Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f$. Suppose both $f$ and $g$ have minimum values at $y = -25$. If the graphs of the two polynomials intersect at the point $(50, -50),$ what is the value of $a + c$?

Solution: \boxed{a + c = -200}


The Badgers play the Cougars in a series of nine basketball games. Each team has an equal chance of winning each game. What is the probability that the Badgers will win at least five games? Express your answer as a common fraction.


Given the value of $\log_{10}{147583}$, find the consecutive integers $c$ and $d$ between which this value lies, and compute $c+d$.


Rectangle $EFGH$ is the base of pyramid $PEFGH$. If $EF = 10$, $FG = 6$, $\overline{PE}\perp \overline{EH}$, $\overline{PE}\perp \overline{EF}$, and $PF = 20$, then what is the volume of $PEFGH$?


Evaluate the expression \((25x^3y) \cdot (4xy^2z) \cdot \left(\frac{1}{(5xyz)^2}\right)\).


In a circle with center $O$, triangle $ABC$ is circumscribed such that $\angle BOC = 150^{\circ}$ and $\angle AOB = 130^{\circ}$. Find the degree measure of $\angle ABC$.

[asy]
pair A,B,C;
draw(Circle((0,0),20),linewidth(0.7));
label("$O$",(0,0),S);
A=(-16,-12);
C=(16,-12);
B=(3,19.7);
draw(A--B--C--cycle,linewidth(0.7));
label("$130^{\circ}$",(0,0),W);
label("$150^{\circ}$",(0,0.3),E);
draw(C--(0,0)--B);
draw(A--(0,0));
label("$A$",A,SW);
label("$B$",B,NE);
label("$C$",C,SE);
[/asy]


A cylindrical cake is $\frac{1}{2}$ of an inch thick and has a diameter of 18 inches. It is cut into 18 equal slices. What is the number of cubic inches in the volume of one slice? Express your answer in terms of $\pi$.


A whole number is said to be "9-heavy" if the remainder when the number is divided by 9 is exactly 6. What is the least four-digit 9-heavy whole number?


Let $x_1, x_2, x_3, x_4$ be positive real numbers such that $2x_1 + 3x_2 + 4x_3 + 5x_4 = 120.$ Find the smallest possible value of
\[x_1^2 + x_2^2 + x_3^2 + x_4^2.\]


If $3a + 4b = 0$ and $a$ is three less than twice $b$, what is $5b$?


How many 5-letter words with at least one vowel can be constructed from the letters A, B, C, D, E, F, G, and I? (Note that A, E, and I are vowels. Any word is valid, not just English language words, and letters may be used more than once.)


How many even divisors of $9!$ are multiples of 15?


In the diagram, $\triangle ABE$, $\triangle BCE$, and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=36$. Find the length of $CE$.


Calculate $1005^2 - 995^2 - 1007^2 + 993^2$ without the use of a calculator.


Joe and JoAnn each bought 15 ounces of coffee in a 20-ounce cup. Joe drank 3 ounces of his coffee and then added 3 ounces of cream. JoAnn added 3 ounces of cream, stirred the coffee well, and then drank 3 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? Express your answer as a fraction.


In triangle $ABC$, where $\angle C = 90^\circ$, $AB = 13$, and $BC = 2\sqrt{10}$. What is $\tan A$?


Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x - \cos x = \sqrt{2}$. Enter all the solutions, separated by commas.


How many numbers are in the list $$ 2, 5, 8, \ldots, 2011, 2014 ?$$


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

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


Two solid cones have the same volume. One cone's height is four times the height of the other. If the radius of the taller cone is 5 units, how many units are in the length of the radius of the shorter cone? Express your answer in simplest radical form.


Suppose $a$ and $b$ are positive integers where $a$ has $4$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$?


For how many integers \( n \) between 1 and 200 is the greatest common divisor of 21 and \( n \) equal to 7?


The polynomial
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
has coefficients that are all integers, and has roots $-3, 6, 10,$ and $-\frac{1}{4}$. If $e$ is a positive integer, what is its smallest possible value?


Let \(a\), \(b\), and \(c\) be the roots of the polynomial equation
\[x^3 - 5x + 7 = 0.\]
Find the monic polynomial, in \(x\), whose roots are \(a + 3\), \(b + 3\), and \(c + 3\).

Solution: \boxed{x^3 - 9x^2 + 22x - 5}


Solve the equation $9^{x+6} = 10^x$ and express the solution in the form $x = \log_d 9^6$. What is $d$?


If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, what is the angle between a vector $\mathbf{c}$ which is perpendicular to $\mathbf{a}$, and $\mathbf{b}$?


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


A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element?


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{1}$, meaning that if the length of the object was originally $x$, then it is now $x + x$. On the second day, he increased the object's new length by $\frac{1}{2}$; on the third day, he increased it by $\frac{1}{3}$; and so on. On the $n^{\text{th}}$ day of performing this magic, Barry makes the object's length exactly 50 times its original length. What is the value of $n$?


Tom cycled a total of $4372_8$ miles in base eight notation. Convert this distance into base ten.


In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 12$ and $CP = 4.$ If $\tan \angle APD = 2,$ then find $AB.$


Let $g(x) = x^3 - 2x^2 + x + 2023$. What is the greatest common divisor of $g(50)$ and $g(52)$?


Determine the domain of the real-valued function $$f(x) = \sqrt{6 - \sqrt{7 - \sqrt{x}}}.$$

Solution: \boxed{[0, 49]}


From the following infinite list of numbers, how many are integers? $$\sqrt{10000}, \sqrt[3]{10000}, \sqrt[4]{10000}, \sqrt[5]{10000}, \sqrt[6]{10000}, \ldots$$


If $x$, $y$, and $z$ are positive numbers such that $xy=30$, $xz=60$, and $yz=90$, what is the value of $x+y+z$?


A polynomial of degree $15$ is divided by $d(x)$ to give a quotient of degree $9$ and a remainder of $5x^4 + 6x^3 - 2x + 7$. What is $\deg d$?


A lamp post is supported by a steel cable which extends from the top of the lamp post to a point on the ground 4 meters away from its base. When Mike walks 3 meters away from the base of the post toward the point where the cable touches the ground, the top of his head just touches the cable. Mike is 1.6 meters tall. How many meters tall is the lamp post?


Let $g$ be a non-constant polynomial such that
\[g(x - 1) + g(x) + g(x + 1) = \frac{[g(x)]^2}{4026x}\]for all nonzero real numbers $x.$  Find the sum of all possible values of $g(1).$


If $81^4 = 27^x$, what is the value of $3^{-x}$? Express your answer as a common fraction.


A girl has the following coins in her purse: 3 pennies, 3 nickels, 1 quarter, and 2 half-dollars. She randomly picks two coins, records the sum of their values, and returns them. She continues to do this multiple times. What is the maximum number of different sums she can record?


Tom and Jerry live 2 miles apart. Tom takes 6 minutes to walk directly to Jerry's house. At the same rate, how long would it take Tom to walk 5 miles to Sam’s house?

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


Find the minimum value of
\[
(\tan x + \cot x)^2 + (\sin x + \cos x)^2
\]
for \(0 < x < \frac{\pi}{2}\).


A student is preparing for a 150-point exam, where the points earned vary directly with the time spent studying. If a student scores 90 points after studying for 2 hours, how much would the student score if they spent 5 hours studying? Assume that for every hour beyond 3 hours, the student gets an additional 10% of the score they would have received at 3 hours.

Solution: \boxed{162 \text{ points}}


Angela has $a$ marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, and Daryl has seven times the number of marbles Caden has. If in total Angela, Brian, Caden, and Daryl have 126 marbles, what is the value of $a$?


Moving along a line in the Cartesian plane, when the $x$-value increases by 2 units, the $y$-value increases by 5 units. Determine how much the $y$-value will increase when the $x$-value increases by 8 units.


Six numbers are 2347, 2573, 2689, 2725, 2839, and 2841. Four of these numbers have a mean of 2666. What is the mean of the other two numbers?


Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its complete factorization.


Solve the congruence $15n \equiv 9 \pmod{47}$.


What is the slope of the line passing through the points \( (1, 3) \) and \( (4, -6) \)? Additionally, calculate the midpoint of the segment connecting these points.

Solution: \boxed{(2.5, -1.5)}


Determine the value of $2a^2 - 3b^2 + 4ab$ if $a = 4$ and $b = -3$.


In a volleyball tournament, Team E competes against Team F and Team G competes against Team H on Saturday. The two winning teams on Saturday face each other for the championship on Sunday, while the losers play for the third and fourth places. Team E has a veteran advantage against Team F, and Team G has a similar advantage against Team H, meaning each advantaged team is more likely to win their Saturday match. Assuming there are no ties and the advantage does translate to a win, what is the total number of possible four-team ranking sequences at the end of the tournament?


In triangle $\Delta ABC$, $AC = BC$, $m\angle DCB = 50^{\circ}$, and line $CD$ is parallel to $AB$. Point $E$ is on extension of $DC$. Determine $m\angle ECA$.


Let
\[
\mathbf{A} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}.
\]
Compute $\mathbf{A}^{93}$.

Solution: \boxed{\mathbf{A}^{93} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}}


The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:3:6$. The coefficients can be written in the form ${n\choose k} : {n\choose k+1} : {n \choose k+2}$. Find the sum of all possible values of $n+k$.


David's quiz scores so far are 85, 88, 90, 82, and 94. What score does he need to get on the sixth quiz to make the arithmetic mean of the six scores equal 90?


The prime factorization of 2560 is $2^8 \times 5^1$. How many of its positive integer factors are perfect squares?


Determine the radius of the circle given by the equation $4x^2 - 8x + 4y^2 - 16y + 20 = 0.$


The admission fee for an exhibition is $ \$30$ per adult and $ \$15$ per child. Last Wednesday, the exhibition collected $ \$2250$ in admission fees from at least one adult and at least one child. Of all the possible ratios of adults to children at the exhibition last Wednesday, which one is closest to $ 1$?


Given the points (4, 15), (7, 25), (13, 42), (19, 57), and (21, 65), what is the sum of the $x$-coordinates of the points that lie in the region above the line $y = 3x + 5$ in the coordinate plane?


Below is the graph of \( y = a \sin (bx + c) + d \) for some positive constants \( a, \) \( b, \) \( c, \) and \( d. \) The graph completes five full cycles between \( x = 0 \) and \( x = 2\pi \). Find \( b. \)


Acme Corporation has released a new type of alphabet soup that includes the vowels A, E, I, O, U, each appearing five times, and a new 'semi-vowel' Y, which appears three times. How many five-letter words can be formed from this new bowl of Acme Extended Vowel Soup?


A list consists of eleven positive integers, of which six integers are given as $3, 5, 7, 9, 1, 8$. What is the largest possible value of the median of this list?


