What is the remainder when $6^{700}$ is divided by $72$?


What is the tenth number in the row of Pascal's triangle that has 100 numbers?


A solid box measures 20 cm by 15 cm by 10 cm. From each corner of this box, a cube measuring 4 cm on each side is removed. What percentage of the original volume is removed?


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

\begin{tabular}{|c|c|c|c|c|}
\hline
& Tokyo & Cairo & Sydney & Paris \\ \hline
Tokyo & & 5900 & 4800 & 6200 \\ \hline
Cairo & 5900 & & 8700 & 2133 \\ \hline
Sydney & 4800 & 8700 & & 10400 \\ \hline
Paris & 6200 & 2133 & 10400 & \\ \hline
\end{tabular}


For what base is the representation of $346_{10}$ a four-digit number whose final digit is odd?


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


The local theater has one ticket window. Seven people, including a family of three who must stay together, line up to buy a ticket. In how many ways can they line up given this condition?


How many integer values of \(n\) satisfy the inequality \(-100 < n^3 + n^2 < 100\)?


Let \( x \) be a real number such that \( x + \frac{1}{x} = 4 \). Define \( S_m = x^m + \frac{1}{x^m} \). Determine the value of \( S_5 \).


The median of a set of consecutive odd integers is 153. If the greatest integer in the set is 167, what is the smallest integer in the set?


What is the base 4 representation of the base 2 number $101001110010_2$?


A street has parallel curbs 60 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 20 feet and each stripe is 50 feet long. Find the distance, in feet, between the stripes.


What is the largest 5-digit integer congruent to 31 modulo 26?


The third term of a geometric sequence of positive numbers is 27, and the ninth term is 3. What is the sixth term of the sequence?


Suppose $\mathbf{a}$ and $\mathbf{b}$ are unit vectors with an angle of $\frac{\pi}{4}$ between them. Calculate the volume of the parallelepiped formed by the vectors $\mathbf{a}, \mathbf{b},$ and $\mathbf{b} + \mathbf{b} \times \mathbf{a}$.


Solution: \boxed{24 \text{ feet}}


Express $\frac{63}{2^3 \cdot 5^4}$ as a terminating decimal.


What is the slope of the line determined by any two solutions to the equation $\frac{4}{x} + \frac{6}{y} = 0$? Express your answer as a common fraction.


In Pascal's Triangle, each number is the sum of the number just above it to the left and the number just above it to the right. Calculate the sum of the numbers in Row 10 of Pascal's Triangle.


A parabola has a vertex at $V = (0, 0)$ and a focus at $F = (0, 2)$. Let $P$ be a point on this parabola in the first quadrant such that the distance from $P$ to the focus $F$ is 50. Find $P$.

Solution: \boxed{(8\sqrt{6}, 48)}


$\triangle ABC$ is inscribed in a circle of radius $r$, with $A$ and $B$ lying diametrically opposite each other. Point $C$ is placed on the circle but not on diameter $AB$. Determine the maximum possible value of $s^2$, where $s = AC + BC$.


In Gridtown, the streets are all $30$ feet wide and the blocks they enclose are all squares of side length $500$ feet. Tom runs on the inner edge of a street surrounding one of these blocks, while Jerry runs on the outer edge of the same street. Calculate how many more feet than Tom does Jerry run for each lap around the block.


Evaluate the series
\[
\sum_{n=1}^\infty \frac{n^3 + 2n^2 - n + 1}{(n+3)!}.
\]


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


There are four points that are $4$ units from the line $y=10$ and $10$ units from the point $(5,10)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


A function $g$ is defined on the complex numbers by $g(z)=(c+di)z,$ where $c$ and $d$ are real numbers. This function has the property that for each complex number $z$, $g(z)$ is equidistant from both $z$ and the origin. Given that $|c+di|=7$, find $d^2.$


Xanthia buys hot dogs that come in packages of 5, and she buys hot dog buns that come in packages of 7. What is the smallest number of hot dog packages she can buy in order to be able to buy an equal number of hot dogs and hot dog buns?


**A square is divided into six congruent rectangles. If the perimeter of each of these six rectangles is 42 inches, what is the perimeter of the square, in inches?**

Solution: \boxed{72 \text{ inches}}


How many four-digit positive integers $x$ satisfy $3874x + 481 \equiv 1205 \pmod{31}$?


A spherical bubble collapses into a right circular cone while maintaining its volume. The height of the cone is 6 times the radius of its base. If the radius of the base of the cone is $3\sqrt[3]{2}$ cm, find the radius of the original sphere before it changed shape.


Evaluate \begin{align*} (5a^2 - 13a + 4)(2a - 3) \end{align*} for $a = 2$.


Evaluate the product $\frac{1}{2}\cdot\frac{4}{1}\cdot\frac{1}{8}\cdot\frac{16}{1} \dotsm \frac{1}{16384}\cdot\frac{32768}{1}$.


Find the matrix $\mathbf{M}$ that scales the first row of any matrix by 5 and triples the second row. Specifically, determine $\mathbf{M}$ such that
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 5a & 5b \\ 3c & 3d \end{pmatrix}.\]
If no such matrix $\mathbf{M}$ exists, then enter the zero matrix.

Solution: \boxed{\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix}}


How many distinct, positive factors does $1320$ have?


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

\(\bullet\) Sequence \(A\): \(3\), \(9\), \(27\), \(81\), \(243\), \(\ldots\)

\(\bullet\) Sequence \(B\): \(100\), \(110\), \(120\), \(130\), \(\ldots\)


The value $2^{10} - 1$ is divisible by how many prime numbers, and what is their sum?


The values of $f$, $g$, $h$, and $j$ are 6, 7, 8, and 9, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$, and $fj$?


Find the area of triangle $ABC$ below which is a right triangle with a $45^\circ$ angle at $A$.

[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (5,0);
R = (0,5);
draw (P--Q--R--P,linewidth(0.9));
label("$A$",P,S);
label("$B$",Q,S);
label("$C$",R,W);
label("$5$",Q/2,S);
label("$5$",R/2,W);
label("$45^\circ$",(-0.2,0.2),S);
[/asy]


The 5-digit number $63\,47\square$ is a multiple of 9. Which digit is represented by $\square$?


A regular octagon $ABCDEFGH$ has its sides' midpoints connected to form a smaller octagon. What fraction of the area of the larger octagon $ABCDEFGH$ is enclosed by the smaller octagon?

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


Determine how many distinct five-digit positive integers make the product of their digits equal 18.


Let $a \star b = 2ab - 3b - a$. If $4 \star y = 80$, find the value of $y$.


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


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


Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle XYZ$ are perpendicular. If $AD = 18$ and $BE = 24$, determine the area of $\triangle XYZ$.


Suppose all four numbers \[-1+\sqrt{5}, \; -1-\sqrt{5}, \; 2-\sqrt{3}, \; 2+\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


A sports conference comprises 16 teams divided equally into two divisions of 8 teams each. How many games are in a complete season for the conference if each team must play every other team in its own division three times and every team in the other division twice?


The function $g(x)$ satisfies
\[xg(y) = yg(x)\]
for all real numbers $x$ and $y.$  If $g(20) = 30,$ find $g(5).$


Let $A = (0,9),$ $B = (6,9),$ and $C = (p,q)$ be three points on the parabola $y = x^2 - 6x + 9,$ where $1 \le p \le 6.$  Find the largest possible area of triangle $ABC.$


Let vectors $\mathbf{u}$ and $\mathbf{v}$ be such that $\|\mathbf{u}\| = 9$ and $\|\mathbf{v}\| = 13$. Find all possible values of $\mathbf{u} \cdot \mathbf{v}$ if the angle between them, $\theta$, is between $\frac{\pi}{6}$ and $\frac{\pi}{3}$.

Solution: \boxed{[58.5, 58.5\sqrt{3}]}


Two standard dice are rolled. What is the expected number of 6's obtained? Express your answer as a common fraction.


Consider two lines: line $l$ parametrized as
\begin{align*}
x &= 2 + 5t,\\
y &=  3 + 4t
\end{align*}
and the line $m$ parametrized as
\begin{align*}
x &= -3 + 5s,\\
y &= 5 + 4s.
\end{align*}
Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.
Assume $\overrightarrow{PA}$ is the projection of $\overrightarrow{BA}$ onto some vector $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ such that $3v_1 + v_2 = 4$. Find $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} \frac{16}{7} \\ -\frac{20}{7} \end{pmatrix}}


A $168 \times 350 \times 390$ rectangular solid is constructed by gluing together $1 \times 1 \times 1$ cubes. An internal diagonal of this solid passes through how many of the $1 \times 1 \times 1$ cubes?


The deli has five kinds of bread, seven kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, turkey, cheddar cheese, and rye bread are each offered at the deli. If Al never orders a sandwich with a ham/cheddar cheese combination nor a sandwich with a rye bread/turkey combination, how many different sandwiches could Al order?


Find the remainder when $123456789012$ is divided by $180$.


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


In a different sequence, the first term is $a_1 = 2000$ and the second term remains $a_2 = 2008$. The values of the remaining terms are set such that:
\[a_n + a_{n + 1} + a_{n + 2} = 2n\]
for all $n \geq 1$. Determine $a_{1000}$.


In the expansion of $(x+1)^{50}$, what is the coefficient of the $x^3$ term?


In writing the integers from 100 through 999 inclusive, how many times is the digit 6 written?


A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid).


Note that $8^2 = 64$, which contains no zeros; $88^2 = 7744$, which contains no zeros; and $888^2 = 788,\!944$, which contains 1 zero. Assuming this pattern continues, how many zeros are in the expansion of $88,\!888,\!888^2$?


How many non-similar convex quadrilaterals have angles whose degree measures are distinct positive integers in arithmetic progression?


Alan is 4 years younger than Bob. Bob is 5 years older than Carl. Donna is two years older than Carl. Emily is the sum of Alan's and Donna's age minus Bob's age. If Bob is 20 years old, how old is Emily?


In the diagram, $O$ is the center of a circle with radii $OP=OQ=r$. What is the perimeter of the shaded region assuming the arc $PQ$ corresponds to exactly half of the circle?

[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(0,-1),S);

fill(Arc((0,0),1,-90,0)--cycle,mediumgray);
draw(Arc((0,0),1,-90,0));
fill((0,0)--(-1,0)--(0,-1)--cycle,white);
draw((-1,0)--(0,0)--(0,-1));
draw((-.1,0)--(-.1,-.1)--(0,-.1));

[/asy]

Solution: \boxed{2r + \pi r}


What is the number of square meters in the area of a circle with a diameter of \(7.5\) meters? Express your answer in terms of \(\pi\).


