An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is a $1$?

Note: In this context, the zeroth row is counted as the first, comprising only the number 1 at the top.


What is the value of $2^4 \times 3^2 \times 5^4 \times 7^2$?


Solution: \boxed{g(x) = -2x^5 + 4x^4 - 12x^3 + 2x^2 + 4x + 4}


Four dice with faces numbered 1 through 6 are stacked as shown. Eight of the twenty-four faces are visible, leaving sixteen faces hidden. What is the total number of dots NOT visible in this view? Assume the visible faces show the numbers 1, 2, 2, 3, 4, 5, 6, 6.


Let $F_n$ be the Fibonacci sequence, where $F_0 = 0$, $F_1 = 1$, and each subsequent term is the sum of the two preceding ones, $F_{n+2} = F_{n+1} + F_n$. Compute the sum:
\[
\sum_{n=0}^{\infty} \frac{F_n}{5^n}.
\]


The numbers $x, y, z, w$ are equal to 1, 5, 2, 4, in some order. Find the largest possible value of
\[xy - yz + zw - wx.\]


What is the greatest common divisor of 1755 and 1242?


How many ordered pairs of real numbers $(x, y)$ satisfy the following system of equations?
\[
\left\{
\begin{aligned}
x + 2y &= 5 \\
\left| |x| - 2|y| \right| &= 2
\end{aligned}
\right.
\]


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


How many of the divisors of $10!$ are larger than $9!$?


A circular piece of paper has a radius 15 cm. Sarah cuts and removes part of this circle as shown in a diagram, then forms a cone by joining two edges of the remaining sector. The cone has a radius of 10 cm and a volume of $500\pi$ cm³. What is the angle measure, in degrees, of the sector Sarah removed?


The graph of $y = f(x)$ is shown as follows, defined by pieces:

1. From $x = -3$ to $x = 0$: \( f(x) = -2 - x \)
2. From $x = 0$ to $x = 2$: \( f(x) = \sqrt{4 - (x - 2)^2} - 2 \)
3. From $x = 2$ to $x = 3$: \( f(x) = 2(x - 2) \)

For certain constants $a$, $b$, and $c$,
\[ g(x) = af(bx) + c. \]
The modified graph of $y = g(x)$ undergoes a horizontal stretch by a factor of 3, then a downward shift by 3 units. Determine the ordered triple $(a,b,c)$ that represents these transformations.

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


Solution: \boxed{2^{103} - 1}


Alice and Carol each have a rectangular sheet of paper. Alice has a sheet of paper measuring 10 inches by 12 inches and rolls it into a tube by taping the two 10-inch sides together. Carol rolls her sheet, which measures 8 inches by 15 inches, by taping the two 15-inch sides together. Calculate $\pi$ times the positive difference of the volumes of the two tubes.


Consider a right triangle with sides 8, 15, and 17. Find the length of the shortest altitude.


I had $\$50$ in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on video games?

[asy]
size(150);
pair A, B, C, D, E, O, W, X, Y, Z, V;
O=(0,0);
A=(.707,.707);
B=(-.259, .966);
C=(-.940, .342);
D=(-.707,-.707);
E=(.940,-.342);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
draw(O--E);
W=(.5,.3);
label("Movies", W, N);
label("$\frac{1}{4}$", W, S);
X=(-.1, .6);
label("Burgers", X, N);
label("$\frac{1}{5}$", X, S);
Y=(-.6,0);
label("Ice Cream", Y, N);
label("$\frac{1}{10}$", Y, S);
Z=(-.1,-.6);
label("Music", Z, N);
label("$\frac{2}{5}$", Z, S);
V=(.6,0);
label("Video Games", V, N);
[/asy]


Determine the digit $A$ such that the five-digit numeral $53A47$ is divisible by 11.


A cutting tool is designed in the shape of a notched circle. The radius of the circle is $\sqrt{80}$ cm, with points $A$, $B$, and $C$ on its boundary such that $AB = 8$ cm and $BC = 3$ cm, forming a right angle at point $B$. Compute the square of the distance from $B$ to the center of the circle.


If altitude $CE$ is $2\sqrt{2}$ centimeters in $\triangle ABC$, find the area of $\triangle ABC$.

[asy] import olympiad; pair A,B,C,E; A = (0,2*sqrt(2)); B = (2,0);
C = foot(A,B,-B); E = foot(C,A,B); draw(A--B--C--A); draw(C--E,dashed);
label("$45^{\circ}$",A-(0.2,0.3),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$E$",E,NE);
draw((0,.2)--(.2,.2)--(.2,0)); draw(E + .2*dir(225)--E + sqrt(2)*.2*dir(180)--E+.2*dir(135));
[/asy]


Louise is designing a custom dress and needs to provide her hip size in millimeters. If there are $12$ inches in a foot and $305$ millimeters in a foot, and Louise's hip size is $42$ inches, what size should she specify in millimeters?