Two sides of a right triangle have the lengths 6 and 8. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth.


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


Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+2mn$ for all positive integers $m$ and $n.$ Find $b_{10}.$


A circle is tangent to the lines $3x - 4y = 20$ and $3x - 4y = -40.$ The center of the circle lies on the line $x - 3y = 15.$ Find the center of the circle.

Solution: \boxed{(-18, -11)}


Let $a > b > c$ be three non-positive integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest absolute value of $b$?


Triangle $XYZ$ has side lengths $XY=7$, $YZ=8$, and $XZ=9$. A pair of ants start simultaneously from $X$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $W$. What is $YW$?

Solution: \boxed{YW = 5}


Five years ago, there were 30 trailer homes on Maple Drive with an average age of 15 years. At that time, a group of trailer homes, which were all 3 years old, was then added to Maple Drive. Today, the average age of all the trailer homes on Maple Drive is 17 years. How many slightly older trailer homes were added five years ago?


Determine $a+b$ such that the value of $\log_{10}{1679616}$ is between the consecutive integers $a$ and $b$.


Given the numbers \(a\), \(b\), \(c\), and \(d\) assigned the values 1, 5, 3, and 4, but not necessarily in that order, determine the largest possible value of the sum of the four products \(ab\), \(bc\), \(cd\), and \(da\).


What are the least common multiple and the greatest common divisor of 180 and 630?


A list of five different positive integers has a median of 3 and a mean of 10. What is the maximum possible value of the list's largest element?


How many integers $n$ are there such that $4 \leq n \leq 15$ and $12321_n$ (the number written as $12321$ in base $n$) is a perfect square?


The integer $y$ has 24 positive factors. The numbers 18 and 20 are factors of $y$. What is $y$?


Let \( b_n = 5^n + 7^n \). Determine the remainder upon dividing \( b_{97} \) by \( 36 \).


A rectangle has a perimeter of 80 meters and each side has an integer length. How many non-congruent rectangles meet these criteria?


Ray climbs a flight of stairs in two distinct ways. When he climbs 6 steps at a time, he has 4 steps left at the top. When he climbs 7 steps at a time, he has 3 steps left at the top. What is the smallest possible value of the number of steps ($n$) in the staircase that is greater than 20?


Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 6.$ Find the minimum value of
\[\frac{9}{x} + \frac{25}{y} + \frac{49}{z}.\]


An infinite geometric series has a common ratio of $\frac{1}{4}$ and its sum is 40. What is the second term of the series?


Write the expression $\frac{5+5c}{7} + 3$ as a single fraction.


In triangle $XYZ$, $XY=15$ and $XZ=9$. The angle bisector of $\angle X$ intersects $YZ$ at point $D$, and point $M$ is the midpoint of $XD$. Let $P$ be the point of intersection of $XZ$ and $BM$. The ratio of $ZP$ to $PX$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


In Santiago's junior football club, there are 18 players on the team, including reserves. Of these, four are goalies. During a particular training session, the coach organizes a challenge to see which goalie can block the most number of penalty kicks. In each round, one goalie defends the goal while the remaining players, including the other goalies, each take turns trying to score.

How many penalty kicks must be taken to ensure that every player has had a chance to take a shot against each of the goalies?


Find the number of real solutions to the equation
\[\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{120}{x - 120} = x.\]


Nina's basketball team needs to select their new uniform design. The ninth-graders will choose the color of the shorts (black, gold, or blue) and the tenth-graders will choose the color of the jersey (black, white, or gold). The two groups will make their decisions independently, and each color option is equally likely to be chosen. What is the probability that the color of the shorts will be different from the color of the jersey? Express your answer as a common fraction.


Serena and Liam each think of a polynomial. Each of their polynomials is monic, has degree 5, and has the same positive constant term and the same coefficient of $z^2$. The product of their polynomials is \[z^{10} + 2z^9 + 3z^8 + 4z^7 + 5z^6 + 6z^5 + 9z^4 + 4z^3 + 10z^2 + 5z + 9.\] What is the constant term of Liam's polynomial?


All triangles have the same value, all circles have the same value, and all squares have the same value. What is the sum of four squares? \begin{align*}
\Delta + \bigcirc + \Delta + \square &= 27\\
\bigcirc + \Delta + \bigcirc + \square &= 25\\
\square + \square + \square + \Delta &= 39
\end{align*}


Calculate the volume in cubic feet of a round swimming pool which is 20 feet in diameter and has a depth that starts at 3 feet on one end and increases linearly to 6 feet at the other end. Express your answer in terms of $\pi$.


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


A rectangle has a perimeter of 80 inches and each side has an integer length. How many non-congruent, non-square rectangles meet these criteria?


If $x$ and $y$ are positive integers less than 20 for which $x + y + xy = 99$, what is the value of $x + y$?


How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit?


Two distinct positive integers $x$ and $y$ are factors of 48. If $x \cdot y$ is not a factor of 48, what is the smallest possible value of $x \cdot y$?


A cylinder has a volume of $72\pi$ $\text{cm}^3$. Calculate the combined volume of a cone with the same height as the cylinder and a hemisphere that fits exactly on the top of the cylinder's open end. The radius of the hemisphere is equal to the radius of the cylinder and the cone. Express your answer in terms of $\pi$.

Solution: \boxed{72\pi \text{ cm}^3}


What is the smallest positive perfect square that is divisible by both 5 and 7?


Compute the domain of the function $$g(x) = \frac{1}{\lfloor x^2 - 9x + 21 \rfloor}.$$

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


In a different hyperbola, the center is at $(1,1),$ one focus is at $(1,8),$ and one vertex is at $(1,-2).$ The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\] Find $h + k + a + b.$

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


Let \( g(x) \) be a function defined for all positive real numbers such that \( g(x) > 0 \) for all \( x > 0 \) and
\[ g(x - y) = \sqrt{g(xy) + 3} \]
for all \( x > y > 0 \). Determine \( g(103) \).


The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(45) = 15,$ find $f(60).$


The exterior angle at one vertex of a triangle is 120 degrees, and the interior angles of the triangle are in the ratio 4:5:6. Find the measure of the largest interior angle of the triangle.


A prism has 27 edges. How many faces does this prism have?


A square and four circles, each with a radius of 7 inches, are placed inside it as previously described. What is the total area, in square inches, of the square?


A Senate committee consists of 4 Democrats and 6 Republicans. All members are distinguishable. In how many ways can they sit around a circular table so that no two Democrats sit next to each other?


A point $(x, y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,3)$, and $(0,3)$. What is the probability that $2x < y$?


A sequence of integers is defined as follows: $a_i = i$ for $1 \le i \le 2,$ and
\[a_i = a_1 a_2 \dotsm a_{i - 1} - 1\]for $i > 2.$  Evaluate $a_1 a_2 \dotsm a_{100} - \sum_{i = 1}^{100} a_i^2.$


For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{30}$ a repeating decimal?


The sequence $9720, 3240, 1080, \ldots$ is created by repeatedly dividing by 3. How many integers are in this sequence?


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


What is the greatest common factor of 36, 60, and 90?


Compute $\tan 30^\circ + 4 \sin 30^\circ.$

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


Find the inverse of the matrix
\[
\begin{pmatrix} a & a+1 \\ -1 & 4 \end{pmatrix}.
\]
where \( a \neq -\frac{15}{2} \).

Solution: \boxed{\begin{pmatrix} \frac{4}{5a+1} & \frac{-a-1}{5a+1} \\ \frac{1}{5a+1} & \frac{a}{5a+1} \end{pmatrix}}


Rationalize the denominator of $\frac{3}{4\sqrt{5} + 3\sqrt{7}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?


Simplify $\sqrt{50x} \cdot \sqrt{18x} \cdot \sqrt{32x}$. Express your answer in simplest radical form in terms of $x$.


In a new game, Jane and her brother each spin a spinner once. The spinner has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.


Express \( 4.\overline{8} \) as a common fraction.


What is the total area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 8 units and lateral edges measuring 10 units?


Triangle $PQR$ is equilateral. Lines $\overline{ST}$, $\overline{UV}$, and $\overline{WX}$ are parallel to $\overline{QR}$, and $PS = SU = UW = WX$. What is the ratio of the area of trapezoid $WXQR$ to the area of triangle $PQR$?


Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number appearing on five slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b \neq a$. Calculate the value of $\frac{q}{p}$.


The degree measures of the angles in a convex 15-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.


What is the least positive integer $n$ such that $4500$ is a factor of $n!$?


An oreo shop now sells $5$ different flavors of oreos, $3$ different flavors of milk, and $2$ different flavors of cookies. Alpha and Gamma decide to purchase some items. Since Alpha is picky, he will order no more than two different items in total, avoiding replicas. To be equally strange, Gamma will only order oreos and cookies, and she will be willing to have repeats of these flavors. How many ways can they leave the store with exactly 4 products collectively?


Suppose we want to divide 12 dogs into three groups, one with 4 dogs, one with 6 dogs, and one with 2 dogs. How many ways can we form the groups such that Fluffy is in the 4-dog group and Nipper is in the 6-dog group?


Given that the volume of the parallelepiped determined by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 4, find the volume of the parallelepiped determined by vectors $\mathbf{a} + 2\mathbf{b}$, $\mathbf{b} + \mathbf{c}$, and $\mathbf{c} + 5\mathbf{a}$.


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


The function
\[ f(x) = \left\{ \begin{aligned}
x+3 & \quad \text{if } x < 5 \\
\sqrt{x-1} & \quad \text{if } x \ge 5
\end{aligned} \right. \]
has an inverse $f^{-1}$. Find the value of $f^{-1}(-2) + f^{-1}(-1) + \dots + f^{-1}(4) + f^{-1}(5)$.


Fully factor the polynomial expression: $3y^2 - 27$.

Solution: \boxed{3(y + 3)(y - 3)}


How many 6-digit numbers have at least two zeros?


What non-zero, real value of \( y \) satisfies \( (3y)^6 = (9y)^5 \)? Express your answer as a common fraction.


What is the positive difference between the median and the mode of the data given in the stem and leaf plot below? In this plot $5|8$ represents $58.$

\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
1 & $2 \hspace{2mm} 3 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 5 \hspace{2mm} 7$ \\ \hline
2 & $2 \hspace{2mm} 2 \hspace{2mm} 2 \hspace{2mm} 6 \hspace{2mm} 6 \hspace{2mm} 8$ \\ \hline
3 & $1 \hspace{2mm} 1 \hspace{2mm} 8 \hspace{2mm} 9 \hspace{2mm} \hspace{1.9mm}$ \\ \hline
4 & $ 0 \hspace{2mm} 1 \hspace{2mm} 2 \hspace{2mm} 3 \hspace{2mm} \hspace{1.9mm}$ \\ \hline
5 & $ 2 \hspace{2mm} 8 \hspace{2mm} 9 \hspace{2mm} \hspace{2mm} \hspace{2mm} \hspace{1.5mm}$ \\\hline
\end{tabular}


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


Consider a similar layout of an array of unit squares, with a center square surrounded by successive rings of squares. The first ring around the center square contains 8 unit squares, the second ring contains 16 unit squares, and so on, adding 8 squares for each subsequent ring. What is the cumulative number of unit squares from the first ring up to and including the $50^{th}$ ring?


Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2014$, and $a^2 - b^2 + c^2 - d^2 = 2014$. Find the number of possible values of $a$.


My school's Chess Club has 24 members. It needs to select 3 officers: president, secretary, and treasurer. Each person can hold at most one office. Two of the members, Alice and Bob, will only serve together as officers. In how many ways can the club choose its officers?


A library has $1870$ books stacked in $55$ boxes. If each box originally contains an equal number of books, and the librarian decides to repack the books so that there are now $36$ books in each new box, how many books will be left over after repacking?


How many times does the digit 9 appear in the list of all integers from 1 to 700?


Solution: \boxed{-12t^6 - 8t^5 + 25t^4 - 21t^3 - 22t^2 + 29t - 15}


Given the quadratic function $y = 2x^2 - 8x + 15$, determine the least possible value of $y$ and find the corresponding value of $x$ that achieves this minimum.

Solution: \boxed{y_{\text{min}} = 7 \text{ at } x = 2}


The graph of the parabola $x = 3y^2 - 9y + 5$ has an $x$-intercept $(a, 0)$ and two $y$-intercepts $(0, b)$ and $(0, c)$. Find $a + b + c$.


Quantities \( a \) and \( b \) vary inversely. When \( a \) is \( 1500 \), \( b \) is \( 0.45 \). Determine the value of \( b \) when \( a \) is \( 3000 \). Express your answer as a decimal to the nearest thousandths.


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


If $A\,345\,B67$ is divisible by $9,$ where $A$ and $B$ each represent a single digit, what is the sum of all possible values of the sum $A+B?$


A Senate committee consists of 6 Democrats and 4 Republicans. In how many ways can they sit around a circular table, assuming all members are distinguishable, and two seatings are considered the same if one is a rotation of the other?


In triangle $DEF$, $DE = 10$, $DF = 15$, and $EF = 17$. The medians $DM$, $EN$, and $FO$ of triangle $DEF$ intersect at the centroid $G$. Let $P$ be the foot of the altitude from $G$ to $EF$. Calculate the length of $GP$.


Triangle $AHI$ is equilateral. $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ with $AB = BD = DF = FH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$ if $\overline{AF} = \frac{5}{6}\overline{AH}$.


Compute the distance between the parallel lines given by
\[
\begin{pmatrix} 4 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -6 \end{pmatrix}
\]
and
\[
\begin{pmatrix} 3 \\ -2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -6 \end{pmatrix}.
\]


A wire is cut into two pieces, one of length $x$ and the other of length $y$. The piece of length $x$ is bent to form a square, and the piece of length $y$ is bent to form a regular octagon. The square and the octagon have equal perimeter. What is $\frac{x}{y}$?


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 48 dips?

Solution: \boxed{22.5 \text{ daps}}


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


Compute \[ \left\lfloor \dfrac {2021^3}{2019 \cdot 2020} - \dfrac {2019^3}{2020 \cdot 2021} \right\rfloor,\]where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


$x$ is a real number such that $x + \frac{1}{x} = 4$. Let $S_m = x^m + \frac{1}{x^m}$. Determine the value of $S_6$.


Define the sequence $\{b_k\}$ by starting with $b_1 = 2$ and having the recurrence relation $b_{m+n} = b_m + b_n + 2mn$ for all positive integers $m$ and $n$. Compute $b_{10}$.


A regional conference involves 4 schools, each having 5 members. To conduct a special presidency meeting, the following conditions must be met:

i. Select one school as the host, which will then send 3 representatives.
ii. Each of the remaining three schools sends 1 representative.

How many different ways can the representatives be chosen for the presidency meeting?


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


April has five different basil plants and three different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?


In how many different ways can 4 men and 5 women be placed into two groups of three people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable.


Find the real root of the equation \[\sqrt{x} + \sqrt{x+4} = 12.\]


Determine the number of positive integers $a$ less than $15$ such that the congruence $ax \equiv 1 \pmod{15}$ has a solution in $x$, and find the sum of all such integers.


Jay and Paul, initially 3 miles apart in a park, start walking in opposite directions. Jay walks at a speed of 1 mile every 20 minutes, and Paul walks at a speed of 3 miles every 40 minutes. After 2 hours, how far apart are Jay and Paul?


A semicircle is drawn with diameter $PQ$ measuring 10 cm. Square $PQRS$ is constructed outside the semicircle, sharing the diameter $PQ$ as one of its sides. If point $N$ is the midpoint of arc $PQ$, what is the length of segment $NR$? Express your answer in simplest radical form.


The chart illustrates the number of home runs hit by the top players in a month. What is the average number of home runs hit by these players?

```
Number of Home Runs   | 5 | 6 | 7 | 9 | 11 |
-----------------------|---|---|---|---|----|
Number of Players      | 7 | 5 | 4 | 3 |  1 |
```


What is the smallest whole number $b$ such that 87 can be expressed in base $b$ using only three digits?


Let \(a\) and \(b\) be positive real numbers such that \(a + b = 2.\) Find the set of all possible values of \(\frac{1}{a} + \frac{1}{b}.\)

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


Find the value of $k$ so that the line $4x + 6y + k = 0$ is tangent to the parabola $y^2 = 32x$.


Convert the point in spherical coordinates \( (\rho, \theta, \phi) = \left( 15, \frac{3\pi}{4}, \frac{\pi}{4} \right) \) to rectangular coordinates.

Solution: \boxed{\left(-\frac{15}{2}, \frac{15}{2}, \frac{15\sqrt{2}}{2}\right)}


A dodecahedron has 12 pentagonal faces and each vertex is shared by 3 faces. How many triangles can be formed using the vertices of a dodecahedron, assuming each vertex can only be used once?


How many 5-letter words with at least two consonants can be constructed from the letters $A$, $B$, $C$, $D$, $E$, and $F$? (Note that $B$, $C$, $D$, and $F$ are consonants, any word is valid, not just English language words, and letters may be used more than once.)


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


In a history class, students are assigned to read a 900-page documentary book. Two students, Ella and John, plan to divide the reading to finish it together equally fast. Ella reads at a pace of one page every 40 seconds, and John reads at a pace of one page every 50 seconds. They decide Ella will start the book from page 1 and John will pick up at the page Ella stops. Determine the last page that Ella should read so that both spend the same amount of time reading.


A pie chart shows the distribution of populations in a group of towns. Determine what percent of the towns have populations between 5,000 and 20,000 residents.
[asy]
draw(Circle((0,0),25),linewidth(1));
draw((-24,-7)--(0,0)--(24,-7),linewidth(1));
draw((0,0)--(7,-24),linewidth(1));
label("40\%",(0,12));
label("35\%",(-10,-10));
label("25\%",(10,-10));
label("\textbf{Distribution of Town Populations}",(0,25),N);
label("\tiny{5,000 to 20,000}",(30,15),N);
label("\tiny{less than 5,000}",(-30,-15),S);
label("\tiny{20,000 or more}",(30,-15),S);
label("\tiny{residents}",(30,13),N);
label("\tiny{residents}",(-30,-17),S);
label("\tiny{residents}",(30,-17),S);
[/asy]


Alice conducted a survey about common misconceptions regarding foxes among a group of people. She found that $75.6\%$ of the participants believed that foxes are typically domestic animals. Among those who held this belief, $30\%$ thought that foxes could be trained like domestic dogs. Since foxes are not typically trainable like domestic dogs, these 27 people were incorrect. How many people in total did Alice survey?


If $p$ and $q$ are positive integers less than $30$, for which $p + q + pq = 119$, what is the value of $p + q$?


Find the distance between the planes \(2x - 4y + 2z = 10\) and \(x - 2y + z = 5.\)


Let $x,$ $y,$ $z$ be nonnegative integers. Define
\begin{align*}
A &= \sqrt{x + 3} + \sqrt{y + 6} + \sqrt{z + 11}, \\
B &= \sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}.
\end{align*}
Find the minimum value of $A^2 - B^2.$


In a recent examination, Professor Chang recorded the score distribution for her 100 students as shown below. Calculate the mean percentage score of the class.

\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&10\\\hline
90&15\\\hline
80&20\\\hline
70&30\\\hline
60&20\\\hline
50&4\\\hline
40&1\\\hline
\end{tabular}


Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = 2|c| > 0$ and
\[az^2 + bz + c = 0.\]Find the largest possible value of $|z|.$

Solution: \boxed{\frac{1 + \sqrt{3}}{2}}


The Big Sixteen Basketball League consists of two divisions, each with eight teams. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled?


In the diagram, three identical circles touch each other, and each circle has a circumference of 72. What is the perimeter of the shaded region formed by the intersection of the three circles?


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


When $5000^{50}$ is expanded out, the result is $1$ followed by how many zeros?


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 $BC=10$. Find the volume of the tetrahedron.


Determine the number of integer values for \( c \) in the range \([0, 500]\) for which the equation
\[ 5 \lfloor x \rfloor + 3 \lceil x \rceil = c \]
has a solution for \( x \).


Find the number of integers $n$ that satisfy
\[30 < n^2 < 200.\]


Given $\overline{AD} \| \overline{FG}$, find the measure of angle $EFG$.

[asy]
import olympiad;

pair A = (-15,20);
pair B = (-12,35);
pair C = (35,50);
pair D = (35,20);
pair E = (14,20);
pair F = (0,0);
pair G = (40,0);

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

label("F", F, W);
label("G", G, ENE);
label("C", C, N);
label("A", A, W);
label("D", D, ENE);
label("E", E, SE);
label("B", B, NW);

draw(scale(20)*anglemark(G, F, C));
draw(shift(E)*scale(35)*shift(-E)*anglemark(B, E, A));
draw(shift(E)*scale(20)*shift(-E)*anglemark(C, E, B));

label("$x$", (6,20), NW);
label("$3x$", (13,25), N);
label("$2x$", (5,0), NE);
[/asy]


The arithmetic mean, geometric mean, and harmonic mean of $x$, $y$, $z$ are $10$, $7$, and $5$ respectively. What is the value of $x^2+y^2+z^2$?


What is the median of the sequence consisting of the first ten positive integers followed by two zeroes? Express your answer as a decimal to the nearest tenth.