Calculate the number of digits in the value of $2^{15} \times 5^{10} \times 12$.


A certain integer has $5$ digits when written in base $7$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?


Find all values of $x$ such that $\arctan x > \arcsin x$.

Solution: \boxed{[-1, 0)}


The graph of \(y = h(x)\) is defined in parts as follows:
1. For \(x \geq -4\) and \(x \leq 1\), \(y = 1 - x\).
2. For \(x \geq 1\) and \(x \leq 3\), \(y = \sqrt{4 - (x - 3)^2} + 1\).
3. For \(x \geq 3\) and \(x \leq 4\), \(y = 2(x - 3) + 1\).

The graph of \(y = j(x)\) is derived from the graph of \(y = h(x)\). First, the graph of \(y = h(x)\) is reflected over the y-axis and then shifted 6 units to the right. What is \(j(x)\) in terms of \(h(x)\)?

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


If $p$, $q$, and $r$ are positive integers such that $\gcd(p,q) = 210$ and $\gcd(p,r) = 770$, what is the smallest possible value of $\gcd(q,r)$?


Determine how many times the graphs \( r = 3 \cos \theta \) and \( r = 6 \sin \theta \) intersect.


Five years ago, there were 30 trailer homes on Maple Street with an average age of 12 years. At that time, a group of brand new trailer homes was then added to Maple Street. Today, the average age of all the trailer homes on Maple Street is 10 years. How many new trailer homes were added five years ago?


Find the base \( b \) such that when the number \( 64_{10} \) is expressed in base \( b \), it has exactly 4 digits.


In triangle $XYZ$, medians $\overline{XM}$ and $\overline{YN}$ intersect at right angles at point $O$. If $XM = 18$ and $YN = 24$, find the area of $\triangle XYZ$.


Below is a portion of the graph of a quadratic function, $y=p(x)=ax^2+bx+c$:
\[ \text{The graph has symmetry such that } p(10) = p(20) \text{ and } p(9) = p(21) \text{, etc., with an axis of symmetry at } x = 15. \]
Given that $p(25) = 9$ and the graph goes through the point $(0, 1)$, calculate $p(5)$.


The equation \( y = -16t^2 + 64t \) describes the height (in feet) of a projectile launched from the ground at 64 feet per second. At what \( t \) will the projectile reach 25 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth.


In the diagram, each circle is divided into two equal areas, and $O$ is the center of the larger circle. The area of the larger circle is $100\pi$. Two smaller circles of equal size lie within the larger circle such that each smaller circle touches $O$ and the perimeter of the larger circle. What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((1,0),1,180,360)--cycle,mediumgray); fill(Arc((-1,0),1,180,360)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((1,0),1));
draw(Circle((-1,0),1));
dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1,-1)--(1,-1));
[/asy]


The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $12$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.


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


Define the function \( g \) on the positive integers as:
\[
g(n) = \left\{
\begin{array}{cl}
n^2 + 20 & \text{if } n \le 12, \\
g(n - 7) & \text{if } n > 12.
\end{array}
\right.
\]
Find the maximum value of function \( g \).


Billy is hiking in Colorado. He walks eastward five miles, then turns $45$ degrees northward and walks eight miles. How far is he from his starting point? Express your answer in simplest radical form.

Solution: \boxed{\sqrt{89 + 40\sqrt{2}}}


Find the remainder when $3x^5 - 2x^3 + 5x - 8$ is divided by $x^2 + 2x + 1$.


Compute the distance between the parallel lines given by
\[\begin{pmatrix} 3 \\ -4 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -14 \\ 0 \end{pmatrix}\]and
\[\begin{pmatrix} 2 \\ -7 \\ 4 \end{pmatrix} + s \begin{pmatrix} 2 \\ -14 \\ 0 \end{pmatrix}.\]


Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. If set $C$ intersects only with set $A$ adding 300 more elements to the union, how many elements are in set $A$?

[asy]
label("$A$", (2,67));
label("$B$", (80,67));
label("$C$", (41,10));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
draw(Circle((44, 27), 22));
label("1200", (44, 45));
label("300", (44, 27));
[/asy]


Eight painters working at the same rate take $2.5$ work-days to finish a job. If six painters are available and they take a rest day for every five consecutive workdays, how many total calendar days will it take them to finish the job, working at the same rate?


Let \(h(x) = x^5 + x^4 + x^3 + x^2 + x + 1.\) What is the remainder when the polynomial \(h(x^{18})\) is divided by the polynomial \(h(x)\)?


Calculate the least number of digits in the repeating block of the decimal expansion of $7/13$.


Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. If $n$ is a multiple of 18, what is the smallest value of $n$ such that $f(n) > 18$?

Solution: \boxed{n = 342}


How many degrees are in the sum of the measures of the nine numbered angles pictured in the diagram? In the diagram, there are three triangles, each sharing sides but not overlapping:
1. Triangle A with angles labeled 1, 2, 3.
2. Triangle B with angles labeled 4, 5, 6.
3. Triangle C, newly added, with angles labeled 7, 8, 9.

Each triangle has its vertices connected such that no vertex lies inside another triangle.


Given a cylinder with a fixed volume \( V \), the total surface area (including the two circular ends) is minimized for a radius \( R \) and height \( H \). Find \( \frac{H}{R} \) when the volume is doubled.


For positive integers $n$, let $g(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point in base $3$. How many positive integer divisors does $g(2023)$ have?


The National Courier Company charges an extra $\$0.15$ in postage if the length of an envelope in inches divided by its height in inches is less than $1.2$ or greater than $2.8$. Additionally, an extra $\$0.10$ is charged if the thickness of the envelope in inches exceeds $0.25$. For how many of these envelopes must the extra postage be paid?
\begin{tabular}[t]{cccc}
Envelope & Length in inches & Height in inches & Thickness in inches\\\hline
A &7 &5 &0.2\\
B &10 &2 &0.3\\
C &7 &7 &0.1\\
D &12 &4 &0.26
\end{tabular}


A cylinder-shaped container of cookies is 4 inches in diameter and 5 inches high and sells for $\$$2.00. If the price of the container increases quadratically with volume, what would be the price for a container that is 8 inches in diameter and 10 inches high?


A circular pizza with a diameter of $16\text{ cm}$ is divided into four equal sectors. Determine the square of the length of the longest line segment that can be drawn within one of these sectors.


Compute
$$\sum_{k=1}^{500} k(\lceil \log_{\sqrt{3}}{k}\rceil- \lfloor\log_{\sqrt{3}}{k} \rfloor).$$


In triangle $DEF$, the measure of $\angle D$ is $88^\circ$. The measure of $\angle E$ is $20^\circ$ more than four times the measure of $\angle F$. What is the measure, in degrees, of $\angle F$?


Let $g$ be a function defined by $g(x) = \frac{px + q}{rx + s}$, where $p$, $q$, $r$ and $s$ are nonzero real numbers, and the function has the properties $g(31)=31$, $g(41)=41$, and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Determine the unique number that is not in the range of $g$.


Find the sum of the distinct prime factors of $7^7 - 7^4$.


When $\sqrt[3]{8000}$ is simplified, the result is $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ is as small as possible. What is $c+d$?


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


We are allowed to remove exactly one integer from the list $$-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,$$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?


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


Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \).


There are three complex numbers $a+bi$, $c+di$, and $e+fi$. Given $b=5$, $e = -2(a+c)$, and the sum of the complex numbers is $4i$. Find $d+f$.


For how many positive integers $n$ does the sum $1+2+\cdots+n$ evenly divide $10n$?


Find $100^{-1} \pmod{101}$, as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)


If 72 cards are dealt to 10 people as evenly as possible, how many people will end up with fewer than 8 cards?


Determine the sum of the squares of the coefficients when the expression $3(x^2-3x+3)-8(x^3-2x^2+x-1)$ is fully simplified.


For $-1<s<1$, let $T(s)$ denote the sum of the geometric series \[20 + 20s + 20s^2 + 20s^3+\cdots .\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b) = 4800$. Find $T(b) + T(-b)$.


How many ways are there to put 5 balls in 4 boxes if the balls are not distinguishable and neither are the boxes?


What is the 150th digit to the right of the decimal point in the decimal representation of $\frac{17}{150}$?


In a right triangle $\triangle PQR$, we know that $\tan Q = 0.5$ and the length of $QP = 16$. What is the length of $QR$?

Solution: \boxed{8 \sqrt{5}}


Find the remainder when \(5x^4 - 9x^3 + 3x^2 - 7x - 30\) is divided by \(3x - 9\).


How many license plates consist of 3 letters followed by 3 digits, where the digits must include exactly one odd number and two even numbers?


The function $g(x)$ satisfies
\[g(x) + 3g(1 - x) = 4x^2\] for all real numbers $x.$ Find $g(4)$.


What is the least positive multiple of 17 that is greater than 450?


In two concentric circles, the radius of the outer circle is twice the radius of the inner circle. What is the area of the gray region, in square feet, if the width of the gray region is now 3 feet instead of 2 feet? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),4),gray);
filldraw(circle((0,0),1),white);

draw((1,0)--(4,0),linewidth(1));

label("$3^{\prime}$",(2.5,0),N);
[/asy]


What is the area enclosed by the graph of $|5x| + |3y| = 30$?


Let \( f(x) = x^2 + px + q \) and \( g(x) = x^2 + rx + s \) be two distinct quadratic polynomials where 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 \( f \), also both \( f \) and \( g \) have the same minimum value. If the graphs of the two quadratic polynomials intersect at the point \( (50,-200), \) what is the value of \( p + r \)?


I have 8 shirts, 4 pairs of pants, and 8 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, plus white, yellow, and red. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?


Triangle $ABC$ has side-lengths $AB = 15, BC = 25,$ and $AC = 20.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects the extensions of $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. Determine the perimeter of $\triangle AMN$.


Let \[f(x) = \left\{
\begin{array}{cl}
\sqrt{x} &\text{ if }x\geq 5, \\
x^2 &\text{ if }x < 5.
\end{array}
\right.\]Find $f(f(f(3)))$.


What is the remainder when the sum of the first 150 counting numbers is divided by 11200?


In how many ways can five people sit in a row of seven chairs if one specific person must always sit in the middle chair?


In trapezoid $ABCD$, the lengths of the bases $AB$ and $CD$ are 10 and 20, respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. The height of the trapezoid from $AB$ to $CD$ is 12 units. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.


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


In a certain ellipse, the endpoints of the major axis are $(3, -5)$ and $(23, -5)$. Also, the ellipse passes through the point $(19, -2)$. Find the area of the ellipse.

