bx \cos cx = 0. \]


Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 11 a.m., and stay for exactly $n$ minutes. The probability that either one arrives while the other is in the cafeteria is $50 \%,$ and $n = p - q\sqrt {r},$ where $p, q,$ and $r$ are positive integers, and $r$ is not divisible by the square of any prime. Find $p + q + r.$


Find the range of the function $f(x) = \sin^4 x + (\sin x \cos x)^2 + \cos^4 x.$

Solution: \boxed{\left[ \frac{3}{4}, 1 \right]}


What is the shortest distance between the circles defined by \(x^2 - 2x + y^2 + 6y + 2 = 0\) and \(x^2 + 6x + y^2 - 2y + 9 = 0\)?

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


What is the greatest two-digit whole number, the product of whose digits is 12, and whose digits are in increasing order?


Four-fifths of the people in a hall are using five-sixths of the desks. The rest of the people are either standing or using floor cushions. If there are 12 desks that are not being used, how many people are in the hall?


Consider a parallelogram where the coordinates of the vertices are $(1, 3)$, $(5, 3)$, $(7, 8)$, and $(3, 8)$. Calculate the perimeter $p$ of the parallelogram and its area $a$. What is the value of the sum $p + a$?

Solution: \boxed{28 + 2\sqrt{29}}


A coin is biased so that the probability of flipping heads is $\frac{1}{4}$ and the probability of flipping tails is $\frac{3}{4}$. You win $\$4$ if the coin shows heads, and lose $\$3$ if it shows tails. What is the expected value, in dollars, of your winnings after one flip of the coin?


A circle is tangent to the lines \(3x + 4y = 24\) and \(3x + 4y = -16.\) The center of the circle lies on the line \(x - 2y = 0.\) Find the center of the circle.

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


Find the polynomial $p(x)$, with real coefficients, such that $p(3) = 10$ and
\[p(x) p(y) = p(x) + p(y) + p(xy) - 2\] for all real numbers $x$ and $y$.

Solution: \boxed{p(x) = x^2 + 1}


Consider a circle with an equilateral triangle and a regular pentagon inscribed in it. The triangle and pentagon share a common vertex. Label the vertices of the triangle as $A$, $B$, and $C$ where $A$ is the shared vertex, and the vertices of the pentagon as $A$, $P_1$, $P_2$, $P_3$, $P_4$ in clockwise order. Calculate the measure of $\angle BAC$ within the triangle.


One line is parameterized by
\[\begin{pmatrix} 2 + s \\ 4 - ks \\ 3 + ks \end{pmatrix}.\]
Another line is parameterized by
\[\begin{pmatrix} t/3 \\ 2 + t \\ 5 - t \end{pmatrix}.\]
If the lines are coplanar, then find \(k\).


Bertha has 8 daughters and no sons. Some of her daughters have 4 daughters, and the rest have none. Bertha has a total of 40 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


Carlos has a collection of marbles. When he divides his marbles into groups of 5, 7, or 11, he has one marble left over in each case. Additionally, when he divides them into groups of 4, he has 2 marbles left. Assuming Carlos has more than one marble, what is the smallest possible number of marbles Carlos could have in his collection?


In the diagram, if $\angle PQR = 60^\circ$ and triangle $PMQ$ is isosceles with $PM = MQ$, what is the measure of $\angle PMN$? Assume that $\angle MPN$ and $\angle PQR$ are opposite angles, and $PR = RQ$.
[asy]
size(6cm);
pair p = (0, 0); pair m = dir(180 - 30); pair n = dir(180 + 30); pair r = 1.3 * dir(30); pair q = 2 * 1.3 * Cos(60) * dir(-30);
label("$M$", m, N); label("$R$", r, N); label("$P$", p, 1.5 * S); label("$N$", n, S); label("$Q$", q, SE);
draw(m--q--r--n--cycle);
add(pathticks(m--p, s=4));
add(pathticks(q--p, s=4));
add(pathticks(n--p, s=4));
add(pathticks(r--p, 2, spacing=0.9, s=4));
add(pathticks(r--q, 2, spacing=0.9, s=4));
[/asy]


Solution: \boxed{\cos 36^\circ = \frac{1 + \sqrt{5}}{4}}


If $5x + 9 \equiv 3 \pmod{19}$, then what is $3x + 18 \equiv \pmod{19}$?


Both roots of the quadratic equation $x^2 - 65x + k = 0$ are prime numbers. How many possible values of $k$ are there?