How many prime numbers between 50 and 100 have a prime remainder when divided by 12?


How many ways are there to put 6 balls into 4 boxes if the balls are not distinguishable, the boxes are not distinguishable, and each box must contain at least one ball?


If $x^{3y} = 8$ and $x = 2$, what is the value of $y$? Express your answer as a common fraction.


In a survey of $150$ people, $120$ like tea, and $100$ like coffee. What is the minimum number of people surveyed who could have said they liked both tea and coffee?


Let $a$, $b$, $c$ be the sides of a triangle, with angles $\alpha$, $\beta$, $\gamma$ opposite them respectively. Given that $a^2 + b^2 + 32c^2 = 2021c^2$, calculate:
\[
\frac{\cot \gamma}{\cot \alpha + \cot \beta}.
\]


A number $x$ is equal to $7 \cdot 36 \cdot 54$. What is the smallest positive integer $y$ such that the product $xy$ has to be a perfect cube?


What is the least positive multiple of 15 for which the product of its digits is also a positive multiple of 15?


A $\textit{palindrome}$ is a number which reads the same forward as backward, like 121 or 2442. What is the smallest natural number that can be subtracted from 56,789 to create a palindrome?


Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=104$. What is $x$?


The integers $r$ and $k$ are randomly selected such that $-5 < r < 8$ and $1 < k < 8$ with $k$ being a prime number. What is the probability that $r \div k$ results in an integer?


Compute without using a calculator: $\dfrac{10!}{7!2!1!}$


Solution: \boxed{45x^3(1 - 3x^4)}


In the diagram, a small circle is inscribed inside a larger circle. The visible gray area within the larger circle is equal to five times the area of the white circular region. Determine the ratio of the radius of the small circle to the radius of the large circle. Express your answer as a common fraction.


For how many values of the digit $A$ is it true that $45$ is divisible by $A$ and $273{,}1A6$ is divisible by $8$?


A palindrome is a number that reads the same forward and backward. How many four-digit perfect squares are palindromes?


What is the value of $\dfrac{(5 \times 6) - 4}{8}$?


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.4\overline{67}$?


Circle $D$ has a radius of 10 cm. Compute the area in square centimeters of the largest possible inscribed triangle having one side as a diameter of circle $D$.


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


Mr. and Mrs. Lee have three children. When they travel in their family van, three people sit in the front row, and the other three sit in the back row. Either Mr. Lee or Mrs. Lee must sit in the driver's seat. How many seating arrangements are possible?


Line $l_1$ has the equation $2x + y = 4$ and goes through point $A = (2, 0)$. Line $l_2$ has the equation $y = 2$ and intersects $l_1$ at point $B$. Line $l_3$ has a positive slope, passes through $A$, and intersects $l_2$ at point $C$. The area of $\triangle ABC$ is $2$. Determine the slope of $l_3$.


Brand X soda now advertises, "We will give you 25% more soda than Brand Y for a total price that is 12% less than Brand Y's price!" What is the ratio of the unit price of Brand X soda to the unit price of Brand Y soda? Express your answer as a common fraction.


Calculate $180 \div \left( 12 + 9 \times 3 - 4 \right)$.


If $a \ge b \ge c > 1$, determine the maximum possible value of $\log_a \left(\frac{a}{bc}\right) + \log_b \left(\frac{b}{ca}\right) + \log_c \left(\frac{c}{ab}\right)$.


Jessica's grandmother's age is fifteen times Jessica's age. If Jessica's grandmother was 60 years old when Jessica was born, how many years old is Jessica now?


Given that three altitudes of triangle PQR have lengths 3 and 9, and the length of the third altitude is an integer, what is the smallest possible length of the third altitude?


Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Then Amal rolls a fair seven-sided die marked with numbers 1 through 7. What is the probability that the product of the two rolls is a multiple of 5?


Let $a$ and $b$ be real numbers. Consider the following five statements:
1. $a^2 + b^2 < (a+b)^2$
2. $ab > 0$
3. $a > b$
4. $a > 0$
5. $b > 0$

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


How many integers are common solutions to the following three inequalities?
\[
\begin{array}{cccc}
(1) & -5x & \geq & 2x+10 \\
(2) & -3x & \leq & 15 \\
(3) & -6x & \geq & 3x+21
\end{array}
\]


Suppose that $x$ and $y$ are positive integers such that $(x-yi)^2 = 15 - 20i$. What is $x-yi$?

Solution: \boxed{5 - 2i}


Expand and simplify the product $$(x^2-3x+3)(x^2+3x+3).$$

Solution: \boxed{x^4 - 3x^2 + 9}


Find the inverse of the function $h(x) = 6 - 7x$.