Solution: \boxed{37.5 \pi}


Find the integer $n$, such that $0 \le n \le 8$, and it satisfies:
\[n \equiv -4567 + x \pmod{9}\]
where $x$ is the smallest positive integer that makes $n$ non-negative.


What is the product of all real numbers that are tripled when added to their reciprocals?


Triangle $PQR$ is an obtuse, isosceles triangle. Angle $P$ measures $30^\circ$. What is the number of degrees in the measure of the largest interior angle of triangle $PQR$?


What is the remainder when \( x^4 - 8x^3 + 12x^2 + 5x - 20 \) is divided by \( x + 2 \)?


Define \( A \diamond B \) as \( A \diamond B = \frac{(A-B)}{5} \). What is the value of \( (7 \diamond 15) \diamond 2 \)?


A line segment has one endpoint at $(10,-5)$. The midpoint of the segment is scaled by a factor of 2 along each axis, resulting in a point at $(12, -18)$. Determine the sum of the coordinates of the other endpoint.


At the beginning of every session of Geometry class, Mr. Slate picks a random student to help him with a demonstration. There are 15 students in his class, and the class meets three times a week. How many different sequences of student helpers are possible in a week?


What is the largest integer less than $\log_3 \frac{3}{2} + \log_3 \frac{4}{3} + \cdots + \log_3 \frac{3010}{3009} + \log_3 \frac{3011}{3010}$?


Define a new operation $@$ by
\[a @ b = \frac{a + b}{1 + ab}.\]
Compute the value of
\[1 @ (2 @ (3 @ (\dotsb @ (999 @ 1000) \dotsb))).\]


Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/37 of the original integer.


Let $a$, $b$, $c$, $x$, $y$, and $z$ be real numbers that satisfy the three equations:
\begin{align*}
  15x + by + cz &= 0 \\
  ax + 25y + cz &= 0 \\
  ax + by + 45z &= 0.
\end{align*}
Suppose that $a \neq 15$, $b \neq 25$ and $x \neq 0$. What is the value of
\[ \frac{a}{a - 15} + \frac{b}{b - 25} + \frac{c}{c - 45} \, ?\]


Let \( O \) be the origin and let \( (a, b, c) \) be a fixed point. A plane with the equation \( x + 2y + 3z = 6 \) passes through \( (a, b, c) \) and intersects the \( x \)-axis, \( y \)-axis, and \( z \)-axis at \( A, B, \) and \( C \), respectively, all distinct from \( O \). Let \( (p, q, r) \) be the center of the sphere passing through \( A, B, C, \) and \( O \). Find
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]


Solution: \boxed{2 + \log_{10} 2.5}


Fifty cards are placed into a box, each bearing a number from 1 to 10, with each number appearing on five cards. Five cards are drawn from the box at random and without replacement. Let $p$ be the probability that all five cards bear the same number. Let $q$ be the probability that four of the cards bear a number $a$ and the fifth bears a number $b$ that is not equal to $a$. What is the value of $q/p$?


What is the total number of digits used when the first 2500 positive even integers are written?


The matrix for projecting onto a certain line $\ell,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{9} & \frac{1}{18} & \frac{1}{6} \\ \frac{1}{18} & \frac{1}{36} & \frac{1}{12} \\ \frac{1}{6} & \frac{1}{12} & \frac{1}{4} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]Find the direction vector of line $\ell.$  Enter your answer in the form $\begin{pmatrix} a \\ b \\ c \end{pmatrix},$ where $a,$ $b,$ and $c$ are integers, $a > 0,$ and $\gcd(|a|,|b|,|c|) = 1.$

Solution: \boxed{\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}}


Determine the positive difference between the roots of the quadratic equation $5x^2 - 9x + 1 = 0$. Express the difference in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are integers with $p$ not divisible by the square of any prime number. Compute $p + q$.


Pirate Pete shares his treasure with Pirate Paul, but this time in a slightly different manner. Pete starts by saying, "One for me, one for you," giving himself one coin and starting Paul's pile with one coin. He then says, "Two for me, and two for you," giving himself two more coins but making Paul's pile two coins in total. The pattern continues with Pete announcing, "Three for me, three for you," and so forth, until Pete gives himself $x$ more coins making Paul’s pile $x$ coins in total. They continue this process until all the coins are distributed, with Pirate Pete having exactly three times as many coins as Pirate Paul. Determine the total number of gold coins they have.


Find the smallest real number $c$ such that \[|x_1| + |x_2| + \dots + |x_9| \geq c|M|\] whenever $x_1, x_2, \ldots, x_9$ are real numbers such that $x_1+x_2+\cdots+x_9 = 10$ and $M$ is the median of $x_1, x_2, \ldots, x_9$.

Solution: \boxed{c = 9}


A square carpet of side length 12 feet is designed with one large shaded square and twelve smaller, congruent shaded squares. If the ratios $12:\text{S}$ and $\text{S}:\text{T}$ are both equal to 4, and $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares, what is the total shaded area?


Find \(89^{-1} \pmod{90}\), as a residue modulo 90. (Give an answer between 0 and 89, inclusive.)


Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \).


Fifty slips are placed into a hat, each bearing a number from 1 to 12, with each number appearing on five slips. Five slips are drawn at random without replacement. Let $p'$ be the probability that all five slips bear the same number. Let $q'$ be the probability that three of the slips bear a number $a$ and the other two bear a number $b$ where $b \ne a$. What is the value of $q'/p'$?


After replacing both the numerator and the denominator of a fraction with new numbers, a student incremented each by 8, resulting in a fraction value of $\frac{2}{5}$. If the original numerator was 3, what was the original denominator?


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


Calculate the sum of the squares of the coefficients of the expanded form of $3(x^4 + 2x^3 + 5x^2 + x + 2)$.


The product of two consecutive page numbers is \(20,412.\) What is the sum of the two page numbers?


I have 6 marbles numbered 1 through 6 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles?


A function \( g(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have
\[ 3g(x) + g\left(\frac{1}{x}\right) = 7x + 6. \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( g(x) = 2005 \). Compute the integer nearest to \( T \).


Determine the numerical value of $k$ such that the equations are valid:
\[\frac{9}{x - y} = \frac{k}{x + z} = \frac{16}{z + y}.\]


Given the system of equations
\[
4x - 2y = a,
\]
\[
6y - 12x = b.
\]
Find \(\frac{a}{b}\), assuming \(b \neq 0\).


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


In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 13, sides $\overline{CD}$ and $\overline{DA}$ both have length 20, and the measure of angle $ADC$ is $120^\circ$. What is the length of diagonal $\overline{AC}$?


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


For a finite sequence \( B = (b_1, b_2, \dots, b_{199}) \) of numbers, the Cesaro sum of \( B \) is defined to be
\[ \frac{T_1 + \cdots + T_{199}}{199}, \]
where \( T_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 199 \).

If the Cesaro sum of the 199-term sequence \( (b_1, \dots, b_{199}) \) is 2000, what is the Cesaro sum of the 200-term sequence \( (2, b_1, \dots, b_{199}) \)?


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

Solution: \boxed{\left( \frac{9}{16}, \infty \right)}


Find the integer $n$, $0 \le n \le 12$, such that \[n \equiv 123456 \pmod{11}.\]


For the given quadrilateral, calculate the number of possible integer values for the length of the diagonal, represented by the dashed line.

[asy]
draw((0,0)--(4,4)--(10,-1)--(6,-7)--cycle,linewidth(0.7));
draw((0,0)--(10,-1),dashed);
label("6",(2,2),NW);
label("12",(7,1.5),NE);
label("14",(8, -4),SE);
label("10",(3,-3.5),SW);
[/asy]


Dots are spaced one unit apart, horizontally and vertically. Consider a polygon formed by joining the dots at coordinates (0,0), (3,0), (6,0), (6,3), (9,3), (9,6), (6,6), (6,9), (3,9), (3,6), (0,6), (0,3), and back to (0,0). What is the number of square units enclosed by this polygon?


The function $g$ defined by $g(x) = \frac{px+q}{rx+s}$, where $p$, $q$, $r$, and $s$ are nonzero real numbers, has the properties $g(13)=13$, $g(61)=61$, and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Find the unique number that is not in the range of $g$.


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


How many solutions does the equation \[\frac{(x-1)(x-2)(x-3)\dotsm(x-120)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-120^2)} = 0\] have for $x$?


The repeating decimal for $\frac{7}{26}$ is $0.abcdabcdabcd\ldots$ What is the value of the expression $3a - b$?


In triangle $ABC$, $AB=AC$ and $D$ is a point on $\overline{AC}$ such that $\overline{BD}$ bisects angle $ABC$. Point $E$ is on $\overline{BC}$ such that $\overline{DE}$ bisects $\angle BDC$. Given $BD=BC$, find the measure, in degrees, of angle $BDE$.


Let a sequence be given where $b_1 = 3$, $b_2 = 9$, and for $n \geq 3$, $b_{n+1} = b_n b_{n-1}$. Calculate $b_{20}$.


How many pairs of positive integers $(x, y)$ satisfy the equation $x^2 - y^2 = 77$?


Determine how many of the first $25$ rows of Pascal's triangle, excluding row $0$ and row $1$, consist entirely of even numbers.


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


Calculate the last three digits of $11^{30}$.


Calculate the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if line $CX$ bisects $\angle ACB$. See the diagram below:
- $BC = 36$ units
- $AC = 45$ units
- The line $CX$ is the angle bisector of $\angle ACB$.
Express your answer as a common fraction.


What is the minimum value of the expression $x^2 + y^2 - 8x + 6y + 20$ for real $x$ and $y$?


The sum of two numbers is 60, and their difference is 10. Calculate their product and the square of their sum.


Mark has $\frac{4}{5}$ of a dollar, Carolyn has $\frac{2}{5}$ of a dollar, and Dave has $\frac{1}{2}$ of a dollar. How many dollars do they have altogether?


Let $p$, $q$, $r$, $s$, and $t$ be positive integers with $p+q+r+s+t=2025$ and let $N$ be the largest of the sums $p+q$, $q+r$, $r+s$, and $s+t$. Determine the smallest possible value of $N$.


Suppose \( q(x) \) is a polynomial such that \( q(x) + (2x^6 + 4x^4 + 5x^3 + 11x) = (10x^4 + 30x^3 + 40x^2 + 8x + 3) \). Express \( q(x) \) as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{-2x^6 + 6x^4 + 25x^3 + 40x^2 - 3x + 3}


Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$?


Given that $x$ is a multiple of $3456$, what is the greatest common divisor of $f(x)=(5x+3)(11x+2)(14x+7)(3x+8)$ and $x$?