The function $f(x)$ satisfies
\[b^2 f(a) = a^2 f(b)\]
for all real numbers $a$ and $b.$ Let's assume $f(3) \neq 0.$ Find
\[\frac{f(7) - f(3)}{f(3)}.\]


What is the greatest possible number of digits in the product of a 4-digit whole number and a modified 3-digit whole number where the 3-digit number is $10^3-3$?


Determine the sum of the series: $3 - 6 + 9 - 12 + 15 - \cdots - 60 + 63 - 66 + 69$.


Suppose \(43 \cdot 67 \cdot 92 \equiv m \pmod{25}\), where \(0 \leq m < 25\).

What is the value of \(m\)?


In Dr. Lee's math class, there are 24 boys and 30 of her students carry rulers to class. If 18 of the students with rulers are girls and there are 5 non-binary students, 3 of whom have rulers, then how many boys did not bring rulers to class?


What is the greatest integer less than or equal to \[\frac{5^{105} + 4^{105}}{5^{99} + 4^{99}}?\]


If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 6, \\
a+b+d &= 2, \\
a+c+d &= 3, \\
b+c+d &= -3,
\end{align*}
what is $ab + cd$?


Consider the following scenario where three stores offer different discounts on a television priced at $149.99$:
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Television $Y$} \\
\hline
Value Market & $\$10$~off the list price~$\$149.99$ \\
Tech Bargains & $30\%$~off the list price~$\$149.99$ \\
The Gadget Hub & $20\%$~off the list price~$\$149.99$ \\
\hline
\end{tabular}
How much cheaper, in cents, is the cheapest store's price compared to the most expensive?


A point has rectangular coordinates $(-5,-7,4)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, -\theta, \phi)$.

Solution: \boxed{(-5, 7, 4)}


My uncle has 12 pieces of art, including 4 prints by Dali. If he hangs the pieces of art in a row in a random order, what is the probability that all four Dali prints will be placed consecutively?


A classroom contains 48 chairs to be arranged in a rectangular array where each row and each column contains at least two chairs, and all the chairs in the room must be included. Each arrangement where the number of rows differs from the number of columns counts as distinct. How many different arrays can be formed?


Find the complex number $z$ such that
\[ |z - 2| = |z + 4| = |z + 2i|.\]

Solution: \boxed{-1 + i}


Simplify $\displaystyle\frac{2-2i}{3+4i}$, where $i^2 = -1.$

Solution: \boxed{-\frac{2}{25} - \frac{14}{25}i}


How many distinct four-digit numbers can be written with the digits $1$, $2$, $3$, $4$, and $5$ if exactly one digit is repeated once?


Find the integer $n,$ $-180 \le n \le 180,$ such that $\cos n^\circ = \cos 430^\circ.$


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


The solution of the equation $9^{x+8} = 16^x$ can be expressed in the form \(x = \log_b 9^8\). What is $b$?


Calculate the number of distinct integers in the list:
\[\left\lfloor \frac{1^2}{500} \right\rfloor, \ \left\lfloor \frac{2^2}{500} \right\rfloor, \ \left\lfloor \frac{3^2}{500} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{500} \right\rfloor.\]


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


A school plans to arrange seating for 360 students at a special event. Students must sit in rows with the same number of students in each row. The setup requires at least 12 rows and each row must have at least 18 students. Determine the sum of all possible values for the number of students in each row, $x$.


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


How many four-digit positive integers have at least one digit that is a 3 or a 7?


In $\triangle ABC,$ $AB=AC=30$ and $BC=28.$ Points $G, H,$ and $I$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $AGHI$?


What is the coefficient of $x^2$ when $-3x^4 - 2x^3 - 4x^2 - 8x + 2$ is multiplied by $2x^3 - 5x^2 + 3x - 1$ and the like terms are combined?


Find the sum of the series $1 + 2\left(\dfrac{1}{3}\right) + 3\left(\dfrac{1}{3}\right)^2 + 4\left(\dfrac{1}{3}\right)^3 + \cdots$.


In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1500$ and $AD = 2800$. Let $\angle A = 40^\circ$, $\angle D = 50^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$.


The time right now is exactly 8:30 a.m. What time will it be in 2423 minutes?

Solution: \boxed{12:53 \text{ a.m.}}


What is the number of degrees in the acute angle formed by the hands of a clock at 9:15?


If $2x - 3y = 15$ and $x + 2y = 8$, what is the value of $x$?