Either increasing the radius of a cylinder by four inches or tripling the height of the cylinder results in the same volume. The original height of the cylinder is four inches. What is the original radius in inches?

Solution: \boxed{r = 2 + 2\sqrt{3}}


What is the greatest common factor of the numbers 2505 and 7350?


Triangle $PQR$ has vertices $P(1, 9)$, $Q(3, 2)$, and $R(9, 2)$. A line through $Q$ cuts the area of $\triangle PQR$ in half and has a $y$-intercept of $1$. Find the slope of this line and then find the sum of the slope and $y$-intercept.


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


Let $b_0 = \sin^2 \left( \frac{\pi}{30} \right)$ and for $n \geq 0$,
\[ b_{n + 1} = 4b_n (1 - b_n). \]
Find the smallest positive integer $n$ such that $b_n = b_0$.


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 $S$.


A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. Consider a month with 31 days and the number of the month having multiple small prime divisors. Determine how many relatively prime dates are in December, given that December has 31 days and the number 12 as its month number which factors into $2^2 \cdot 3$.


Let $p(x)$ be a monic quartic polynomial such that $p(2) = 7,$ $p(3) = 12,$ $p(4) = 19,$ and $p(5) = 28.$ Find $p(6).$


At a cafeteria, Elaine wants to purchase a meal including an entree, a dessert, and a drink. The list below shows her options in each category. Calculate the total number of distinct meals Elaine can buy considering her options:
\[
\begin{tabular}{|c|c|c|}
\hline
\textbf{Entrees} & \textbf{Drinks} & \textbf{Desserts} \\
\hline
Hamburger & Soda & Pie \\
\hline
Salad & Coffee & Cake \\
\hline
Veggie Wrap & Tea &  \\
\hline Meatball Sub &  &  \\
\hline
\end{tabular}
\]


At the start of her hike, Jenna can walk at a speed of 25 miles per hour. As she gets exhausted, her speed drops to 10 miles per hour. If she hikes a total of 140 miles in a total time of 8 hours, for how many hours was Jenna walking at her initial speed of 25 miles per hour? Express your answer as a common fraction.


In polar coordinates, the point $\left(-3, \frac{7\pi}{6}\right)$ is equivalent to what other point, using the standard polar coordinate representation? Enter your answer in the form $(r,\theta)$, where $r > 0$ and $0 \le \theta < 2 \pi.$

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


In 1950, there were 500,000 cases of a particular disease reported in a country. In 2000, there were 1,000 cases reported. How many cases of the disease would have been reported in 1975 if the number of cases reported from 1950 to 2000 decreased linearly?


The graph of $y = ax^2 + bx + c$ has a minimum value of 32, and it passes through the points $(2,0)$ and $(-4,0)$. Find $a + b + c$.


Let \( a \), \( b \), and \( c \) 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 \( a + 2b + 3c \).


Solve for $s$: $s = \displaystyle\frac{\sqrt{7^2+24^2}}{\sqrt{49+16}}$. Express as a common fraction.


The fifth and sixth terms of a geometric sequence are 48 and 64, respectively. What is the first term of the sequence?


In triangle $XYZ,$ $\cot X \cot Z = \frac{1}{4}$ and $\cot Y \cot Z = \frac{1}{8}.$ Find $\tan Z.$


Our school's girls volleyball team now has 16 players, including a set of twins, Beth and Bonnie, who must either both be on the team or neither. In how many ways can we choose 6 starters when selecting one twin requires selecting the other?


The average age of the 50 members of a sports club is 22 years. There are 30 girls, 15 boys, and 5 coaches. The average age of the girls is 18 and the average age of the boys is 20. The coaches consist of 3 head coaches and 2 assistant coaches. The average age of the head coaches is 30. What is the average age of the assistant coaches?


Let $x^2 + bx + c = 0$ be a quadratic equation whose roots are each three less than the roots of $5x^2 - 4x - 9 = 0$. Determine the value of $c$.


What is the value of \( x + y \) if the sequence \( 3, ~9, ~15, \ldots, ~x, ~y, ~33 \) is an arithmetic sequence?


Let \( \omega \) be a nonreal root of \( z^4 = 1.\) Find the number of ordered pairs \( (a,b) \) of integers such that \( |a \omega + b| = \sqrt{2}. \)


You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you eat one-third of the pizza. On each successive trip, you eat half of the remaining pizza. After four trips to the refrigerator, what fractional part of the pizza have you eaten?


What is the area, in square units, of a triangle with vertices at \( A(2, 3), B(9, 3), C(5, 12) \)?


Simplify
\[
\frac{2 + 2\sin x - 2\cos x}{2 + 2\sin x + 2\cos x}.
\]

Solution: \boxed{\tan \frac{x}{2}}


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?


Cylinder X's height is triple the radius of cylinder Y, and cylinder X's radius is half the height of cylinder Y. If the volume of cylinder X is three times the volume of cylinder Y, express the volume of cylinder X in terms of the height \( h \) of cylinder Y as \( M \pi h^3 \). Find the value of \( M \).


Similarly, by restricting the domain of the function $g(x) = 3x^2 - 6x - 8$ to an interval, we can make it invertible. What is the largest such interval that includes the point $x=2$?

Solution: \boxed{[1, \infty)}


It is currently $8\!:\!15\!:\!00 \text{ a.m.}$. What time will it be in $7455$ seconds? (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)


You have a rectangular prism box with length $x + 5$ units, width $x - 7$ units, and height $x^2 + x + 30$ units. For how many positive integer values of $x$ is the volume of the box less than 800 units?


A rectangular band formation is a formation with $m$ band members, arranged such that $x$ members in each of $r$ rows results in five members left over. If the director increases the number of members in each row by 2 and reduces the number of rows by 3, there are exactly enough places in the new formation for each band member. What is the largest number of members the band could have, given that the total is fewer than 100?


How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one eleventh of $N$?


Determine the integer $n$ such that
\[\frac{\tan \frac{\pi}{7} + i}{\tan \frac{\pi}{7} - i}\]
is equal to $\cos \frac{2n \pi}{14} + i \sin \frac{2n \pi}{14}$ for some integer $n$ between 0 and 13 inclusive.


What is the sum of all integer solutions to the inequality $|n| < |n-3| < 10$?


Milton spilled some ink on his homework paper again. This time, he can't read the coefficient of $x$, but he remembers that the quadratic equation has two distinct negative, integer solutions. What is the sum of all possible distinct integers that could be under the ink stain if the constant term he can faintly make out is 24?


Simplify $\dfrac{5+6i}{2-3i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary).

Solution: \boxed{\dfrac{-8}{13} + \dfrac{27}{13}i}


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


Determine the value of the coefficient $b$ in the polynomial $Q(x) = x^4 - 3x^3 + bx^2 - 12x + 24$ such that $x+2$ is a factor of $Q$.


Determine all values of $m$ so that the domain of
\[ c(x) = \frac{3mx^2 - 4x + 1}{7x^2 - 6x + m} \]
is the set of all real numbers.

Solution: \boxed{\left(\frac{9}{7}, \infty\right)}


A point $(4\sqrt{5},d+5)$ is $4d$ units away from the origin. Determine the smallest possible value of $d$.


Given the rational function $\frac{r(x)}{s(x)}$ where $s(x)$ is quadratic and $r(x)$ is linear, and knowing that $r(2)=2$, $s(1)=1$, and the graph has a vertical asymptote at $x=3$ and a hole at $x=0$, find $r(x) + s(x)$.

Solution: \boxed{-\frac{1}{2}x^2 + \frac{5}{2}x}


Let's say there is a traveler named Marco who plans to make a trip from city $D$ to $E$, then from $E$ to $F$, and finally from $F$ to $D$. The cities are arranged in a right-angled triangle with $D$ and $F$ being 4000 km apart, and $D$ to $E$ being 5000 km apart. Marco has two options for each part of his journey:
- taking a bus, which costs him $0.20$ per kilometer, or
- taking an airplane, which includes a $150$ booking fee plus $0.12$ per kilometer traveled.
Determine the total kilometers Marco will travel for his entire trip.

Solution: \boxed{12000 \text{ km}}


In a revised magic square, the sum of the three entries in any row, column, or diagonal must be the same value. This magic square now has a few different entries as shown below.
[asy]
size(2cm);
for (int i=0; i<=3; ++i) draw((i,0)--(i,3)^^(0,i)--(3,i));
label("$x$",(0.5,2.5));label("$21$",(1.5,2.5));
label("$50$",(2.5,2.5));label("$3$",(0.5,1.5));
[/asy]
Given these entries, find value of $x$.


What is the sum of the first 15 odd positive integers starting from 5?


For all real numbers $x$ except $x=0$ and $x=1$ the function $f(x)$ is defined by
\[f \left( \frac{x}{x - 1} \right) = \frac{1}{x}.\] Suppose $0\leq t\leq \frac{\pi}{2}$. What is the value of $f(\csc^2t)$?

Solution: \boxed{\cos^2 t}


Find the number of positive integers $n$ such that
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) > 0.\]


What digit must replace $\square$ in the 5-digit number $71\,36\square$ so that it becomes a multiple of 5?


For what values of $k$ does the equation $(3x+6)(x-4) = -33 + kx$ have exactly one real solution?

Solution: \boxed{-6 + 6\sqrt{3}, -6 - 6\sqrt{3}}


How many non-empty subsets $T$ of $\{1,2,3,\ldots,17\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $T$.
$(2)$ If $T$ contains $k$ elements, then $T$ contains no number less than $k+1$.


A rectangular room measures 10 meters by 8 meters. How many square yards of carpet are needed to cover the floor of the room?


Let $ f(x) = x^4 - x^3 + x^2 - x + 1$. Suppose $ g$ is a quartic polynomial such that $ g(0) = -1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(16)$.


Suppose that $n+1, n+2, n+3, n+4, n+5, n+6$ are six consecutive integers.

Determine a simplified expression for the sum of these six consecutive integers and also find their average.

Solution: \boxed{n + 3.5}


What is the smallest positive integer $n$ such that $5n \equiv 2024 \pmod{26}$?


Quadrilateral ABCD is a parallelogram. Inside triangle BCD, angle DCB measures 75 degrees. If triangle BCD is isosceles with BC = CD, what is the degree measure of angle A?


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


Let $G_n$ be the double of Fibonacci sequence values, that is, $G_n = 2 \cdot F_n$ where $F_n$ are Fibonacci numbers with $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute
\[
\sum_{n=0}^{\infty} \frac{G_n}{5^n}.
\]

Solution: \boxed{T = \frac{10}{19}}


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

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


What is the $100^{\mathrm{th}}$ odd positive integer, and what even integer directly follows it?


Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \cos 675^\circ.$


What is the positive difference between the two largest prime factors of $175616$?


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


Factor $x^2 + 6x + 9 - 100x^4$ into two quadratic polynomials with integer coefficients.