Construct a stem-and-leaf plot and find the positive difference between the median and the mode of the data given below:
\begin{tabular}{|c|c|}
\hline
\textbf{Tens} & \textbf{Units} \\
\hline
2 & $0 \hspace{2mm} 0 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 1$ \\
\hline
3 & $4 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 5 \hspace{2mm} 7 \hspace{2mm} 9$ \\
\hline
4 & $1 \hspace{2mm} 3 \hspace{2mm} 5 \hspace{2mm} 7 \hspace{2mm} 7$ \\
\hline
\end{tabular}


Find the area of the triangle bounded by the $y$-axis and the lines $y - 4x = -3$ and $4y + x = 16$.


Compute
\[\sin^6 0^\circ + \sin^6 1^\circ + \sin^6 2^\circ + \dots + \sin^6 90^\circ.\]


The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?


If $B$ is an angle such that $\tan B + \sec B = 3,$ find all possible values of $\cos B.$


How many integers from 1 through 9999, inclusive, do not contain any of the digits 2, 3, 4, 5, or 8?


Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \cos 390^\circ.$


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


What is the coefficient of \(a^3b^3\) in \((a+b)^6\left(c+\dfrac{1}{c}\right)^6\)?


Compute the exact value of the expression $\left|3\pi - | 3\pi - 10 | \right|$. Write your answer using only integers and $\pi$, without any absolute value signs.

Solution: \boxed{6\pi - 10}


An isosceles trapezoid has bases of lengths 25 units (AB) and 13 units (CD), with the non-parallel sides (AD and BC) both measuring 13 units. How long is the diagonal AC?


How many integers from 1 through 999, inclusive, do not contain the digits 1, 2, 4, or 8?


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


Find the quadratic polynomial, with real coefficients, which has $3 + 4i$ as a root, and where the coefficient of $x$ is $8$.

Solution: \boxed{-\frac{4}{3}x^2 + 8x - \frac{100}{3}}


Given that $a+b=2$ and $a^3+b^3=16$, find the value of $ab$.


What is the smallest possible value of the sum $\lvert x + 3\rvert + \lvert x + 6\rvert + \lvert x + 8\rvert$ as $x$ varies?


How many digits are located to the right of the decimal point when $\frac{5^7}{10^5 \cdot 8}$ is expressed as a decimal?


Copying a page costs 3.5 cents. There is a special discount of 5 cents total if you copy at least 400 pages. How many pages can you copy for $\$25$ if you meet the discount requirement?


The graph of the equation $y = \frac{x}{x^3 + Ax^2 + Bx + C}$, where $A, B, C$ are integers, displays three vertical asymptotes at $x = -3, 0, 4$. Calculate $A + B + C$.


You have $7$ red shirts, $5$ blue shirts, $8$ green shirts, $10$ pairs of pants, $10$ green hats, $6$ red hats, and $7$ blue hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?


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


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


If $\frac{1}{8}$ of $2^{32}$ is $4^x$, then what is the value of $x$?


$ABCDEFGHIJ$ is a rectangular prism with $AB = 2$, $BC = 3$, $AE = 4$. Find $\sin \angle GAC$.

[asy]
import three;
triple A, B, C, D, EE, F, G, H;
A = (0, 0, 0);
B = (2, 0, 0);
C = (2, 3, 0);
D = (0, 3, 0);
EE = (0, 0, 4);
F = B + EE;
G = C + EE;
H = D + EE;
draw(B--C--D--A--cycle);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F);
draw(C--G);
draw(D--H);
draw(dashed, G--A--C);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, NE);
label("$D$", D, NW);
label("$E$", EE, S);
label("$F$", F, E);
label("$G$", G, N);
label("$H$", H, W);
[/asy]


The function \( g(x) \) satisfies
\[
g(3^x) + (x + 1)g(3^{-x}) = 3
\]
for all real numbers \( x \). Find \( g(3) \).


A circular cylindrical post with a circumference of 6 feet has a string wrapped around it, spiraling from the bottom of the post to the top. The string evenly loops around the post exactly three full times, starting at the bottom edge and finishing at the top edge. The height of the post is 15 feet. What is the length, in feet, of the string?


Consider the 2004th, 2005th, and 2006th rows of Pascal's triangle. Let $(a_i)$, $(b_i)$, and $(c_i)$ represent the sequence, from left to right, of elements in these rows, respectively, with the leftmost element occurring at $i = 0.$ Compute
\[
\sum_{i = 0}^{2005} \frac{b_i}{c_i} - \sum_{i = 0}^{2004} \frac{a_i}{b_i}.
\]


Determine how many integers between 1 and 1200 inclusive can be expressed as the difference of the squares of two nonnegative integers.


An ellipse has foci at \((8, 1)\) and \((8, 9)\), and it passes through the point \((17, 5)\). Write the equation of this ellipse in the standard form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\] where \(a\), \(b\), \(h\), and \(k\) are constants, with \(a\) and \(b\) positive. Find the ordered quadruple \((a, b, h, k)\).

Solution: \boxed{(9, \sqrt{97}, 8, 5)}


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


The sum of two numbers is $24$ and their difference is $8$. What is the sum of the squares of their sum and difference?


Find the maximum value of the expression:
\[
\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_6 + \cos \theta_6 \sin \theta_7 + \cos \theta_7 \sin \theta_1,
\]
over all real numbers $\theta_1,\theta_2,\theta_3,\theta_4,\theta_5, \theta_6, \theta_7$.


The quadratic equation $x^2 + mx + n = 0$ has roots that are three times those of $x^2 + px + m = 0$, and none of $m, n, p$ is zero. Determine the value of $\frac{n}{p}$.


Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^4$ for all $x \ge 1.$  If $g(64) = 64,$ then compute $[g(8)]^4.$


A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, of which $32$ are triangular and $12$ are quadrilaterals. Determine the number of space diagonals inside $Q$.


Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after two turns?


In a conference room, each row can seat either 9 or 10 people. Fifty-four people need seating with every seat filled. How many rows must seat exactly 10 people?


In how many ways can the digits of $85,\!550$ be arranged to form a 5-digit number? (Remember, numbers cannot begin with 0.)


What is the greatest integer less than 200 for which the greatest common factor of that integer and 72 is 9?


A box contains 8 white balls and 7 black balls. Seven balls are drawn out of the box at random. What is the probability that they all are white?


If $f(x) = ax + b$ and $f^{-1}(x) = bx + a + c$ where $c$ is an additional constant and $a, b, c$ are real numbers, what is the value of $a + b + c$?


Solution: \boxed{1663232 \cdot 16^{10}}


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_4b_5 = 21$, what is $b_3b_6$?


Find the integer \( n \), \( -180 < n < 180 \), such that \( \tan n^\circ = \tan 1500^\circ \).

Solution: \boxed{n = 60}


In the diagram, $PQRS$ is a trapezoid with an area of $18.$ $RS$ is three times the length of $PQ.$ What is the area of $\triangle PQS?$

[asy]
draw((0,0)--(1,4)--(4,4)--(12,0)--cycle);
draw((4,4)--(0,0));
label("$S$",(0,0),W);
label("$P$",(1,4),NW);
label("$Q$",(4,4),NE);
label("$R$",(12,0),E);
[/asy]


After collecting coins for a year, Sarah has a jar containing 157 quarters and 342 dimes. Mark has a jar containing 211 quarters and 438 dimes. They decide to combine their coins and roll them into rolls containing 25 quarters and 40 dimes each. Calculate the total value, in dollars, of the quarters and dimes that cannot be rolled.


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


I take variable $c$, triple it, and add six. I subtract $6c$ from this new expression, and divide the resulting difference by three. What is my final expression in simplest form?

Solution: \boxed{-c + 2}


$P, Q, R,$ and $S$ are distinct positive integers such that the product $PQ = 120$, the product $RS = 120$, and $P - Q = R + S$. What is the value of $P$?


If \(a+b=\frac{9}{17}\) and \(a-b=\frac{1}{51}\), what is the value of \(a^2-b^2\)? Express your answer as a common fraction.


A biologist wants to estimate the population of a fish species in a large pond. On April 1, she captures and tags 120 fish, then releases them back into the pond. On August 1, she captures another sample of 150 fish, of which 5 are found to be tagged. She estimates that 30% of the fish present on April 1 have left the pond and that 50% of the fish in the August sample were not in the pond in April. Using this information, how many fish were in the pond on April 1?


The polynomial $3x^3 + bx + 15$ has a factor of the form $x^2 + px + 1$. Find $b$.


Suppose $a,b,$ and $c$ are integers such that $4b = 10 - 3a + c$. Determine how many of the numbers from the set $\{1, 2, 3, 4, 5, 6\}$ must be divisors of $3b + 15 - c$.


Simplify and write the result as a common fraction: $$\sqrt{\sqrt[4]{\left(\frac{1}{65536}\right)^2}}$$


Find the number of integers \( n \) that satisfy
\[15 < n^2 < 121.\]


How many integers $n$ satisfy $(n-3)(n+5)<0$?


What is the area of the circle defined by $x^2 - 8x + y^2 - 16y + 68 = 0$ that lies above the line $y = 4$?


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 316 hours, 59 minutes, and 59 seconds, the time will be $A:B:C$. What is the value of $A + B + C$?


Let
\[\mathbf{A} = \begin{pmatrix} 5 & 2 \\ -12 & -5 \end{pmatrix}.\]
Compute $\mathbf{A}^{50}.$

Solution: \boxed{\mathbf{A}^{50} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}


There are 25 gremlins, 20 imps, and 10 sprites at the Horizon Fantasy Meetup. The imps continue to avoid shaking hands among themselves. However, the sprites only shake hands with each other and with the gremlins, but not with the imps. Each pair of creatures shakes hands only once. How many handshakes occurred at this event?


A square is divided into four congruent strips, as shown. If the perimeter of each of these four strips is 40 inches, what is the perimeter of the square, in inches?
[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((0,1)--(4,1));
draw((0,2)--(4,2));
draw((0,3)--(4,3));
[/asy]


Find the remainder when \(x^4\) is divided by \(x^2 + 4x + 1\).

Solution: \boxed{-56x - 15}


The sides of a right triangle measure 8, 15, and 17. Determine the distance between the centers of the triangle's inscribed circle and the circumscribed circle.


Choose two different prime numbers between $20$ and $30$. Calculate their product, the sum, and the sum of their squares. What number can be obtained by the following expression: $$xy - (x + y) - (x^2 + y^2)?$$


A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 616, and the sum of the lengths of its 12 edges is 160. What is $s$?


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


What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $76$?


What is the area, in square units, of a triangle with vertices at $A(2, 3), B(7, 3), C(4, 9)$?