Solution: \boxed{\frac{6 - x}{7}}


When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers all of which are nonprime?


**Given**: Let $a$ and $b$ be positive integers such that \( a > b \). If the expression  \( E = \left| \frac{a+2b}{a-b} + \frac{a-b}{a+2b} \right| \), find the smallest possible value of \( E \).


Simplify $\frac{1}{1+\sqrt{3}} \cdot \frac{1}{1+\sqrt{3}}$.

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


It takes five painters working at the same rate 0.8 work-days to finish a job. If only four painters are available, how many work-days will it take them to finish the job, working at the same rate? Express your answer as a mixed number.


The measures of the angles of a hexagon are in the ratio 3:3:3:4:5:6. What is the number of degrees in the measure of the largest angle?


Laura constructs a cone for an art project. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Laura needs to find the smallest cube-shaped box to transport her cone safely to the art gallery. What is the volume of this box, in cubic inches?


Let \(m\) be the smallest integer whose cube root is of the form \(n+r\), where \(n\) is a positive integer and \(r\) is a positive real number less than \(1/500\). Find \(n\).


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


Mr. and Mrs. Lopez, along with their two children and two grandparents, are traveling in a minivan that has one driver's seat, three seats in the middle row, and two seats in the back row. Either Mr. Lopez, Mrs. Lopez, one of the grandparents, or the other grandparent must sit in the driver's seat. How many seating arrangements are possible?


Modern widescreen television screens typically have a ratio of $16:9$. What is the horizontal length (in inches) of a "50-inch" widescreen television?


Shown below are rows 1, 2, and 3 of Pascal's triangle.

\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]
Let \((a_i)\), \((b_i)\), \((c_i)\) be the sequence, from left to right, of elements in the 3005th, 3006th, and 3007th rows, respectively, with the leftmost element occurring at \(i = 0.\) Compute
\[
\sum_{i = 0}^{3006} \frac{b_i}{c_i} - \sum_{i = 0}^{3005} \frac{a_i}{b_i}.
\]


Evaluate the sum $$2 + 333 + 5555 + 77777 + 999999 + 11111111 + 222222222$$ and find the value of $n$ such that the sum is congruent to $n$ modulo 11, where $0 \leq n < 11$.


Let \( y_1, y_2, y_3 \) be positive real numbers such that \( 2y_1 + 3y_2 + 4y_3 = 120 \). Find the smallest possible value of
\[ y_1^2 + y_2^2 + y_3^2. \]


Three of the four endpoints of the axes of an ellipse are, in some order, \[(1, 3), \; (6, -1), \; (11, 3).\] Find the distance between the foci of the ellipse.


The probability it will rain on Saturday is $80\%$, on Sunday is $30\%$, and on Monday, it increases to $50\%$ if it rains on Sunday and decreases to $10\%$ if it does not rain on Sunday. Assume the probability of rain on Saturday and Sunday remains independent. What is the probability it will rain on both Sunday and Monday, expressed as a percent?


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


In Pascal's Triangle, each entry is the sum of the two entries directly above it. Find the row where three consecutive entries have the ratio $2:3:4$.


A circle is tangent to the lines $3x - 5y = 15$ and $3x - 5y = -45.$ The center of the circle lies on the line $x - 3y = 0.$ Find the center of the circle.

Solution: \boxed{\left(-\frac{45}{4}, -\frac{15}{4}\right)}


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


Define a function $g$ on the set of ordered pairs of positive integers with the following properties:
\[
g(x,x) = x^2,
\]
\[
g(x,y) = g(y,x),
\]
\[
(x + y) g(x,y) = yg(x, x + y).
\]
Calculate $g(10,44)$.


Find the solution to the equation $15x + 2 \equiv 7 \pmod{20}$, where the solution has the form $x \equiv a \pmod{m}$ for some positive integers $m \geq 2$ and $a < m$. What is $a + m$?


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


It costs 8 cents to copy 6 pages. How many pages can you copy for \$50?


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


Let \(x, y, z\) be positive real numbers such that \(x + y + z = 5.\) Find the minimum value of
\[\frac{9}{x} + \frac{4}{y} + \frac{25}{z}.\]


The current time is 3 o'clock. What time will it be 3333 hours from now on a 12-hour clock?


Solution: \boxed{\log_5 \left(\frac{53}{3}\right)}


At a college of 2500 students, each student studies either German or Russian, and some study both. The number who study German is between 70% and 75% of the college population, and the number who study Russian is between 40% and 45%. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.


Find the smallest composite number that has no prime factors less than 20.


How many two-digit positive integers are congruent to 2 (mod 4)?