Solution: \boxed{(-10x^2 + x + 3)(10x^2 + x + 3)}


Determine the constant $k$ such that the quadratic equation $5kx^2 + 30x + 10 = 0$ has exactly one solution. Find this solution.


Determine the slope of the line and the coordinates of the midpoint between points \((1, -7)\) and \((-4, 3)\).

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


A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction.


Find the remainder when $1234567890123$ is divided by $180$.


Determine the number of angles between 0 and $2\pi$, other than integer multiples of $\frac{\pi}{2}$, such that $\sin \theta$, $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order when the relations are modified to $\sin^2 \theta = \cos^4 \theta$ and $\cos^4 \theta = \sin \theta$.


What is the sum of the last two digits of $7^{25} + 13^{25}?$


The operation $\Diamond$ is redefined as $a \Diamond b = ab^{3} - b^2 + 1$. What is the value of $3 \Diamond 2$?


How many natural numbers between 10 and 90 are relatively prime to 18?


A point is chosen randomly on the number line between 0 and 1, and is colored blue. Another point is then chosen randomly on the number line between 0 and 1, and is colored red. What is the probability that the number of the red point is greater than the number of the blue point, but less than three times the number of the blue point?


What is the sum of the integers greater than 10 and less than 25, excluding those divisible by 4?


What is the volume, in cubic inches, of a rectangular box whose faces have areas of $40$ square inches, $10$ square inches, and $8$ square inches?


In a similar tournament setup, the top 6 bowlers have a playoff. First #6 bowls #5, and the loser gets the 6th prize. The winner then bowls #4, and the loser of this match gets the 5th prize. The process continues with the previous winner bowling the next highest ranked bowler until the final match, where the winner of this match gets the 1st prize and the loser gets the 2nd prize. How many different orders can bowlers #1 through #6 receive the prizes?


Two concentric circles are centered at point Q. The sides of a \(30^\circ\) angle at Q form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a \(24^\circ\) angle at Q. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your answer as a common fraction.


Find all values of $k$ such that the quadratic expression
\[x^2 - (k - 5)x + k + 2 > 0\]
is positive for all $x$.

Solution: \boxed{\left(7 - 4\sqrt{2}, 7 + 4\sqrt{2}\right)}


How many factors of 12,000 are perfect squares?


How many 3-digit numbers have the property that the units digit is at least three times the tens digit?


A rectangular prism measures 12-inches by 16-inches by 24-inches. What is the length, in inches, of the diagonal connecting point A and point B?

Solution: \boxed{4\sqrt{61} \text{ inches}}


What is the smallest positive integer that can be written in the form $3003m + 55555n$, where m and n are integers, and find such m and n?


Assuming license plates must follow specific alpha-numeric formats, Texas issues plates with a format "AB 1234 C", where 'A', 'B', and 'C' are letters, and '1234' are digits. California, however, issues plates in the format "1ABC 234", where 'A', 'B', 'C' are letters and '1', '234' are digits. Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates can Texas issue than California.


Given a right triangle with two sides having lengths 6 and 7, what is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth.


There are 5 representatives from each of 5 different companies at a conference. During a networking event, each person shakes hands with every other attendee except their own company's representatives. How many handshakes occur?


What is the largest digit that is never found in the units place of an odd number?


When the greatest common divisor and least common multiple of two integers are multiplied, the product is 252. How many different values could be the greatest common divisor of the two integers?


The three-digit positive integer $N$ has a ones digit of 4. What is the probability that $N$ is divisible by 4? Express your answer as a common fraction.


Consider a trapezoidal field where it's planted uniformly with wheat. The trapezoid has the following measurements: side $AB$ is 150 m, base $AD$ (the longest side) is 300 m, and the other base $BC$ is 150 m. The angle at $A$ is $75^\circ$, and the angle at $B$ is $105^\circ$. At harvest time, all the wheat is collected at the point nearest to the trapezoid's perimeter. What fraction of the crop is brought to the longest side $AD$?


Find the number of real solutions of the equation
\[\frac{x}{50} = \sin x\]
for \( x \geq 0 \).


In a women's doubles tennis tournament, there were four teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner, except one player who did not shake hands with anyone from one specific team. What is the number of handshakes that occurred?


How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$?


Consider the function $g(x) = x^2 + 4\sqrt{x}$. Evaluate $3g(3) - g(9)$.

Solution: \boxed{-66 + 12\sqrt{3}}


What is the smallest positive value of $x$ such that $x + 8765$ results in a palindrome?


Solution: \boxed{27x^{12} - 343y^9}


Eight students participate in a banana eating contest. The graph shows the number of bananas eaten by each participating student. Alice ate the most bananas and Tom ate the fewest. How many more bananas than Tom did Alice eat?

[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[10];
for( int i = 0 ; i < 10 ; ++i ){

yaxis[i] = (0,i);

draw(yaxis[i]--yaxis[i]+(17,0));
}
draw((0,0)--(0,9));
draw((17,9)--(17,0));

fill((1,0)--(1,4)--(2,4)--(2,0)--cycle,grey);
fill((3,0)--(3,7)--(4,7)--(4,0)--cycle,grey);
fill((5,0)--(5,8)--(6,8)--(6,0)--cycle,grey);
fill((7,0)--(7,5)--(8,5)--(8,0)--cycle,grey);
fill((9,0)--(9,3)--(10,3)--(10,0)--cycle,grey);
fill((11,0)--(11,2)--(12,2)--(12,0)--cycle,grey);
fill((13,0)--(13,6)--(14,6)--(14,0)--cycle,grey);
fill((15,0)--(15,1)--(16,1)--(16,0)--cycle,grey);
label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("8",yaxis[8],W);
label("9",yaxis[9],W);
label("Students/Participants",(8.5,0),S);
label("Results of a Banana Eating Contest",(8.5,9),N);
label(rotate(90)*"$\#$ of Bananas Eaten",(-1,4.5),W);
[/asy]


Compute the product of the roots of the equation \[x^3 - 15x^2 + 50x + 35 = 0.\]


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


If $\overline{AD} \| \overline{FG}$, how many degrees are in angle $EFG$?

[asy]
import olympiad;

pair A = (-15,20);
pair B = (-12,35);
pair C = (35,50);
pair D = (35,20);
pair E = (14,20);
pair F = (0,0);
pair G = (40,0);

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

label("F", F, W);
label("G", G, ENE);
label("C", C, N);
label("A", A, W);
label("D", D, ENE);
label("E", E, SE);
label("B", B, NW);

draw(scale(20)*anglemark(G, F, C));
draw(shift(E)*scale(35)*shift(-E)*anglemark(B, E, A));
draw(shift(E)*scale(20)*shift(-E)*anglemark(C, E, B));

label("$x$", (6,20), NW);
label("$3x$", (13,25), N);
label("$2x$", (5,0), NE);
[/asy]


What is the area, in square units, of a triangle with sides of $5, 5,$ and $6$ units? Express your answer in simplest radical form.


What is the smallest positive base ten number that requires seven digits for its binary representation?


Four prime numbers are randomly selected without replacement from the first twelve prime numbers. What is the probability that the sum of the four selected numbers is even?


Let $x$ be a complex number such that $x^7=1$ and $x\neq 1$. Compute the sum
\[
\frac{x^2}{x-1} + \frac{x^4}{x^2-1} + \frac{x^6}{x^3-1}.
\]


Solution: \boxed{-6t^5 + 5t^4 - t^3 - 24t^2 + 23t - 5}


Tom rolls 8 fair 6-sided dice. What is the probability that at least two dice show the same number?


Determine the value of \(x\) that satisfies \(\sqrt[3]{x\sqrt[4]{x^5}} = 4\).

Solution: \boxed{x = 2^{8/3}}


What is the least prime factor of $5^6 - 5^4 + 5^2$?


Consider the series $S = 1 + 3x + 5x^2 + \dots$. If $S = 4$, find $x$.

Solution: \boxed{\frac{9 - \sqrt{33}}{8}}


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


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


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


Solve for $x$ in the equation $x(5x-11) = -6$ and express the solutions in the form $\frac{m+\sqrt{n}}{p}$ and $\frac{m-\sqrt{n}}{p}$, where $m$, $n$, and $p$ have a greatest common divisor of 1. Find the sum $m+n+p$.


Simplify $\sqrt{18} \times \sqrt{32} \times \sqrt{2}$.


Ray climbs up a flight of stairs using two methods. When he jumps 6 steps at a time, there are 4 steps left at the top. When he uses 7 steps at a time, there are 5 steps left at the top. What is the smallest possible value of \(n\) that is greater than 20?


What is the sum of the digits of the base-2 expression for $315_{10}$?


The line \( y = 5 \) intersects the graph of \( y = 5x^2 + 2x - 2 \) at points \( C \) and \( D \). Determine the distance between \( C \) and \( D \).


Simplify $\sqrt[3]{40x^5y^9z^{14}}$ and determine the sum of the exponents of the variables that are outside the radical.


Let $\omega$ be a nonreal root of $z^3 = 1.$ Find the number of ordered pairs $(a,b)$ of integers such that $|a \omega - b| = 1.$


The integer $y$ has 24 positive factors. The numbers 18 and 28 are factors of $y$. What is the smallest possible value of $y$?


A dog and two kittens together weigh 36 pounds. The dog and the larger kitten together weigh exactly three times as much as the smaller kitten, and the dog and the smaller kitten together weigh exactly the same as the larger kitten.


The quadratic $12x^2+16x-21$ has two real roots. What is the sum of the squares of these roots?


How many different counting numbers will each leave a remainder of 7 when divided into 65?


An infinite geometric series has a first term of 400 and a sum of 2500. What is its common ratio?


Integers $a$ and $b$ with $a > b > 0$ satisfy $a + b + ab = 119$. What is $a$?


Rationalize the denominator of $\displaystyle \frac{5}{2+\sqrt{5}}$.

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


Let $q(x)$ be a monic quartic polynomial such that $q(1) = 3$, $q(2) = 6, $ $q(3) = 11$, and $q(4) = 18$. Find $q(5)$.


For a certain hyperbola
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]
where $a > b$, the angle between the asymptotes is $45^\circ$. Find the value of $\frac{a}{b}$.

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


Ben rolls 6 fair 12-sided dice, with each die numbered from 1 to 12. What is the probability that exactly three of the dice show a prime number?


Let $g(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2010}.$ There are exactly 1010 distinct values among
\[|s_1|, |s_2|, \dots, |s_{2010}|.\] What is the minimum number of real roots that $g(x)$ can have?


A club has 25 members and needs to select 4 officers: president, secretary, treasurer, and spokesperson. Each person can hold at most one office. Three of the members, Alice, Bob, and Carol, insist on the condition that either all three are officers or none of them are. How many ways can the club select its officers under this condition?