Solution: \boxed{15 \text{ square units}}


Let \[f(x) = \left\{
\begin{array}{cl}
-x + 3 & \text{if } x \le 0, \\
2x - 5 & \text{if } x > 0.
\end{array}
\right.\]How many solutions does the equation $f(f(x)) = 6$ have?


Let $Q(x)$ be a polynomial such that when $Q(x)$ is divided by $x-10$, the remainder is $20$, and when $Q(x)$ is divided by $x-7$, the remainder is $10$. What is the remainder when $Q(x)$ is divided by $(x-10)(x-7)$?

Solution: \boxed{\frac{10}{3}x - \frac{40}{3}}


A frustum of a right circular cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate both the lateral surface area and the volume of the frustum.


If an integer is divisible by 6 and the sum of its last two digits is 15, then what is the product of its last two digits?


The sum of the first 1010 terms of a geometric sequence is 300. The sum of the first 2020 terms is 540. Find the sum of the first 3030 terms.


We define a function $f(x)$ such that $f(13)=33$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and

$f(b)=3b+1$ if $b$ is odd

$f(b)=\frac{b}{2}$ if $b$ is even.

What is the smallest possible number of integers in the domain of $f$?


Determine the remainder when $1 + 3 + 3^2 + \cdots + 3^{1000}$ is divided by $500$.


A whole number is considered "11-heavy" if the remainder when the number is divided by 11 is greater than 7. What is the least four-digit "11-heavy" whole number?


Simplify $\frac{5^6 + 5^3}{5^5 - 5^2}$. Express your answer as a common fraction.


Alice and Bob 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 $x$. The product of their polynomials is \[x^{10} + 4x^9 + 6x^8 + 4x^7 + 5x^6 + 10x^5 + 5x^4 + 6x^3 + 4x^2 + 4x + 9.\] What is the constant term of Bob's polynomial?


John scored 84, 88, and 95 on his first three biology exams. If he scores 92 on his fourth exam, how much will his average score change?


What is the greatest integer less than 200 for which the greatest common divisor of that integer and 18 is 6?


If the odds for pulling a prize out of the box are $5:6$, what is the probability of not pulling out the prize? Express your answer as a common fraction.


Thirteen girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Dana, starts with the ball, skips the next four girls, and throws to the sixth girl, who then skips the next four girls and throws the ball following the same pattern. If the throwing pattern continues, including Dana's initial throw, how many total throws are necessary for the ball to return to Dana?


Find the number of real solutions of the equation
\[
\frac{x}{50} = \sin x.
\]


The equation $x^2 - 2x = i$ has two complex solutions. Determine the product of their real parts.

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


A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Calculate the length of the longer side of the rectangle, in centimeters. Express your answer in terms of $\pi$.

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


Find the greatest common divisor of 108 and 450.


How many digits are located to the right of the decimal point when $\frac{5^7}{10^5 \cdot 15625}$ is expressed as a decimal?


What is the base $2$ representation of $96_{10}$?


Find the matrix $\mathbf{N}$ such that
\[\mathbf{N} \mathbf{v} = \begin{pmatrix} 7 \\ 2 \\ -8 \end{pmatrix} \times \mathbf{v}\] for all vectors $\mathbf{v}.$

Solution: \boxed{\begin{pmatrix} 0 & 8 & 2 \\ -8 & 0 & -7 \\ -2 & 7 & 0 \end{pmatrix}}


Let \(x\) and \(y\) be two distinct positive real numbers, and \(n\) a positive integer. We define sequences \((P_n), (G_n)\) as follows. First, \(P_1\) is \( \sqrt[n]{\frac{x^n + y^n}{2}} \) and \( G_1 \) is \( \sqrt{x \cdot y} \). Then for \( n \ge 2 \), \( P_n \) and \( G_n \) are:
\[ P_{n+1} = \sqrt[n]{\frac{(\sin P_n)^n + (\sin G_n)^n}{2}} \text{ and } G_{n+1} = \sqrt{\sin P_n \cdot \sin G_n} \] respectively.

Determine if the sequences \( (P_n) \) and \( (G_n) \) converge and identify their limits if they exist.


I have six apples and eight oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.)


Convert $\rm{BAC}_{16}$ to a base 10 integer.


What is the sum of all the two-digit primes that are greater than 20 but less than 90 and are still prime when their two digits are interchanged?


Compute $\sqrt{75} \cdot \sqrt{45} \cdot \sqrt{20}$.


Find the greatest possible value of $k$ if the roots of the equation $x^2 + kx + 8 = 0$ differ by $\sqrt{85}$.


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

Solution: \boxed{\frac{5 - \sqrt{33}}{2}}


In a right triangle $ABC$ with $\angle A = 90^\circ$, the lengths of $AB = 8$ and $BC = 10$. Find $\sin C$.


If the three lines $4y-3x=2$, $x+3y=3$ and $8x-12y=9$ are drawn in the plane, how many points will lie at the intersection of at least two of the three lines?


What is the base-10 integer 729 when expressed in base 7?


The arithmetic mean of 15 scores is 75. When the highest and lowest scores are removed, the new mean becomes 77. If the highest of the 15 scores is 98, what is the lowest score?


In triangle $ABC$, $AB = 6$, $BC = 8$, and $CA = 10$.

Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to $B$ than it is to either $A$ or $C$?


How many positive integers less than 500 are congruent to 5 (mod 13)?


What is the least positive multiple of 36 for which the product of its digits is also a positive multiple of 9?


Consider that there exists a real number $m$ such that the equation
\[\begin{pmatrix} 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 6 \\ -2 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} + s \begin{pmatrix} -3 \\ m \end{pmatrix}\]
does not have any solutions in $t$ and $s$. Find $m$.


My school's math club has 7 boys and 10 girls. We need to select a team of 6 people to send to the state math competition, ensuring that the team includes at least one previously competing boy and girl. Out of these, 1 boy and 1 girl have competition experience. In how many ways can I select the team to have 3 boys and 3 girls including the experienced members?


A "super ball" is dropped from a window 25 meters above the ground. On each bounce, it rises to 80% of the height of the preceding bounce. How far does the ball travel until it reaches the high point after hitting the ground for the fourth time?

Solution: \boxed{132.84 \text{ meters}}


Suppose $(v_n)$ is a sequence of real numbers satisfying
\[ v_{n+2} = 2v_{n+1} + v_{n} \]
with $v_4 = 15$ and $v_7 = 255$. Find $v_5$.


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


Tom’s graduating class has 300 students. At the graduation ceremony, the students will sit in rows with the same number of students in each row. If there must be at least 12 rows and at least 18 students in each row, what is the sum of all possible values of $x$ where $x$ is the number of students in each row?


If I choose four cards from a standard $52$-card deck, without replacement, what is the probability that I will end up with one card from each suit, in a sequential order (e.g., clubs, diamonds, hearts, spades)?


A portion of the graph of $y = H(x)$ is shown below. The distance between grid lines is $1$ unit. Compute $H(H(H(H(H(2)))))$.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}
};
rr_cartesian_axes(-5,5,-5,5);
real h(real x) {return ((x-2)*(x-2)/4 - 2);}
draw(graph(h,2-sqrt(2*8),2+sqrt(2*8),operator ..), red);
[/asy]


Let $b_1, b_2, b_3,\dots$ be an increasing arithmetic sequence of integers. If $b_3b_4 = 72$, what is $b_2b_5$?


Joann consumed a total of 154 lollipops over seven days. Each day after the first, she ate four more lollipops than she had the previous day. How many lollipops did she eat on the fifth day?


Suppose that $p$ is prime and $1014_p + 307_p + 114_p + 126_p + 7_p = 143_p + 272_p + 361_p$. How many possible values of $p$ are there?


Solve for $t$: $5 \cdot 5^t + \sqrt{25 \cdot 25^t} = 50$.


What is half of the absolute value of the difference of the squares of 15 and 12?


I have four distinct mystery novels, four distinct fantasy novels, four distinct biographies, and four distinct science fiction novels. I'm planning a long trip and want to take two books from different genres. How many possible pairs can I choose?


An integer greater than 9 and less than 100, which is divisible by 7, is randomly chosen. What is the probability that its digits are different?


Determine the number of digits in the value of $2^{15} \times 5^{10} \times 3$.


For how many digits $C$ is the positive four-digit number $1C35$ a multiple of 5?


A round cake has a diameter of $12$ cm and is cut into six equal-sized sector-shaped pieces. Calculate the square of the length ($l^2$) of the longest line segment that may be drawn in one of these pieces.

Solution: \boxed{36 \text{ cm}^2}


When you simplify $\sqrt[3]{40a^6b^7c^{14}}$, what is the sum of the exponents of the variables that are outside the radical?


Evaluate $\log_4 128\sqrt2$. Express your answer as an improper fraction.


What is $\log_{5}{3120}$ rounded to the nearest integer?


A collection of five positive integers has a mean of 4.6, unique mode 5, and median 5. If a 10 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth.


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


Points $P$ and $Q$ are midpoints of two adjacent sides of a rectangle with length $l$ and width $w$. What fraction of the interior of the rectangle is shaded? Consider the triangle formed by the vertex at the intersection of uncut sides and points $P$ and $Q$ to be unshaded.

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


Let $n$ be the number of ordered quadruples $(x_1, x_2, x_3, x_4)$ of positive even integers that satisfy $\sum_{i=1}^4 x_i = 104$. Find $\frac{n}{100}$.


I have fifteen books, of which six are written by one author, and nine by another. How many different pairs can I choose such that each pair contains books by different authors?


Jenny considers the quadratic equation $x^2 - sx + p$ with roots $r_1$ and $r_2$. She finds out that all power sums of the roots up to 2023 equate to $s$, i.e., $r_1 + r_2 = r_1^2 + r_2^2 = \cdots = r_1^{2023} + r_2^{2023} = s$. Jenny is curious to find out the maximum possible value of $\frac1{r_1^{2024}}+\frac1{r_2^{2024}}$.


Line \( l_1 \) has equation \( 4x - 3y = 2 \) and passes through \( A = (-2, -3) \). Line \( l_2 \) has equation \( y = 2 \) and intersects line \( l_1 \) at point \( B \). Line \( l_3 \) has a positive slope, goes through point \( A \), and intersects \( l_2 \) at point \( C \). The area of \( \triangle ABC \) is \( 6 \). Find the slope of \( l_3 \).


Given that $x$ is a multiple of $32515$, what is the greatest common divisor of $g(x) = (3x+5)(5x+3)(11x+7)(x+17)$ and $x$?


During her summer break, Julie works for 48 hours per week for 10 weeks, earning a total of $\$5000$. For the school year, which lasts 40 weeks, she hopes to earn a total of $6000$. At the same rate of pay, how many hours per week must she work during the school year?