I planned to work for 25 hours a week for 15 weeks to earn $\$4500$ for a vacation. However, I ended up being sick for the first three weeks of the period and couldn't work at all. How many hours per week must I work for the remaining weeks if I still want to fund my vacation?


In triangle $ABC$, where $AB = \sqrt{34}$ and $AC = 5$, the angle $B$ is $90^\circ$. Calculate $\tan A$.


In a local debate club, there are 5 teams with 8 members each. Each of the 5 teams rotates to organize monthly debates. During each debate, each team selects three members to be on the debate organizing committee, except the organizing team, which selects four members. How many possible 16-member debate organizing committees can be formed?


How many three-digit whole numbers have at least one 5 or at least one 8 as digits?


For how many integers $n$ between 1 and 200 is the greatest common divisor of 91 and $n$ equal to 13?


The numbers $x,$ $y,$ $z,$ $w$ are equal to 1, 3, 5, 7, in some order. Find the largest possible value of
\[xy + yz + zw + wx.\]


Find the area of the region enclosed by the graph of $|x-80| + |y| = \left|\frac{x}{2}\right|$.


An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $20$, and one of the base angles is $\arcsin(0.6)$. Find the area of the trapezoid.


Convert $253_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$


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


Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(xg(y) - x) = xy - g(x) \] for all \( x, y \).
Determine the number of possible values for \( g(3) \) and the sum of all possible values of \( g(3) \). Compute the product of these two quantities.


Consider a rectangular grid with vertical and horizontal segments forming a 4 by 3 rectangle from point $A$ to point $B$. You can only move to the right (R) or down (D). How many different routes are there from point $A$ to point $B$?

[asy]
unitsize(0.06inch);
draw((0,0)--(12,0)--(12,16)--(0,16)--cycle);
draw((4,0)--(4,16));
draw((8,0)--(8,16));
draw((0,4)--(12,4));
draw((0,8)--(12,8));
draw((0,12)--(12,12));
dot((0,16));
dot((12,0));
label("$A$",(0,16),NW);
label("$B$",(12,0),SE);
[/asy]


Let $T$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{3}}{2} \le x \le \frac{2}{\sqrt{3}}.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in T$ such that $z^n = 1.$


Consider a pentagonal prism with seven faces, fifteen edges, and ten vertices. One of its faces will be used as the base for a new pyramid. Calculate the maximum value of the sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid).


What is the remainder when the sum of the first 105 counting numbers is divided by 1000?


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


Determine the tens digit of $19^{2021}$.


A cube is painted so that one side is blue, three sides are red, and two sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube.


In triangle $ABC$, $\angle BAC = 90^\circ$. If $\angle ABC = 2y^\circ$ and $\angle BCA = y - 10^\circ$, find the value of $y$.


Suppose that $x$ is a positive multiple of $4$. If $x^3$ is less than $8000$, what is the greatest possible value of $x$?


Calculate the length of the diagonal of a rectangle where one side is $30\sqrt{2}$ cm and the other side is $30(\sqrt{2} + 2)$ cm. Express your answer in simplest form.

Solution: \boxed{\sqrt{7200 + 3600\sqrt{2}}}


Evaluate $$\lceil\sqrt{8}\rceil + \lceil\sqrt{9}\rceil + \lceil\sqrt{10}\rceil + \cdots + \lceil\sqrt{35}\rceil.$$


Define a $\emph{smooth integer}$ as an even integer that is greater than 10, less than 100, and such that the sum of its digits is 10. What fraction of all smooth integers is divisible by 7? Express your answer as a common fraction.


A circle rests in the interior of the parabola with equation $y = x^2 + 1,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?


Given the equation $z^2 - 4z + 29 = 0$, determine how many possible unique values there are for $|z|$.


The length of a rectangular playground exceeds three times its width by 30 feet, and the perimeter of the playground is 730 feet. What is the area of the playground in square feet?

Solution: \boxed{23,554.6875 \text{ square feet}}


Solve the inequality
\[\frac{x - 2}{x^2 + 4x + 13} \ge 0.\]

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


If \(a+b = 8\) and \(a-b = 4\), what is the value of \(a^2 - b^2\)? Also, find the value of \(a^3 - b^3\).

Solution: \boxed{a^2 - b^2 = 32, \quad a^3 - b^3 = 208}


Consider the polynomials \[f(x) = 2 - 15x + 4x^2 - 5x^3 + 6x^4\] and \[g(x) = 4 - 3x - 7x^3 + 10x^4.\] Find \( c \) such that the polynomial \( f(x) + cg(x) \) has degree 3.


Suppose that \(a\) and \(b\) are positive integers such that \(a-b=8\) and \(\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 16\). Find the smallest possible value of \(b\).


The base nine numbers $125_9$ and $33_9$ need to be multiplied and the result expressed in base nine. What is the base nine sum of the digits of their product?