Given a torus (doughnut-shaped object) with an inner radius of $3$ and an outer radius of $5$ sitting on a flat table. Determine the radius of the largest spherical ball that can be placed on top of the center of the torus such that the ball touches the horizontal plane. Assume the torus is formed by revolving the circle (center at $(4,0,1)$ and radius $1$) around the $z$-axis. The spherical ball has its center on the $z$-axis and touches either the table or the torus.


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

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


A torus (donut) having inner radius $3$ and outer radius $5$ sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center of the torus so that the ball still touches the horizontal plane? (If the $xy$-plane is the table, the torus is formed by revolving the circle in the $xz$-plane centered at $(4,0,1)$ with radius $1$ about the $z$-axis. The spherical ball has its center on the $z$-axis and rests on either the table or the donut.)


Solution: \boxed{2009 + \frac{1}{2010}}


How many prime numbers have squares between 2500 and 4900?


Solve the following system of equations:
\[
4x - 7y = -9,
\]
\[
5x + 3y = -11.
\]
Express your answer as an ordered pair $(x, y)$.

Solution: \boxed{\left(\frac{-104}{47}, \frac{1}{47}\right)}


Bertha has 8 daughters and no sons. Some of her daughters have 4 daughters, and the rest have none. Bertha has a total of 28 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


A researcher's hourly tea consumption is inversely proportional to the total number of hours he spends analyzing data. On a particular Sunday, he spent 12 hours analyzing data and consumed 1.5 liters of tea. On Wednesday, he decides to spend only 8 hours on data analysis. How much tea did he consume on Wednesday, assuming his total daily activity hours, including data analysis, are fixed at 16 hours?

Solution: \boxed{2.25 \text{ liters}}


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

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


Calculate the percentage of five-digit numbers that have at least one repeated digit (the repeated digits do not need to be adjacent). Express your answer as a decimal to the nearest tenth.


A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 4$?


Find the largest prime divisor of $24^2 + 75^2$.


If $x+y=\frac{11}{17}$ and $x-y=\frac{1}{143}$, what is the value of $x^2-y^2$? Express your answer as a common fraction.


How many positive factors of 150 are also multiples of 15?


If 60 cards are dealt to 9 people as evenly as possible, how many people will end up with fewer than 7 cards?


Consider a magic square where the sum of the three entries in any row, column, or diagonal is the same. Below is a part of a magic square with some entries known. Find $y$.
[asy]
size(2cm);
for (int i=0; i<=3; ++i) draw((i,0)--(i,3)^^(0,i)--(3,i));
label("$y$",(0.5,2.5));label("$23$",(1.5,2.5));
label("$104$",(2.5,2.5));label("$5$",(0.5,1.5));
[/asy]


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


Henry's little brother now has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all of the stickers across the sheets of paper, given that only the number of stickers on each sheet matters?


Find the length of the longest side of the quadrilateral formed by the following system of inequalities:
$$
\begin{cases}
x + 2y \leq 6 \\
x + y \geq 3 \\
x \geq 0 \\
y \geq 0
\end{cases}
$$


Simplify $\dfrac{5+6i}{3+i}$ in the form $a+bi$, where both $a$ and $b$ are expressed as improper fractions, if needed.

Solution: \boxed{\dfrac{21}{10} + \dfrac{13}{10}i}


The sum of the areas of two squares is 85, while the difference in their areas is 45. Find the sum of their perimeters.

Solution: \boxed{4\sqrt{65} + 8\sqrt{5}}


Given that \(7^{-1} \equiv 55 \pmod{101}\), find \(49^{-1} \pmod{101}\), as a residue modulo 101. (Answer should be between 0 and 100, inclusive.)


Jane and her brother each spin a spinner once. The spinner has six congruent sectors, labeled from 1 to 6. If the non-negative difference of their numbers is less than or equal to 2, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.


In Longtown, the streets are all $30$ feet wide and the blocks they enclose are rectangles with sides $500$ feet and $300$ feet in length. Jenny runs around the block along the $500$-foot side of the street, while Jeremy runs on the opposite side of the street. How many more feet than Jenny does Jeremy run for every lap around the block?


Find the radius of the circle with the equation $x^2 + 8x + y^2 - 10y + 41 = 0$.


Determine the tens digit of $19^{1987}$.


Evaluate the determinant of the matrix:
\[
\begin{vmatrix} 1 & a+b & b+c \\ 1 & a+2b & b+2c \\ 1 & a+3b & b+3c \end{vmatrix}
\]


What is the sum of the last two digits of $9^{20} + 11^{20}$?


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

Solution: \boxed{\log_2 n = 949}


Find the $x$-intercept point of the line $4x + 7y = 28$.


How many positive factors of 180 are perfect squares?


A square piece of paper has each corner cut off to form an isosceles right triangle, resulting in a smaller rectangle within it. If $AB = 24$ units, what is the total area of the four removed triangles?


A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be either less than 7 or even? Express your answer as a common fraction.


The Modified Little Twelve Basketball Conference now includes three divisions, with four teams in each division. Each team plays each of the other teams in its own division three times and every team in the other two divisions twice. How many conference games are now scheduled?


In the seven-digit integer $4B6,\!825,\!1$, what is the largest digit $B$ so that the seven-digit integer will be divisible by 3?


Three plastic bottles can be recycled to create a new bottle. Starting with 729 plastic bottles, how many new bottles can eventually be made from these, considering that new bottles can be recycled again into even newer bottles? Do not include the original 729 bottles in your count.


If Nancy selects one letter at random from the English alphabet, what is the probability that the letter is found in the word `MATHEMATICS'?


Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction.


George purchases a sack of apples, a bunch of bananas, a cantaloupe, a carton of dates, and a package of eggs for $ \$ 30$. A carton of dates costs three times as much as a sack of apples, and the price of a cantaloupe is equal to the price of a sack of apples minus a bunch of bananas. A package of eggs costs as much as the sum of the price of a bunch of bananas and a cantaloupe. How much would it cost George to purchase a bunch of bananas, a cantaloupe, and a package of eggs?


In a chess tournament, the top 4 players compete in a sequential knockout format. Player #4 plays against Player #3, and the loser secures the 4th place. The winner then plays against Player #2, where the loser gets the 3rd place. Finally, the winner from this game plays against Player #1, and the winner secures the 1st place, leaving the loser with the 2nd place. Determine in how many different orders the players can finish from 1st to 4th place.


How many distinct terms are in the expansion of \[(a+b+c)(d+e+f+g+h+i)?\]


What is the remainder when \(6x^8 - 2x^7 - 10x^6 + 3x^4 + 5x^3 - 15\) is divided by \(3x-6\)?


Consider a list of ten positive integers where six of them are $3, 5, 1, 4, 9, 6$. What is the largest possible value of the median of this list?


Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 10? Express your answer as a common fraction.


Three circles have the same center O. Point X divides segment OP in the ratio 1:3, with X being closer to O. Calculate the ratio of the area of the circle with radius OX to the area of the circle with radius OP. Next, find the area ratio of a third circle with radius 2*OX to the circle with radius OP.


What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\text{th}}$ roots of unity?


A circle with center $A$ and radius 4 inches is tangent at point $C$ to a circle with center $B$. The center $B$ is 5 inches away from the center $A$. If the radius of the larger circle is 9 inches, find the area of the shaded region that lies outside the smaller circle but inside the larger circle.


A wizard concocts a mystical potion requiring a magical plant and a specific gemstone. There are four magical plants and six gemstones available for the potion. Two specific gemstones are incompatible with one of the plants, while one other gemstone cannot be used with two different plants. How many valid combinations of plants and gemstones can the wizard use to make his potion?


Let the polynomial \( g(x) = x^4 + px^3 + qx^2 + rx + s \) have real coefficients, and have \( 3i \) and \( 1 + 2i \) as zeros. Determine the value of \( p + q + r + s \).


In trapezoid $ABCD$ with $\overline{BC} \parallel \overline{AD}$, let $BC = 900$ and $AD = 1800$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$.


In the circle with center $O$, the measure of $\angle RIP$ is $60^\circ$ and $OR = 12$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\pi$.


What is the sum of $\left(\dfrac{1}{4}\right) + \left(\dfrac{1}{4}\right)^2 + \left(\dfrac{1}{4}\right)^3 + \left(\dfrac{1}{4}\right)^4 + \left(\dfrac{1}{4}\right)^5$?


Find the greatest common divisor of 8650, 11570, and 28980.


Our football team has expanded to 12 members. Out of these, 3 are strong enough to play as offensive lineman, and 4 are fast enough to be considered as running backs. Given that all other positions can be played by any player, in how many ways can we choose a starting lineup consisting of a quarterback, a running back, an offensive lineman, and a wide receiver?


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


How many 0's are present to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^3 \cdot 5^7}$?


Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?


A wizard is preparing a magic elixir. He needs to choose one of four magical herbs and one of six enchanted stones. However, one specific stone does not work with three of the herbs. All other combinations are valid. How many valid combinations can the wizard use to prepare his elixir?


How many integers between $200$ and $250$ have three different digits in increasing order?


Calculate the perimeter of a triangle with vertices at points $P(2, 3)$, $Q(2, 10)$, and $R(8, 6)$.

Solution: \boxed{7 + 2\sqrt{13} + 3\sqrt{5}}


The point $O$ is the center of a circle circumscribing $\triangle ABC$. Given that $\angle BOC = 120^\circ$ and $\angle AOB = 150^\circ$, determine the degree measure of $\angle ABC$.


Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(x - 1)^2 + (x + 2)^2}
\]
over all real numbers \(x\).


Let $T$ be the set of 12-tuples $(b_0, b_1, \dots, b_{11})$, where each entry is 0 or 1, so $T$ contains $2^{12}$ 12-tuples. For each 12-tuple $t = (b_0, b_1, \dots, b_{11})$ in $T$, let $q_t(x)$ be the polynomial of degree at most 11 such that
\[q_t(n) = b_n\] for $0 \le n \le 11.$ For example, $q(x) = q_{(0,1,0,0,1,0,1,0,0,0,1,0)}(x)$ is the polynomial of degree at most 11 such that $q(0) = q(2) = q(3) = q(5) = q(7) = q(8) = q(9) = q(11) = 0$ and $q(1) = q(4) = q(6) = q(10) = 1.$

Find
\[\sum_{t \in T} q_t(12).\]


Consider a line segment connecting the points $(-3, 9)$ and $(4, 10)$. This line segment can be parameterized by the equations
\[
x = at + b, \quad y = ct + d,
\]
where \(0 \le t \le 1\), and \(t = 0\) corresponds to the point \((-3,9)\). Find \(a^2 + b^2 + c^2 + d^2\).