What is the smallest integer value of $x$ for which $7 + 3x < 26$?


How many ways are there to put 5 distinguishable balls into 3 distinguishable boxes, provided that one of the boxes, Box C, must contain at least 2 balls?


A square and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$? Express your answer as a common fraction.


When four standard dice are tossed, the numbers $a, b, c, d$ are obtained. Find the probability that $abcd = 2$.


If the product $(4x^2 - 3x + 6)(9 - 3x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $8a + 4b + 2c + d$ evaluated at $x=3$.


In triangle $ABC$, $AB = 8$, $AC = 17$, and $BC = 15$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. The centroid divides the median into the ratio $2:1$ with the longer segment being from the vertex to the centroid. Let $P$ be the foot of the altitude from $G$ to side $BC$. Find the length of segment $GP$.

Solution: \boxed{GP = \frac{8}{3}}


What is the 52nd number in the row of Pascal's triangle that contains 55 numbers?


How many three-digit whole numbers have no 5's, 7's, 8's, or 9's as digits?


Determine how many positive integral values of $b$ ensure that $x = 3$ is the only positive integer solution of the system of inequalities:
$$
\begin{cases}
3x > 4x - 4 \\
4x - b > -8
\end{cases}
$$


In rectangle $ABCD,$ $AB=15$ and $AC=17.$ What is the area of rectangle $ABCD?$ Additionally, find the length of the diagonal $BD.$


What is the base ten equivalent of $54321_7$?


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


In a right triangle, the two legs have lengths of 48 inches and 55 inches. Calculate both the area of the triangle and its hypotenuse.

Solution: \boxed{73 \text{ inches}}


A school has eight identical copies of a specific textbook. At any given time, some of these copies are in the school's storage and some are distributed to students. How many different ways are there for some of the books to be in storage and the rest to be distributed to students if at least one book is in storage and at least one is distributed?


A 4x4x4 cube is assembled from 64 unit cubes. Only one unit square on each of the six faces of the cube is painted. How many of the 64 unit cubes have no paint on them?


Determine the number of complex numbers \( z \) that satisfy \( |z| = 1 \) and
\[
\left| \frac{z^2}{\overline{z}^2} + \frac{\overline{z}^2}{z^2} \right| = 1.
\]


We are allowed to remove exactly one integer from the list $$-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,$$ and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12?


Let $a$, $b$, $c$ be integers, and let $\omega$ be a complex number such that $\omega^4 = 1$ and $\omega \neq 1$. Also, let $a = 2b - c$. Find the smallest possible value of
\[
|a + b\omega + c\omega^3|.
\]


Compute the number of degrees in the smallest positive angle $y$ such that
\[6 \sin y \cos^3 y - 6 \sin^3 y \cos y = \frac{3}{2}.\]


A line is parameterized by a parameter $t,$ so that the vector on the line at $t = 1$ is $\begin{pmatrix} 4 \\ 5 \end{pmatrix},$ and the vector on the line at $t = 5$ is $\begin{pmatrix} 12 \\ -11 \end{pmatrix}.$ Find the vector on the line at $t = -3.$

Solution: \boxed{\begin{pmatrix} -4 \\ 21 \end{pmatrix}}


Given vectors $\mathbf{a}$ and $\mathbf{b},$ let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\] Among all such vectors $\mathbf{p},$ there exists constants $t$ and $u$ such that $\mathbf{p}$ is at a fixed distance from $t \mathbf{a} + u \mathbf{b}.$  Enter the ordered pair $(t,u).$

Solution: \boxed{\left(\frac{9}{8}, -\frac{1}{8}\right)}


Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be vectors such that \(\|\mathbf{a}\| = 3\), \(\|\mathbf{b}\| = 2\), \(\|\mathbf{c}\| = 5\), and
\[
\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}.
\]
Find the smallest possible angle between \(\mathbf{a}\) and \(\mathbf{c}\) in degrees.


Find the coefficient of $x$ when the expression $5(2x - 5) + 3(6 - 3x^2 + 4x) - 7(3x - 2)$ is simplified.


There is an unlimited supply of congruent equilateral triangles made of colored paper.  Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown.  Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are eight different colors of triangles from which to choose, how many distinguishable large equilateral triangles can be constructed?


What is the smallest positive integer $n$ such that $\frac{n}{n+53}$ is equal to a terminating decimal?


The equation $x^2 - 6x + 11 = 23$ has two solutions, $c$ and $d$, where $c \geq d$. Determine the value of $3c + 2d$.

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


What is the sum of the digits of the base $8$ representation of $888_{10}$?


A pet store has 20 puppies, 10 kittens, 12 hamsters, and 5 birds. Alice, Bob, Charlie, and Dana each want to buy a pet. For variety, they each want a different kind of pet. How many ways can Alice, Bob, Charlie, and Dana buy pets and leave the store satisfied?


Simplify $\sqrt{50y^3} \cdot \sqrt{18y} \cdot \sqrt{98y^5}$.


Points $C$ and $D$ are on the parabola $y = -3x^2 + 2x + 5$, and the origin is the midpoint of $\overline{CD}$. Find the square of the length $\overline{CD}$.


The line \( y = 4x - 7 \) is parameterized by the form
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ -3 \end{pmatrix} + t \begin{pmatrix} 3 \\ m \end{pmatrix}.
\]
Determine the ordered pair \( (s, m) \).

Solution: \boxed{(1, 12)}


Let $Q(x)$ be a polynomial such that when $Q(x)$ is divided by $x-17$, the remainder is $15$, and when $Q(x)$ is divided by $x-13$, the remainder is $8$. What is the remainder when $Q(x)$ is divided by $(x-17)(x-13)$?

Solution: \boxed{\frac{7}{4}x - \frac{59}{4}}


A ball is dropped from 20 feet, bouncing back three-quarters of the distance it just fell each time. Determine the number of bounces needed for the ball to first reach a height less than 2 feet.


The positive five-digit integers that use each of the digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\text{th}}$ integer in the list?


Let $P$ be the parabola with equation $y = x^2 + 3x + 1$ and let $Q = (10, 50)$. Determine the values of $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s$?


A piece of string fits exactly around the perimeter of a rectangle whose area is 180. The length of one side of the rectangle is three times the length of the other side. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string?


Solution: \boxed{4i - 2}


Of the numbers 1, 2, 3, ..., 20, which number less than 20 has the greatest number of divisors and is also a composite number?


At CleverCat Academy, cats are trained to perform three tricks: jump, fetch, and spin. Here is the available data:
\begin{tabular}{l@{\qquad}l}
60 cats can jump & 25 cats can jump and fetch \\
35 cats can fetch & 18 cats can fetch and spin \\
40 cats can spin & 20 cats can jump and spin \\
12 cats can do all three & 15 cats can do none
\end{tabular}
How many cats are in the academy?


How many positive three-digit integers with each digit greater than 6 are divisible by 12?


Let \( f(x) = x^3 - x^2 + 2x + 2000 \). What is the greatest common divisor of \( f(50) \) and \( f(51) \)?


A square with sides 8 inches is illustrated. If $Q$ is a point such that the segments $\overline{QA}$, $\overline{QB}$, and $\overline{QC}$ are equal in length, and segment $\overline{QC}$ is perpendicular to segment $\overline{FD}$, find the area, in square inches, of triangle $AQB$. [asy]
pair A, B, C, D, F, Q;
A = (0,0); B= (8,0); C = (4,2); D = (8,8); F = (0,8); Q = (4,4);
draw(A--B--D--F--cycle);
draw(C--Q); draw(Q--A); draw(Q--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$Q$",Q,NW);label("$F$",F,NW);
label("$8''$",(4,0),S);
[/asy]


Let \( x = 1 + \frac{\sqrt{3}}{1 + \frac{\sqrt{3}}{1 + \dots}} \). Find the value of \( \frac{1}{(x+2)(x-3)} \). When your answer is in the form \( \frac{A+\sqrt{B}}{C} \), where \( A \), \( B \), and \( C \) are integers, and \( B \) is not divisible by the square of a prime, what is \( |A|+|B|+|C| \)?


How many positive divisors of $180$ are not divisible by 2?


In triangle $ABC$, $\angle C = 90^\circ$, $AB = 13$, and $BC = 5$. What is $\cos B$?

Solution: \boxed{\cos B = \frac{5}{13}}


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


The lattice shown is continued for $12$ rows, each containing $7$ consecutive numbers. The first number in each row starts from $1$ in Row 1, and increases by $8$ as the row number increases. What will be the fifth number in the $12$th row?


What is the domain of the function $g(x) = \log_3(\log_4(\log_5(\log_6x)))$?

Solution: \boxed{(7776, \infty)}


For some constants $a$ and $b,$ let \[g(x) = \left\{
\begin{array}{cl}
ax + b & \text{if } x < 3, \\
9 - 2x & \text{if } x \ge 3.
\end{array}
\right.\]
The function $g$ has the property that $g(g(x)) = x$ for all $x.$ Additionally, $g(x)$ is continuous at $x = 3$. What is $a + b?$


In the NBA finals between the Warriors and the Nets, a team is declared the champion after winning 4 games. If the Warriors win each game with a probability of $\frac{1}{4}$ and there are no ties, what is the probability that the Nets will win the NBA finals but that the series will need all seven games to be decided? Express your answer as a fraction.


Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.)


The deli has five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich includes one type of bread, one type of meat, and one type of cheese. Turkey, roast beef, Swiss cheese, and rye bread are each available at the deli. If Al never orders a sandwich with a turkey/Swiss cheese combination nor a sandwich with a rye bread/roast beef combination, how many different sandwiches could Al order?


A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam observes the light for a random five-second interval. What is the probability that the light changes from one color to another during his observation?


Let $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$. Find the vector $\mathbf{v}$ that satisfies $2\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $3\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}$.

Solution: \boxed{\begin{pmatrix} \frac{7}{6} \\ -\frac{2}{3} \\ \frac{7}{6} \end{pmatrix}}


How many positive divisors does \( 8! \) have?


The marble statue of George Washington in a museum has a height of 120 feet. A scale model of the statue has a height of 6 inches. How many feet of the statue does one inch of the model represent?


In how many ways is it possible to arrange the digits of 11120 to get a four-digit multiple of 5?


How many positive integers smaller than $1{,}000{,}000$ are powers of $3$, but are not powers of $9$?


In parallelogram $ABCD$, angle $B$ measures $135^\circ$. What is the number of degrees in the measure of angles $C$ and $A$?


What is the $156$th digit after the decimal point when $\frac{37}{740}$ is expressed as a decimal?


The sequence 2,000,000; 1,000,000; 500,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?