Let $f(x)$ be a monic quartic polynomial such that $f(-2)=-4$, $f(1)=-1$, $f(-3)=-9$, and $f(5)=-25$. Find $f(2)$.


The equation $y = -16t^2 + 40t + 50$ describes the height (in feet) of a ball thrown upwards at $40$ feet per second from $50$ feet above the ground. Find the time (in seconds) when the ball will hit the ground. Your answer should be in the form of a common fraction.

Solution: \boxed{\frac{5 + 5\sqrt{3}}{4}}


Let $m=\underbrace{33333333}_{\text{8 digits}}$ and $n=\underbrace{666666666}_{\text{9 digits}}$.

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


Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two.  If $TA = TB = TC = 10$, what is the volume of pyramid $TABC$?


Consider a scenario where motorcycle license plates in Hilltown each contain three sections. The first section is a letter chosen from the set $\{B, F, J, N, T\}$, the second section is a number from $\{1, 2, 3\}$, and the third section is a letter from the set $\{A, E, I, O, U\}$. Hilltown decided to update its license plate format by adding one new letter to the first set and one new number to the second set. What is the largest possible number of ADDITIONAL license plates that can be made with these changes?


How many numbers are in the list $165, 159, 153, \ldots, 30, 24?$


In Flatville, vehicle license plates each contain four characters: two letters followed by two numbers. The first letter is chosen from the set $\{B, F, J, N, R\}$, the second letter from $\{E, I, U\}$, the first number can be any digit from 0 to 4, and the second number any digit from 5 to 9.

When Flatville needed more license plates, they added two new digits and one new letter. The new digits are added to the first number set and the new letter to the second letter set. What is the largest possible number of ADDITIONAL vehicle license plates that can be made by these additions?


Let $p,$ $q,$ $r,$ $x,$ $y,$ and $z$ be positive real numbers such that $p + q + r = 2$ and $x + y + z = 1$. Find the maximum value of:
\[\frac{1}{p + q} + \frac{1}{p + r} + \frac{1}{q + r} + \frac{1}{x + y} + \frac{1}{x + z} + \frac{1}{y + z}.\]


Simplify the following expression: $(10x^{12} + 8x^9 + 5x^7) + (11x^{9} + 3x^7 + 4x^3 + 6y^2 + 7x + 9).$ Express your answer as a polynomial with the terms in decreasing order.

Solution: \boxed{10x^{12} + 19x^9 + 8x^7 + 4x^3 + 6y^2 + 7x + 9}


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


Triangle $DEF$ is isosceles with angle $D$ congruent to angle $F$. The measure of angle $F$ is three times the measure of angle $E$. Determine the number of degrees in the measure of angle $D$.


A convex pentagon has interior angles with measures $2x+2$, $3x$, $4x$, $5x$, and $6x-2$ degrees. Calculate the measure of the largest angle.


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


Find the remainder when $5^{5^{5^5}}$ is divided by 500.


Two circles of radius 12 cm overlap such that each circle's center lies on the circumference of the other circle. Determine the length of the common chord.

Solution: \boxed{12\sqrt{3} \text{ cm}}


Determine the residue of $1237 \pmod{17}$. Your answer should be an integer in the range $0,1,2,\ldots,15,16$.


A sphere is inscribed in a regular tetrahedron. What is the ratio of the volume of the inscribed sphere to the volume of the tetrahedron? Express your answer as a common fraction in terms of $\pi$.

Solution: \boxed{\frac{\pi \sqrt{3}}{18}}


A square and a circle intersect so that each side of the square contains a chord of the circle, with the length of each chord being half the radius of the circle. Determine the ratio of the area of the square to the area of the circle, and express your answer as a simplified fraction in terms of \(\pi\).


Determine the value of $x$ such that:
\[5 + \frac{5 + x}{3} + \frac{5 + 2x}{3^2} + \frac{5 + 3x}{3^3} + \dotsb = 15.\]


Determine the number of ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ equals $8^{30}$.


In a special event, ten baseball players are set to be seated in a row for a photo session. Three are from the Cubs, three are from the Red Sox, two are from the Yankees, and two are from the Dodgers. If teams insist on members sitting together, how many ways can these ten players be seated?


If $0 \le p \le 1$ and $0 \le q \le 1$, define $H(p, q)$ by:
\[
H(p, q) = -4pq + 7p(1-q) + 5(1-p)q - 6(1-p)(1-q).
\]
Define $J(p)$ to be the maximum of $H(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $J(p)$?


Find the coefficient of the $x^3$ term in the expansion of the product \((3x^3 + 4x^2 + 5x + 6)(7x^3 + 8x^2 + 9x + 10)\).