Compute the largest integer $k$ such that $2010^k$ divides $2010!^2$.


Simplify the fraction $\dfrac{126}{11088}$.


Derek has a collection of stamps that he intends to arrange evenly into $Z$ groups, each containing the same number of stamps, $W$. There must be more than one stamp in each group, and no group can contain all the stamps. Calculate the smallest possible number of stamps Derek could have if there are 7 possible values for $W$.


When the binary number $110110011011_2$ is divided by 8, what is the remainder (give your answer in base 10)?


Thirty tiles are numbered 1 through 30 and are placed into box $C$. Thirty other tiles numbered 21 through 50 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 20 and the tile from box $D$ is either odd or greater than 40? Express your answer as a common fraction.


Find the positive solution to
\[
\sqrt{x + \sqrt{x + \sqrt{x + \dotsb}}} = \sqrt{x \sqrt{x \sqrt{x \dotsm}}}.
\]


Determine the value of $n$ for which 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.


What is the smallest positive integer $n$ such that the sum of the digits to the right of the decimal point in the decimal form of $\frac{1}{3^n}$ exceeds 15?


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


Two cylindrical cans have a volume of 500 cubic units each. The height of one can is four times the height of the other. If the radius of the taller can is 5 units, what is the radius of the shorter can? Express your answer in simplest radical form.


Find all solutions to the inequality \[\frac{x^2}{(x-5)^2} \ge 0.\] (Give your answer in interval notation.)

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


Suppose $m$ and $n$ are positive integers such that $\gcd(m,n)$ is divisible by exactly $5$ distinct primes and $\mathop{\text{lcm}}[m,n]$ is divisible by exactly $30$ distinct primes.

If $m$ has fewer distinct prime factors than $n$, then $m$ has at most how many distinct prime factors?


If I have four boxes arranged in a $2$ x $2$ grid, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains at most one digit, and exactly two boxes remain empty?


The arithmetic mean of these six expressions is 30. What is the value of $y$? $$y + 10 \hspace{.5cm} 20 \hspace{.5cm} 3y \hspace{.5cm} 18 \hspace{.5cm} 3y + 6 \hspace{.5cm} 12$$


Solution: \boxed{L \times W = 52 \times 77 = 4004}


A whole number is said to be "7-heavy" if the remainder when the number is divided by 7 is greater than 4. What is the least three-digit 7-heavy whole number?


The number $n$ is a three-digit positive integer and is the product of three factors: $x$, $y$, and $10x+y$, where $x$ is a prime digit, $x + y$ is not a multiple of 3, and $10x+y$ forms a prime number. What is the largest possible value of $n$ given these conditions?


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


A wire is cut into two pieces, one of length \(a\) and the other of length \(b\). The first piece \(a\) is bent to form a rectangle where the length is twice the width. The second piece \(b\) is bent to form a circle. Both the rectangle and the circle have equal areas. What is \(\frac{a}{b}\)?


For what base is the representation of $512_{10}$ a four-digit number whose final digit is odd?


The function \( g(x) \) satisfies the functional equation
\[ g(x + y) = g(x) + g(y) \]
for all real numbers \( x \) and \( y \), and it is given that \( g(3) = 4 \). Find \( g(10) \).


Find the product of all integer divisors of $180$ that also divide $20$.


Determine the domain of the function $$ f(x) = \frac{1}{x+7} + \frac{1}{x^2+7} + \frac{1}{x^3+7}~? $$

Solution: \boxed{(-\infty, -7) \cup (-7, -\sqrt[3]{7}) \cup (-\sqrt[3]{7}, \infty)}


When the decimal point of a certain positive decimal number is moved three places to the right, the new number is three times the reciprocal of the original number. What is the original number?


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


The value of $y$ varies inversely as $x^2$. When $x=3$, $y=2$. What is the value of $x$ when $y=18$?

Solution: \boxed{x = 1}


A 12-foot by 15-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile contains a pattern of four white quarter circles of radius 1/2 foot centered at each corner of the tile, and additionally, a white circle of radius 1/4 foot at the center. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

Solution: \boxed{180 - \frac{225\pi}{4}}


Evaluate the determinant:
\[
\begin{vmatrix}
\sin \alpha \sin \beta & \sin \alpha \cos \beta & \cos \alpha \\
\cos \beta & -\sin \beta & 0 \\
\cos \alpha \sin \beta & \cos \alpha \cos \beta & -\sin \alpha
\end{vmatrix}.
\]


What is the remainder when \(98^{50}\) is divided by 100?


How many integers $-13 \leq n \leq 13$ satisfy $(n-3)(n+5)(n+9) < 0$?


What is the least positive integer that satisfies the following conditions?

a) When divided by 7, the remainder is 6.

b) When divided by 11, the remainder is 10.

c) When divided by 13, the remainder is 12.


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

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


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. Provided that the sum of the numbers on the vertices is equal to $2002$, what is the sum of the numbers written on the faces if the sum of one pair of opposite faces equals 22?


Calculate the quotient of $2301_5 \div 22_5$ in base $5$.


Let \( b_1, b_2, b_3, \dots \) be an increasing arithmetic sequence of integers. If \( b_3b_4 = 21 \), what is \( b_2b_5 \)?


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


How many rows of Pascal's Triangle contain the number $53$?


In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Y = \frac{3}{4}$. If $YZ = 30$ and an altitude is dropped from $Z$ to $XY$ at point $W$, determine the length of segment $XY$ and the altitude $ZW$.


What is the least common multiple of the numbers 1056 and 792, and then add 100 to your result?


When the base-10 integers 350 and 1500 are expressed in base 2, and their binary forms are added, how many digits does the resulting binary sum have?


An architectural firm uses the formula $L = \frac{50T^3}{H^3}$ to calculate the dynamic load on cylindrical columns. If $T = 3$ and $H = 6$, what is the value of $L$?


Terrell usually lifts two 25-pound weights 10 times. If he uses two 20-pound weights instead, how many times must Terrell lift them in order to lift the same total weight?


I have two 30-sided dice. Each die has 6 blue sides, 8 yellow sides, 10 green sides, and 6 purple sides. If I roll both dice, what is the probability they come up showing the same color?


Six balls are numbered 1 through 6 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. He then randomly chooses a ball from the bowl again and looks at its number. What is the probability that the product of the two numbers will be odd and greater than 15? Express your answer as a common fraction.


Find all real values of $x$ that satisfy $\frac{x^2+2x^3-3x^4}{x+2x^2-3x^3} \ge -1.$ (Give your answer in interval notation.)

Solution: \boxed{[-1, -\frac{1}{3}) \cup (-\frac{1}{3}, 0) \cup (0, 1) \cup (1, \infty)}


Let $0 \le a \le 1$, $0 \le b \le 1$, and $c = 1$. Find the maximum value of
\[
\sqrt{ab} + \sqrt{(1 - a)(1 - b)}
\]


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


Engineers use the formula \( L=\frac{36T^5}{H^3} \) to find the compression resistance of cylindrical columns. If \( T = 3 \) and \( H = 9 \), what is the value of \( L \)?


Simplify $$(2x^3 - 5x^2 + 8x - 9) + (3x^4 - 2x^3 + x^2 - 8x + 6).$$ Express your answer as a polynomial with the terms in order by decreasing degree.

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


Find the coefficient of the $x^2$ term in the expansion of $(2x^3 + 5x^2 - 3x + 1)(3x^2 - 9x - 5)$.


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


What is the value of $x$ if the square root of $x$ raised to the fourth power equals 256?


Suppose that the angles of triangle $PQR$ satisfy
\[\cos 2P + \cos 2Q + \cos 2R = 1.\]
Two sides of the triangle have lengths 7 and 24. Find the maximum length of the third side.


Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 25$, $b_1 = 56$, $b_n = 0$, and
$$ b_{k+1} = b_{k-1} - \frac{7}{b_k} $$
for $k = 1, 2, \dots, n-1$. Find $n$.


In 2021, the sum of the year's digits is 5. What is the first year after 2021 in which the sum of the digits is 5 again?


Suppose that all four of the numbers \[3 - \sqrt{8}, \;5+\sqrt{11}, \;16 - 3\sqrt{7}, \;-\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


In a laboratory, Michael is preparing a buffer solution by mixing 0.05 liters of solution A with 0.03 liters of solution B. This gives him a total of 0.08 liters of the buffer. He needs to prepare 0.64 liters of this buffer for an experiment. How many liters of solution B will he use?

Solution: \boxed{0.24 \text{ liters}}


Line segment $\overline{AB}$ is extended past $A$ to a point $Q$ such that the ratio $BQ:QA = 5:2$. Then,
\[
\overrightarrow{Q} = x \overrightarrow{A} + y \overrightarrow{B}
\]
for some constants $x$ and $y$. Find the ordered pair $(x, y)$.

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


Each of the numbers $a_1,$ $a_2,$ $\dots,$ $a_{100}$ is $\pm 1.$ Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 100} a_i a_j.\]


Consider an arithmetic sequence $\{b_n\}$ with the following conditions:
$$
b_{51}+b_{52}+ \cdots +b_{150}=300 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{250}=500.
$$
Find the value of $b_2 - b_1$?


What is the radius of the circle inscribed in triangle $PQR$ if $PQ = PR = 8$ and $QR = 10$? Express your answer in simplest radical form.


Find all solutions to the inequality \[\frac{(3x-4)(x+1)}{x} \ge 0.\] (Give your answer in interval notation.)

Solution: \boxed{[-1, 0) \cup [\frac{4}{3}, \infty)}


In a triangle, the area is numerically equal to twice the perimeter. Find the radius of the inscribed circle.


Monica tosses a fair 8-sided die. If the roll results in a prime number, Monica wins that amount in dollars. If the roll is composite, she wins nothing. If she rolls 1, she loses $\$4$, but if she rolls an 8, she wins $\$1$. What is the expected value of her winnings on one die toss?


Find the greatest common divisor of 1239 and 1980.


Mateen needs to walk 960 meters to finish crossing his rectangular garden. He discovers that he must walk the length 24 times or walk its perimeter 8 times to cover this distance. Determine the length of the garden if its width is half of its length.


Determine the units digit of the number $G_n = 2^{3^n} + 1$, where $n$ is a natural number. Specifically, find the units digit of $G_{1000}$.


Pentagon ABCDE has a vertical line of symmetry. The lengths AE and BD are each increased to 5 units, and the total area of the pentagon is now 50 square units. Determine the $y$-coordinate of vertex C. [asy]
unitsize(2mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=4;

pair A=(0,0), B=(0,5), C=(2.5, y_c), D=(5,5), E=(5,0);
pair[] dots={A,B,C,D,E};

draw(B--C--D--E--A--B);
dot(dots);

axes(Arrows(4));

label("A(0,0)",A,SW);
label("E(5,0)",E,SE);
label("D(5,5)",D,NE);
label("C",C,NE);
label("B(0,5)",B,NW);
[/asy]


If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 11 dips, how many daps are equivalent to 66 dips?