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


A jar contains $35\frac{2}{3}$ tablespoons of peanut butter and $18\frac{1}{2}$ tablespoons of jelly. If one serving of peanut butter is $2\frac{1}{6}$ tablespoons and one serving of jelly is 1 tablespoon, how many servings of each does the jar contain? Express your answer as mixed numbers.


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


If $7a+3b=0$ and $a$ is three less than $b$, what is $8b$?


A student, Jim, needs to earn 30 biology exercise points. For the first 6 points, each point requires 1 exercise. For the next 6 points, each point needs 2 exercises. After this, for each subsequent group of 6 points, the number of exercises required per point increases by 1. How many exercises in total must Jim complete to earn all 30 points?


What is the 12th term in the geometric sequence starting with 5 and each subsequent term being multiplied by $\frac{2}{5}$?


What is the smallest positive integer $n$ such that all the roots of $z^6 - z^3 + 1 = 0$ are $n^{\text{th}}$ roots of unity?


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


A quadratic is given as $x^2+bx+\frac16$, where $b$ is a negative number. It can be rewritten using the completion of the square as $(x+p)^2+\frac{1}{18}$. Determine the value of $b$.


Find the smallest positive integer \(b\) for which \(x^2 + bx + 4032\) factors into a product of two binomials, each having integer coefficients.


How many distinct, positive factors does $1320$ have?


The quadratic equation $ax^2 + 30x + c = 0$ has exactly one solution. If $a+c=35$, and $a < c$, find the ordered pair $(a, c)$.

Solution: \boxed{\left(\frac{35 - 5\sqrt{13}}{2}, \frac{35 + 5\sqrt{13}}{2}\right)}


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


Peter borrows $2000$ dollars from John, who charges an interest of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Peter will owe more than three times as much as he borrowed?


Let $p, q, r, s, t, u, v,$ and $w$ be distinct elements in the set
\[
\{-6, -4, -1, 0, 3, 5, 7, 10\}.
\]
What is the minimum possible value of
\[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]


Yan is at a location between his home and the market. He can walk directly to the market, or he can walk back home and then ride his bicycle to the market. Yan rides 9 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the market?


Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere shifted by $(s,t,u)$ that passes through $A,$ $B,$ $C,$ and $O.$ If $(s,t,u) = (1,1,1)$ find:
\[\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.\]


Jeff continues to play with the Magic 8 Ball, this time asking it 7 questions. Each time he asks a question, the probability of receiving a positive answer is still 2/5. What is the probability that he receives exactly 3 positive answers?


Completely factor the following expression: \[(16y^6 + 36y^4 - 9) - (4y^6 - 9y^4 + 9).\]

Solution: \boxed{3(4y^6 + 15y^4 - 6)}


When the base-8 number $12345_8$ is written in base 10, what is the decimal equivalent?


A cube with an edge length of 5 units has the same volume as a square-based pyramid with base edge lengths of 10 units and a height of $h$ units. What is the value of $h$?


Many states have started using a sequence of three letters followed by a sequence of four digits as their standard license-plate pattern. Calculate the probability that such a license plate will contain at least one palindrome (either the three-letter arrangement reads the same left-to-right and right-to-left, or the four-digit arrangement reads the same left-to-right and right-to-left). Express your answer as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m+n$.


For how many digits $C$ is the positive four-digit number $1C34$ a multiple of 4?


What is the value of the expression $(25 + 9)^2 - (25^2 + 9^2)$?


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


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


The letters of the alphabet are each assigned a random integer value, and $T=15$. The value of a word comes from the sum of its letters' values. If $BALL$ is 40 points, $LAB$ is 25 points, and $ALL$ is 30 points, what is the value of $B$?


What is the measure, in degrees, of the angle formed by the hour hand and the minute hand of a 12-hour clock at 3:25?


Let $a$ and $b$ be nonzero real numbers such that $a^2 + b^2 = 10ab$. Find the value of $\left|\frac{a+b}{a-b}\right|$.


In a triangle, two of the side lengths are 10 and 12, and the angle between them is $150^\circ$. Find the length of the third side.

Solution: \boxed{c = \sqrt{244 + 120\sqrt{3}}}


The coefficients of the polynomial
\[a_{12} x^{12} + a_{11} x^{11} + \dots + a_1 x + a_0 = 0\]
are all integers, and its roots $r_1, r_2, \dots, r_{12}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{12} + a_1 x^{11} + \dots + a_{11} x + a_{12} = 0\]
are also $r_1, r_2, \dots, r_{12}$. Find the number of possible multisets $S = \{r_1, r_2, \dots, r_{12}\}$.


One of the asymptotes of a hyperbola has equation $y = -2x + 4$. The foci of the hyperbola have the same $x$-coordinate, which is $-3$. Find the equation of the other asymptote of the hyperbola, giving your answer in the form "$y = mx + b$".

Solution: \boxed{y = 2x + 16}


Calculate the product of $0.\overline{4567}$ and $11$, and express your result as a fraction in simplified form.


Susan wants to determine the average and median number of candies in a carton. She buys 9 cartons of candies, opens them, and counts the number of candies in each one. She finds that the cartons contain 5, 7, 8, 10, 12, 14, 16, 18, and 20 candies. What are the average and median number of candies per carton?


The positive difference between the two roots of the quadratic equation $5x^2 - 2x - 15 = 0$ can be written as $\frac{\sqrt{p}}{q}$, where $q$ is an integer and $p$ is an integer not divisible by the square of any prime number. Find $p + q$.


Find the distance from the point $(2,3,1)$ to the line described by:
\[
\begin{pmatrix} 8 \\ 10 \\ 12 \end{pmatrix} + s \begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix}.
\]


Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 4 \sqrt{3}.$ Find $|\alpha|.$


Consider the following table showing the salaries of employees at a particular company:

\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO & 1 & \$150{,}000 \\\hline
General Manager & 3 & \$100{,}000 \\\hline
Manager & 12 & \$80{,}000 \\\hline
Supervisor & 8 & \$55{,}000 \\\hline
Worker & 35 & \$30{,}000 \\\hline
\end{tabular}

According to the table, how many dollars are in the median value of the 59 salaries paid to this company's employees?


In triangle $ABC,$ $AB = 4,$ $AC = 7,$ $BC = 9,$ and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Find $\cos \angle BAD.$


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


A \(90^\circ\) rotation around the origin in the clockwise direction is applied to \(4 + 2i\). What is the resulting complex number?

Solution: \boxed{2 - 4i}


Compute \[\left\lfloor \dfrac{101^3}{99 \cdot 100} - \dfrac{99^3}{100 \cdot 101} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


Sixty cards are placed into a box, each bearing a number 1 through 15, with each number represented on four cards. Four cards are drawn from the box at random without replacement. Let \(p\) be the probability that all four cards bear the same number. Let \(q\) be the probability that three of the cards bear a number \(a\) and the other bears a number \(b\) that is not equal to \(a\). What is the value of \(q/p\)?


Suppose that $a,b,$ and $c$ are positive integers satisfying $(a+b+c)^3 - a^3 - b^3 - c^3 = 270$. Find $a+b+c$.


Let \(S\) be the set of all nonzero real numbers. Define a function \(f : S \to S\) such that for all \(x, y \in S\) with \(x + y = 1\),
\[ f(x) + f(y) = f(xyf(x + y)) \]
Determine the number of possible values of \(f(3)\) and the sum of all possible values of \(f(3)\). Calculate \(n \times s\) where \(n\) is the number of possible values and \(s\) is the sum of those values.


Evaluate the polynomial \[ p(x) = x^4 - 3x^3 - 9x^2 + 27x - 8, \] where $x$ is a positive number such that $x^2 - 3x - 9 = 0$.

Solution: \boxed{\frac{65 + 81\sqrt{5}}{2}}


If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 10? Express your answer as a common fraction.


Determine how many positive odd integers greater than 1 and less than $200$ are square-free.


Suppose there are 6 red plates, 5 blue plates, and 3 green plates. If I randomly select two plates to serve dinner on, what is the probability that they are both the same color?


A three-row triangle is constructed using unit rods and connectors. The first row uses 3 rods and 4 connectors. Each subsequent row has 3 more rods and one additional connector than the previous row. Determine how many pieces would be needed to create a ten-row triangle.


The sequence $8820, 2940, 980, \ldots$ is made by repeatedly dividing by 3. How many integers are in this sequence?


Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{AB}$ at $P$ and line segment $\overline{AC}$ at $Q$, forming $\triangle APQ$ with area 18. Compute $s$.

Solution: \boxed{10 - 2\sqrt{15}}


Find the least common multiple and the greatest common divisor of 1365 and 910.


Factor the following expression: $75x^{19} + 165x^{38}$.

Solution: \boxed{15x^{19}(5 + 11x^{19})}


All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the tenth number in that list?


A $270^\circ$ rotation around the origin in the counter-clockwise direction is applied to $-4 + 2i.$ Followed by a scaling of the result by a factor of 2. What is the resulting complex number?

Solution: \boxed{4 + 8i}


$ABCD$ is a trapezoid where the measure of base $\overline{AB}$ is three times the measure of the base $\overline{CD}$. Additionally, the perimeter of trapezoid $ABCD$ is 36 units. Point $E$ is the intersection of the diagonals. If the length of diagonal $\overline{AC}$ is 18 units, determine the length of segment $\overline{EC}$.


The area of the parallelogram generated by the vectors $\mathbf{u}$ and $\mathbf{v}$ is 12. Find the area of the parallelogram generated by the vectors $3\mathbf{u} - 2\mathbf{v}$ and $4\mathbf{u} + 5\mathbf{v}$.


Determine the remainder when \(1 + 5 + 5^2 + \cdots + 5^{1002}\) is divided by \(500\).


Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.001, 0.01).$


In right triangle $ABC$, $\angle B = \angle C$ and $AB = 10$. What is the area of $\triangle ABC$?


Given the points $(7, -9)$ and $(1, 7)$ as the endpoints of a diameter of a circle, calculate the sum of the coordinates of the center of the circle, and also determine the radius of the circle.


Find the number of positive integers $n \le 2000$ such that $10n$ is a perfect square.


Jane and her brother each spin this spinner once. The spinner has six congruent sectors, numbered 1 through 6. If the non-negative difference of their numbers is less than or equal to 3, Jane wins; otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.


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


Carol hypothesizes that the length of her exercise session the night before a test and her score on the test are inversely related. On a previous test, after exercising for 45 minutes, she scored 80 points. Carol aims for an average score of 85 over two tests. To the nearest tenth, how long should Carol exercise the night before her next test so that her hypothesis remains valid and she achieves her score goal?