A library has eight identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least 2 books are in the library and no more than 6 are in the library?


Simplify the expression: $(3y-2)\cdot(5y^{12} + 4y^{11} + 6y^9 + 3y^8)$ and express your answer as a polynomial with the degrees of the terms in decreasing order.

Solution: \boxed{15y^{13} + 2y^{12} - 8y^{11} + 18y^{10} - 3y^9 - 6y^8}


What is the sum of all integers in the set $\{1,2,3,4,5,6,7,8\}$ that are primitive roots $\pmod{11}$?


The hyperbolas \[\frac{x^2}{9} - \frac{y^2}{16} = 1\] and \[\frac{y^2}{25} - \frac{(x-4)^2}{M} = 1\] have the same asymptotes. Find $M.$


What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 10 units to the square of its numerical value of its perimeter, in units squared? Express your answer as a common fraction in simplest radical form.


Solution: \boxed{\log_{9} 58}


What is the matrix \(\mathbf{N}\) that performs the transformation which sends square \(ABCD\) to square \(A''B''C''D''\)? Consider that \(ABCD\) is first rotated by \(30^\circ\) clockwise and then translated by the vector \(\begin{pmatrix} 2 \\ -1 \end{pmatrix}\).

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


Convert $156_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$


The area of the base of a hemisphere is $81\pi$. What is the total surface area of the sphere obtained by adding a top circular lid to the hemisphere?


Determine how many positive real solutions exist for the equation $x^{8} + 5x^7 + 10x^6 + 729x^5 - 379x^4 = 0$.


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

Solution: \boxed{\mathbf{Q} = \begin{pmatrix} \frac{1}{26} & \frac{3}{26} & -\frac{2}{13} \\ \frac{3}{26} & \frac{9}{26} & -\frac{6}{13} \\ -\frac{2}{13} & -\frac{6}{13} & \frac{8}{13} \end{pmatrix}}


Subtract $261_9$ from the sum of $374_{9}$ and $625_{9}$. Express your answer in base $9$.


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


Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b > 1$, $b$ is odd, $a$ is a prime number, and $a^b = n$. How many oddly powerful integers are there less than $500$?


Two circles, each with radius 3, are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles?

Solution: \boxed{\frac{9\pi - 18}{2}}


The expression \(x^2 + 21x + 110\) can be written as \((x + a)(x + b)\), and the expression \(x^2 - 19x + 88\) written as \((x - b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?


Let $c$ and $d$ be real numbers. One of the roots of the polynomial $x^3 + cx + d = 0$ is $2 + i \sqrt{2}$. Find $c + d$.


Let tetrahedron $ABCD$ have $AD=BC=30$, $AC=BD=40$, and $AB=CD=50$. For any point $X$ in space, suppose $g(X)=AX+BX+CX+DX$. Determine the least possible value of $g(X)$, expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Report the sum $p+q$.


A paperboy delivers newspapers to 11 houses along Main Street. He may skip delivering to any house, but he must not skip three consecutive houses to avoid losing his job. Determine the number of possible delivery sequences available to the paperboy under these conditions.


The graph of the equation \(x + 3y + 5 = 0\) is perpendicular to the graph of the equation \(bx + 3y + 5 = 0\). What is the value of \(b\)?


A line is parameterized by a parameter $t$. The vector on the line at $t = 0$ is $\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}$, and the vector on the line at $t = 1$ is $\begin{pmatrix} 4 \\ 1 \\ -1 \end{pmatrix}$. Find the vector on the line at $t = -1$ after scaling the direction vector by 2.

Solution: \boxed{\begin{pmatrix} -2 \\ 10 \\ 20 \end{pmatrix}}


Let the function \( g : \mathbb{R} \to \mathbb{R} \) satisfy the equation
\[ g(x) + 2g(2 - x) = 4x^3 - x^2 \]
for all \( x \). Find \( g(5) \).


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


Square $XYZW$ has area $144$. Point $P$ lies on side $\overline{XW}$, such that $XP = 2WP$. Points $Q$ and $R$ are the midpoints of $\overline{ZP}$ and $\overline{YP}$, respectively. Quadrilateral $XQRW$ has an area of $20$. Calculate the area of triangle $RWP$.


Let $F_n$ be the Fibonacci sequence, defined as $F_0 = 0$, $F_1 = 1$, and for $n \geq 0$, $F_{n+2} = F_{n+1} + F_n$. Compute the sum:
\[\sum_{n=0}^{\infty} \frac{F_n}{3^n}.\]


How many three-digit numbers are there in which the hundreds digit is greater than the tens digit which is in turn greater than the ones digit?


Arc \(DE\) is a quarter-circle with center \(F\). The shaded region \(DEF\) is "rolled" along a straight board \(XY\) until it reaches its original orientation for the first time with point \(F\) landing at point \(F'\). If \(EF = \frac{3}{\pi}\) cm, what is the length of the path that point \(F\) travels?