Solution: \boxed{22.5 \text{ daps}}


What is the smallest whole number $b$ such that 80 can be expressed in base $b$ using only three digits?


Jenny places a total of 30 red Easter eggs in several green baskets and a total of 45 orange Easter eggs in some blue baskets. Each basket must contain at least 5 eggs. Determine the number of eggs Jenny placed in each basket.


Determine the positive value of $x$ that satisfies:
\[\log_7 (x - 3) + \log_{\sqrt{7}} (x^4 - 3) + \log_{\frac{1}{7}} (x - 3) = 5.\]

Solution: \boxed{\sqrt[4]{49\sqrt{7} + 3}}


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


If $x\%$ of five-digit numbers have at least one repeated digit, then what is $x$? Express your answer as a decimal to the nearest tenth.


You have a square with vertices at $(2,1)$, $(5,1)$, $(2,4)$, and $(5,4)$. A line joining $(2,1)$ and $(5,3)$ divides the square into two regions. What fraction of the area of the square is above this line?


Let $x$ and $y$ be nonzero real numbers, and let $k$ be any real number except 0. Consider $m$ and $M$ the minimum value and maximum value of
\[ \frac{|kx + y|}{|kx| + |y|}. \]
Find $M - m$.


The square root of $3x$ is greater than 4 and less than 5. How many integer values of $x$ satisfy this condition?


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


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


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


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

[asy]
draw((0,0)--(2,5)--(8,5)--(15,0)--cycle);
draw((8,5)--(0,0));
label("$D$",(0,0),W);
label("$A$",(2,5),NW);
label("$B$",(8,5),NE);
label("$C$",(15,0),E);
[/asy]


A unit cube has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and for $1 \leq i \leq 4,$ vertices $Q_i$ and $Q_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$, $\overline{Q_1Q_4}$, $\overline{Q_1'Q_2'}$, $\overline{Q_1'Q_3'}$, and $\overline{Q_1'Q_4'}$. Determine the side length of the octahedron if each vertex of the octahedron divides its respective segment in the ratio 3:1.


Find all angles $\theta,$ $0 \le \theta \le 2 \pi,$ such that for all real numbers $x,$ $0 \le x \le 1,$
\[
x^2 \cos \theta - x(1 - x) + (1 - x)^2 \sin \theta \ge 0.
\]

Solution: \boxed{\left[ \frac{\pi}{12}, \frac{5\pi}{12} \right]}


A circular cake with a diameter of $20\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Let $m$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $m^2$?


John started collecting baseball cards. He collected 12 cards the first day. Each subsequent day he collected 10 more cards than he had collected the previous day. If he collected baseball cards for 7 consecutive days, what was the average number of cards collected per day?


A modified deck of cards has 60 cards divided into 4 suits, each of which now has 15 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, making a total of 30 red cards, and the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black, also totaling 30 black cards. The cards in the deck are placed in random order. What is the probability that the first three cards drawn from the deck are all the same color?


The quadratic $x^2 - 16x + 64$ can be written in the form $(x+b)^2 + c$, where $b$ and $c$ are constants. What is $b+c$?


A number in the set $\{50, 51, 52, 53, ... , 500\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? Express your answer as a common fraction.


Allison, Brian, and Lily each have a 6-sided cube. All faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5, and 6. Two of the faces on Lily's cube have a 1, two have a 4, and two have a 6. All three cubes are rolled. What is the probability that Allison's roll is greater than each of Brian's and Lily's rolls? Express your answer as a common fraction.


Solve for $x>0$ in the following arithmetic sequence: $2^2, x^2, 5^2, \ldots$.


What value of $x$ will give the minimum value of $3x^2 - 18x + 28 + \sqrt{9 - x^2}$?


In the equation $\frac{1}{7} + \frac{7}{x} = \frac{16}{x} + \frac{1}{16}$, what is the value of $x$?


How many values of $x$, $-25<x<150$, satisfy $\cos^2 x + 3\sin^2 x = 1?$  (Note: $x$ is measured in radians.)


How many distinct ordered pairs of positive integers $(m, n)$ are there such that the sum of the reciprocals of $m$ and $n$ equals $\frac{1}{6}$?


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


There are seven unmarked envelopes on a table, each with a letter for a different person. If the mail is randomly distributed among these seven people, with each person receiving one letter, what is the probability that exactly four people get the right letter?


Determine all real numbers $x$ for which the function $$g(x) = \frac{1}{2+\frac{1}{1+\frac{1}{x-1}}}$$ is undefined, and find their sum.


Find the largest value of $c$ such that $3$ is in the range of $f(x)=x^2-7x+c$.


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

Solution: \boxed{-1 + \frac{7}{2}i}


If $a$, $b$ are nonzero real numbers such that $a^2+b^2=5ab$, find the value of $\left|\frac{a-b}{a+b}\right|$.


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


The number $1025$ can be written as $23q + r$, where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$?


Thirty teams play a tournament where every team plays every other team exactly once. Each game results either in a win or a loss with a $50\%$ chance for either outcome. Calculate the probability that all teams win a unique number of games. Express your answer as $\frac{m}{n}$ where $m$ and $n$ are coprime integers and find $\log_2 n$.


If the probability that a baby born in a certain hospital will speak in the next day is 1/3, what is the probability that at least 3 babies out of a cluster of 7 babies will speak tomorrow?


Let $a,$ $b,$ and $c$ be complex numbers such that $|a| = |b| = |c| = 1$ and
\[\frac{a^3}{bc} + \frac{b^3}{ac} + \frac{c^3}{ab} = -3.\]
Find all possible values of $|a + b + c|$.


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


For how many ordered pairs of positive integers $(x, y)$, with $x < y$, is the harmonic mean of $x$ and $y$ equal to $9^{15}$?


Expand the following product: $\frac{3}{4}\left(\frac{8}{x^2} - 5x^3\right)$.

Solution: \boxed{\frac{6}{x^2} - \frac{15x^3}{4}}


For how many different digits $n$ is the three-digit number $23n$ divisible by $n$?


Given a point with rectangular coordinates $(x, y, z)$ corresponds to spherical coordinates $(4, \frac{3\pi}{7}, \frac{\pi}{8})$, find the spherical coordinates of the point with rectangular coordinates $(-x, y, z)$. Present your answer in the form $(\rho, \theta, \phi)$, with conditions $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$

Solution: \boxed{\left(4, \frac{4\pi}{7}, \frac{\pi}{8}\right)}


The values of a function $g(x)$ are given in the table below.

\begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $g(x)$ & 4 & 3 & 1 & 5 & 2 \\ \hline
\end{tabular}
If $g^{-1}$ exists, then what is $g^{-1}(g^{-1}(g^{-1}(3)))$?


In an isosceles right triangle, a square is inscribed such that two of its vertices are on one leg and one vertex on each of the hypotenuse and the other leg. If the area of this square is found to be $625 \text{cm}^2$, find the area of another square inscribed in the same triangle with one vertex each on the hypotenuse and one leg, and two vertices on the other leg. [Optionally, include a diagram similar to the original for clarity.]

Solution: \boxed{625 \text{cm}^2}


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

Solution: \boxed{-1 - x}


Let $g(x) = x^2 + x + 2023$. What is the greatest common divisor of $g(50)$ and $g(51)$?


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


Planet Yvette orbits its sun in an elliptical path with the sun situated at one of the foci. The closest point of Yvette to its sun (perigee) is 3 astronomical units (AU), and its furthest point (apogee) is 15 AU away. Calculate the distance of Yvette from its sun when it is one-quarter way along its orbit from perigee towards apogee.

Solution: \boxed{5 \text{ AU}}


What is the sum of all integer solutions to $4<(x-3)^2<64$?


Find the point where the line passing through points $(2, 3, 1)$ and $(6, 0, 7)$ intersects the $xz$-plane.

Solution: \boxed{(6, 0, 7)}


Calculate the sum of $1452_6$ and $2354_6$. Express your answer in base 6.


Let set $\mathcal{K}$ be a 60-element subset of the set $\{10, 11, 12, \ldots, 150\}$. Let $U$ be the sum of the elements of $\mathcal{K}$. Find the number of possible values for $U$.


Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, follows the rule: at each vertex, it randomly chooses one of the three edges with equal probability and crawls to the opposite vertex. Let $p = \frac{n}{2187}$ be the probability that the bug is back at vertex $A$ after crawling exactly 8 meters. Find the value of $n$.


At the Clevercat Academy, cats can learn to do three tricks: jump, fetch, and spin. The data provided is as follows:
- 40 cats can jump
- 20 cats can jump and fetch
- 25 cats can fetch
- 10 cats can fetch and spin
- 30 cats can spin
- 15 cats can jump and spin
- 7 cats can do all three tricks
- 5 cats can do none of the tricks

How many cats are in the academy?


A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-4, 5),$ $(-3, 0),$ and $(s, 3).$ Find $s^2.$


Compute the smallest positive integer \( n \) such that \( n + i \), \( (n + i)^2 \), and \( (n + i)^4 \) are the vertices of a triangle in the complex plane whose area is greater than 1000.


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


In quadrilateral $PQRS,$ $PQ = 6,$ $QR = 10$, and $RS = 25$ units. Both angle $Q$ and angle $R$ are right angles. Determine the length of segment $PS$.


Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $E$ and $AE$ at points $C$ and $D$, respectively. Additionally, let point $F$ be on the tangent at $B$ such that $CF$ is perpendicular to $AB$. If $\angle BAE = 30^\circ$, find $\angle CED$, in degrees.


The fourth and eighth terms of a geometric sequence are 2 and 162, respectively. What is the 12th term of this progression?


Let $l, m, n, p$ be real numbers, and let $A, B, C$ be points such that the midpoint of $\overline{BC}$ is $(l,0,0),$ the midpoint of $\overline{AC}$ is $(0,m,0),$ and the midpoint of $\overline{AB}$ is $(0,0,n).$ Additionally, let $p$ be the z-coordinate of all vertices $A, B, C$. Find
\[
\frac{AB^2 + AC^2 + BC^2}{l^2 + m^2 + n^2 + 3p^2}.
\]


Let $T$ be the set of all positive integers that have five digits in base $2$. What is the sum of all of the elements in $T$, when expressed in base $2$?


The line $y = -\frac{2}{3}x + 6$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $R(u,v)$ is on line segment $PQ$. If the area of $\triangle POQ$ is twice the area of $\triangle POR$, what is the value of $u \cdot v$?


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

Solution: \boxed{18x + 12}


When the base-16 number $B1234_{16}$ is written in base 2, how many base-2 digits (bits) does it have?