A basketball player made the following number of successful free throws in 10 successive games: 8, 17, 15, 22, 14, 12, 24, 10, 20, and 16. He attempted 10, 20, 18, 25, 16, 15, 27, 12, 22, and 19 free throws in those respective games. Calculate both the median number of successful free throws and the player's best free-throw shooting percentage game.


Let $A$ equal the number of four-digit odd numbers. Let $B$ equal the number of four-digit multiples of 3. Find $A+B$.


How many 4-letter words can we make from the letters A, B, C, D, and E, if we are allowed to repeat letters, and we must use each of the letters A and E at least once?


In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written?


Let $g(n)$ be a function that, given an integer $n$, returns an integer $m$, where $m$ is the smallest possible integer such that $m!$ is divisible by $n$. Given that $n$ is a multiple of 21, what is the smallest value of $n$ such that $g(n) > 21$?

Solution: \boxed{n = 483}


What is the least positive whole number divisible by four different prime numbers, none of which is smaller than 5?


What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 8?


For all integers $n$ greater than 1, define $a_n = \dfrac{1}{\log_n 5000}$. Let $b = a_2 + a_3 + a_5$ and $c = a_{8} + a_{9} + a_{10} + a_{12}$. Find $b - c.$

Solution: \boxed{-\log_{5000} 288}


In right triangle $DEF$ where $\angle D = 90^\circ$, and side lengths are $DE = 9$ and $EF = 40$. Find $\cos D$.


How many positive integers smaller than $500{,}000$ are powers of $3$, but are not powers of $27$? It may help to know that $3^6 = 729$.


If $p$, $q$, and $r$ are positive integers satisfying $pq+r = qr+p = rp+q = 47$, what is the value of $p+q+r$?


A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$?


What is the remainder when $2011 \cdot 1537$ is divided by $450$?


The diameter, in inches, of a sphere with three times the volume of a sphere of radius 6 inches can be expressed in the form $c\sqrt[3]{d}$ where $c$ and $d$ are positive integers and $d$ contains no perfect cube factors. Compute $c+d$.


Let $\mathbf{c} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$ and $\mathbf{d} = \begin{pmatrix} 6 \\ 1 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{c},$ and $\mathbf{d}.$


Let $\triangle XYZ$ be a right triangle with $Y$ as the right angle. A circle with diameter $YZ$ intersects side $XZ$ at $W$. If $XW = 3$ and $YW = 9$, find the length of $WZ$.


The equation $x^2 - 2x = i$ has two complex solutions. Determine the product of their real parts.

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


A rectangle is termed "great" if the number of square units in its area equals three times the number of units in its perimeter, and it has integer side lengths. Calculate the sum of all different possible areas of such great rectangles.


Suppose you have a cube with a side length of 2x. A regular octahedron is formed by joining the centers of adjoining faces of this larger cube. Calculate the ratio of the volume of this octahedron to the volume of the cube.


Suppose $\cos Q = 0.6$ in the right-angled triangle below. If the length of $QP$ is 18 units, find the length of $QR$.


How many three-digit whole numbers have no 5's, 7's, or 9's as digits?


How many zeros are in the expansion of $999,\!999,\!999,\!997^2$?


Consider the following expansion using the binomial theorem: $(1+0.1)^{2000}$. This can be expanded as:
\[\sum_{k=0}^{2000} {2000 \choose k}(0.1)^k = B_0 + B_1 + B_2 + \cdots + B_{2000},\]
where $B_k = {2000 \choose k}(0.1)^k$. Identify the value of $k$ for which $B_k$ is the largest.


In a right triangle with integer length sides, the hypotenuse has length 65 units. What is the length of the shorter leg?


Let \( r_1, r_2, \) and \( r_3 \) be the roots of the polynomial equation
\[ x^3 - 4x^2 + 5x + 12 = 0. \]
Find the monic polynomial, in \( x \), whose roots are \( 3r_1, 3r_2, \) and \( 3r_3 \).

Solution: \boxed{x^3 - 12x^2 + 45x + 324}


Compute \[ \left\lfloor \dfrac {2023^3}{2021 \cdot 2022} - \dfrac {2021^3}{2022 \cdot 2023} \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$


What is the measure of the smaller angle between the hands of a 12-hour clock at 3:40 pm, and also find the larger angle? Express your answer as a decimal to the nearest tenth of a degree.

Solution: \boxed{230.0 \text{ degrees}}


A triangle has vertices at coordinates (3,3), (7,7), and (8,3). What is the number of units in the length of the longest side of this triangle?


In how many ways can I arrange 4 different math books and 6 different history books on my bookshelf, if I require a math book on both ends and two specific history books must not be adjacent?


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


Consider the system of equations
\[
4x - 3y = a,
\]
\[
6y - 8x = b.
\]
Assuming \( b \neq 0 \), find the value of \(\frac{a}{b}\).


If we let $g(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $j$ exist such that $1 \le j \le 5041$ and $g(j) = 1 + \sqrt{j} + j$?


Points $C$ and $D$ are on a graph where both have the same $x$-coordinate of 7, but different $y$-coordinates. Point $C$ has a $y$-coordinate of 5 and point $D$ has a $y$-coordinate of -3. What is the value of the $x$-coordinate where this line intersects the $x$-axis?


Find \( h(x) \), with terms in order of decreasing degree, if
\[ 12x^4 + 5x^3 + h(x) = -3x^3 + 4x^2 - 7x + 2. \]

Solution: \boxed{-12x^4 - 8x^3 + 4x^2 - 7x + 2}


Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to her friend Rachel. Tracy and her mom then each ate 20 candies from what Tracy had left. Finally, Tracy’s brother took anywhere from two to six candies, leaving Tracy with ten candies. How many candies did Tracy have at the start?


If I roll a fair, regular ten-sided die six times, what is the probability that I will roll the number $1$ exactly four times?


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


A circular piece of paper with a radius of 5 inches has a section cut out, forming a \(225^\circ\) sector. This sector is then used to create a right circular cone by gluing points A and B together. Determine the circumference of the cone's base, expressed in terms of \(\pi\).


Jason borrowed money from his parents to buy a new surfboard. His parents have agreed to let him work off his debt by babysitting under the following conditions: his first hour of babysitting is worth $\$1$, the second hour worth $\$2$, the third hour $\$3$, the fourth hour $\$4$, the fifth hour $\$5$, the sixth hour $\$6$, the seventh hour $\$7$, the eighth hour $\$1$, etc. If he repays his debt by babysitting for 39 hours, how many dollars did he borrow?


Carlos and Nina play a game where Carlos picks an integer between 1 and 4500 inclusive. Nina divides 4500 by that integer and checks if the result is an integer. How many integers can Carlos pick so that the quotient Nina receives is an integer? Additionally, Carlos should only pick numbers that are divisible by 3.


There are 60 chips in a box. Each chip is either small or large. If the number of small chips is greater than the number of large chips by twice a prime number, what is the greatest possible number of large chips?


In the increasing sequence of positive integers $a_1,$ $a_2,$ $a_3,$ $\dots,$ it's given that
\[a_{n + 2} = a_{n + 1} + a_n\] for all $n \ge 1$. If $a_7 = 210,$ find $a_9$.


Consider the cubic polynomial $p(x) = x^3 - (a^2 + b^2 + c^2)x^2 + (a^4 + b^4 + c^4)x - abc$ whose roots are real numbers, and the matrix
\[ \begin{pmatrix} a^2 & b & c \\ b^2 & c & a \\ c^2 & a & b \end{pmatrix} \]
is not invertible. List all possible values of
\[ \frac{a^2}{b^2 + c} + \frac{b^2}{a^2 + c} + \frac{c^2}{a^2 + b^2} \]


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


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

Solution: \boxed{a = -\frac{55}{3}}


Factor $x^2 + 6x + 9 - 64x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2 + bx + c)(dx^2 + ex + f)$, with $a < d$.

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


Let $P_1$ be a regular $r~\mbox{gon}$ and $P_2$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1$ is $\frac{61}{60}$ as large as each interior angle of $P_2$. What is the largest possible value of $s$?


Each of $a_1,$ $a_2,$ $\dots,$ $a_{50}$ is equal to $1$ or $-1.$  Find the minimum positive value of
\[\sum_{1 \le i < j \le 50} a_i a_j.\]


Let $r$ and $s$ be the two distinct solutions to the equation $$\frac{5x-15}{x^2+3x-18}=x+3.$$ If $r > s$, what is the value of $r - s$?


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


Suppose $a, b, c$ are positive numbers such that $abc = 1$, $a + \frac{1}{c} = 7$, and $b + \frac{1}{a} = 31$. Find $c + \frac{1}{b}$.


Find the maximum integer value of
\[
\frac{3x^2 + 9x + 21}{3x^2 + 9x + 7}
\]
where $x$ is a real number.


Let \(a_1, a_2, \dots\) be a sequence of positive real numbers such that
\[a_n = 15a_{n-1} - 2n\] for all \(n > 1\). Find the smallest possible value of \(a_1\).


Tom has a red marble, a green marble, a blue marble, a purple marble, and four identical yellow marbles. How many different groups of two marbles can Tom choose?


Find the greatest common divisor of $7429$ and $13356$.


If $n = 2^{12} \times 3^{15} \times 5^9$, how many of the natural-number factors of $n$ are multiples of 300?


Given \( w \) and \( z \) are complex numbers such that \( |w+z|=2 \) and \( |w^2+z^2|=8 \), find the smallest possible value of \( |w^3+z^3| \).


Find the number of complex numbers $z$ satisfying $|z| = 1$ and
\[\left| \frac{z^3}{\overline{z}^3} + \frac{\overline{z}^3}{z^3} \right| = 3.\]


A hexagon is formed by joining, in order, the points $(0,0)$, $(1,1)$, $(3,1)$, $(4,0)$, $(3,-1)$, $(1,-1)$, and back to $(0,0)$. The perimeter of the hexagon can be written in the form $a + b\sqrt{2} + c\sqrt{5}$, where $a$, $b$, and $c$ are whole numbers. Find $a+b+c$.


Select two distinct numbers simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smaller one divides the larger one and both numbers are either both even or both odd?


At a gathering, there are 8 married couples. Each person is seated in a circular arrangement and shakes hands with every other person except for his or her spouse and the person directly next to them (neighbors). How many handshakes occur?


When the graph of $y = 2x^2 - x + 5$ is shifted seven units to the right and then shifted three units up, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$.


Consider you are given $a > 1$. Determine all possible values of $a$ if $f(g(a)) = 12$, where $f(x) = x^2 + 2$ and $g(x) = x^2 + 2$.

Solution: \boxed{a = \sqrt{\sqrt{10} - 2}}


What is the modulo $7$ remainder of the sum $1+2+3+4+\ldots+140?$