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


I have 7 marbles numbered 1 through 7 in a bag. Suppose I take out two different marbles at random, one of which must either be a 1 or a 7. What is the expected value of the sum of the numbers on the marbles?


Below is the graph of $y = a \tan bx$, where $a$ and $b$ are positive constants. Determine $ab$.

[asy]import TrigMacros;
size(250);
real g(real x)
{
    return 3*tan(2*x);
}
draw(graph(g, -pi/4 + 0.01, pi/4 - 0.01),blue);
draw(graph(g, 3*pi/4 + 0.01, 5*pi/4 - 0.01),blue);
limits((-pi,-6),(pi,6),Crop);
draw((-pi/4,-6)--(-pi/4,6),dashed);
draw((pi/4,-6)--(pi/4,6),dashed);
trig_axes(-pi,pi,-6,6,pi/6,1);
layer();
rm_trig_labels(-7, 7, 6);

label("$3$", (0,3), E);
label("$6$", (0,6), E);
label("$-3$", (0,-3), E);
label("$-6$", (0,-6), E);
[/asy]


In a different lemonade recipe, the ratio of water to lemon juice is $5$ parts water to $2$ parts lemon juice. If I decide to make $2$ gallons of lemonade, how many quarts of water are needed? Note: there are four quarts in a gallon.


Convert the point \((\rho,\theta,\phi) = \left( 15, \frac{5\pi}{4}, \frac{\pi}{4} \right)\) in spherical coordinates to rectangular coordinates.

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


Determine the maximum number of real roots for the polynomial
\[ x^n + x^{n-1} + \dots + x + k = 0, \]
where \(n\) is a positive integer and \(k\) is a non-zero real number.


How many perfect squares are factors of 450?


For how many integer values of \( n \) between 1 and 500 inclusive does the decimal representation of \( \frac{n}{1800} \) terminate?


What is the remainder when \( 8x^3 - 20x^2 + 28x - 30 \) is divided by \( 4x - 8 \)?


The Happy Valley Kennel has now 5 chickens, 2 dogs, and 4 cats. In how many ways can the 11 animals be placed in a row of 11 cages, such that all of the animals of each type are in adjacent cages? (Two animals of the same species are considered distinguishable.)


In a bag, there are 7 purple chips, 6 green chips, and 5 orange chips. One chip is drawn from the bag and put back in, then a second chip is drawn. What is the probability that the two selected chips are of different colors?


If $(x+3)(x-4)=20$, find the sum of the possible values of $x$.


Judy goes shopping and buys $7$ carrots at $\$2$ each, $4$ bottles of milk at $\$3$ each, $3$ pineapples at $\$5$ each (currently on a 50% discount), $1$ bag of flour at $\$8$, and a $\$10$ pack of cookies. The store offers a 10% discount on the total cost for purchases over $40$, before any other discounts. Judy also uses a $\$5$ coupon for orders over $25$. How much does she pay this week?


Solve the inequality
\[\frac{x^2 - 9}{x + 3} < 0.\]

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


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


Write the following expression as a polynomial: $$(3x^2+4x+8)(x+2)-(x+2)(x^2+5x-72)+(4x-15)(x+2)(x+6).$$

Solution: \boxed{6x^3 + 20x^2 + 6x - 20}


Sarah rolls a six-sided dice 10 times. The dice is biased and has a $\frac{1}{3}$ probability of landing on six and a $\frac{2}{3}$ probability of landing on any number that is not six. What is the probability that she rolls exactly 4 sixes?


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


How many factors of 1200 are perfect squares?


Calculate the residue of $240 \cdot 15 - 21 \cdot 9 + 6 \pmod{18}$.


The measures of the interior angles of a triangle are in a 3:4:9 ratio. If the largest angle is decreased by $x$ degrees, and it still maintains a triangle with the sum of angles being 180 degrees, find $x$ such that the smallest angle doubles its initial value.


Find the sum of all integers $n$ such that $n^2 - 17n + 72$ is a perfect square and $\dfrac{24}{n}$ is an integer.


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


If the system of equations
\[
4x - 2y = c,
\]
\[
6y - 12x = d,
\]
has a solution, find \(\frac{c}{d}\), assuming \(d \neq 0\).


Two spinners are divided into fifths and sixths, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? The numbers on the first spinner are 3, 5, 6, 7, and 9. The numbers on the second spinner are 2, 4, 6, 8, 9, and 11.


For all positive integers $n$, let $g(n) = \log_{3003} n^2$. Calculate $g(3) + g(11) + g(13)$.


Each term of a sequence, after the first term, is the square of the term preceding it. If the first term is 3 and the second term is 9, what is the 6th term?


Lisa constructs a medium-sized obelisk for her garden decoration. The obelisk has a total height of 15 inches and a rectangular base measuring 8 inches by 10 inches. Lisa needs the smallest cube-shaped box to transport her obelisk securely. What should be the volume of this box, in cubic inches?

Solution: \boxed{3375 \text{ cubic inches}}


The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 5$ at the points $C$ and $D$. Determine the distance between $C$ and $D$ and express this distance in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers.


In triangle $XYZ$, $\angle X = 90^\circ$ and $\sin Y = \frac{3}{5}$. Find $\cos Z$.


Add $25_7 + 54_7.$ Express your answer in base 7.


The circles whose equations are $x^2 + y^2 - 6x + 4y - 20 = 0$ and $x^2 + y^2 - 16x + 8y + 40 = 0$ intersect in the points $C$ and $D$. Compute the slope of $\overline{CD}$.


I have 8 shirts and 8 hats, both coming in the colors tan, black, blue, gray, white, yellow, red, and green. I have 6 pairs of pants available in tan, black, blue, gray, red, and green. I refuse to wear an outfit where all three items are the same color or the pants and the hat are of the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?


Consider two solid spherical balls, one centered at $\left( 0, 0, 5 \right),$ with radius 3, and the other centered at $(0,0,0)$ with radius $2$. How many points $(x,y,z)$ with only integer coefficients are there in the intersection of the balls?


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


What is the sum of the different prime factors of $245,700?$


In triangle $DEF,$ $\sin D = \frac{4}{5}$ and $\cos E = \frac{12}{13}.$ Find $\cos F.$

Solution: \boxed{\cos F = -\frac{16}{65}}


Let $T$ be the set of all positive integers that have five digits in base $2$. What is the sum of all the elements in $T$, when expressed in base $10$?


Compute $\displaystyle \sum_{n=3}^\infty \sum_{k=1}^{n-2} \frac{k}{3^{n+k}}$.


What integer is closest to the value of $\sqrt[3]{5^3+9^3}$?


The quadratic $x^2 + 1800x + 2700$ can be written in the form $(x+b)^2 + c$, where $b$ and $c$ are constants. What is $\frac{c}{b}$?


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


In a right triangle $PQR$, the sides $PQ$, $PR$, and $QR$ measure $15$, $9$, and $12$ units respectively. What is the distance from $R$ to the midpoint of the segment $PQ$?


Fully factor the following expression: $4x^2 - 36$.


The center of a circle has coordinates $(3, -7)$. The circle is reflected about the line $y=-x$. What are the $x,y$ coordinates of the center of the image circle? State the $x$ coordinate first.

Solution: \boxed{(7, -3)}


What is the product of the digits in the base 8 representation of $8675_{10}$?


Let $O$ be the origin. Determine the scalar $k$ such that for any points $P$, $Q$, $R$, and $S$ satisfying the vector equation
\[4 \overrightarrow{OP} - 3 \overrightarrow{OQ} + 6 \overrightarrow{OR} + k \overrightarrow{OS} = \mathbf{0},\]
the points $P$, $Q$, $R$, and $S$ are guaranteed to be coplanar.


In a school's Science Club, there are 18 members eligible for election to 6 different officer positions. Among these candidates, 8 have served as officers before. Determine how many different groups of officers can be elected such that at least 2 past officers are included.


What is the median number of moons for the listed celestial bodies, taking into account both planets and prominent dwarf planets? Include the distance from the sun for each celestial body. Use the following data:
\begin{tabular}{c|c|c}
Celestial Body & Number of Moons & Distance from Sun (AU)\\
\hline
Mercury & 0 & 0.39\\
Venus & 0 & 0.72\\
Earth & 1 & 1.00\\
Mars & 2 & 1.52\\
Ceres & 0 & 2.77\\
Jupiter & 16 & 5.20\\
Saturn & 23 & 9.58\\
Uranus & 15 & 19.22\\
Neptune & 2 & 30.05\\
Pluto & 5 & 39.48\\
Haumea & 2 & 43.13\\
Makemake & 1 & 45.79\\
Eris & 1 & 67.67\\
\end{tabular}


The polynomial $3x^3 + cx + 9$ has a factor of the form $x^2 + qx + 1.$ Find $c.$


Stuart draws a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the larger circle, each tangent to the smaller one. If $m\angle ABC = 80^\circ$, how many segments will he draw before returning to his starting point at $A$?


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


Given vectors \(\mathbf{u}\) and \(\mathbf{z}\) such that \(\|\mathbf{u}\| = 5,\) \(\|\mathbf{z}\| = 8,\) and \(\mathbf{u} \cdot \mathbf{z} = 20,\) find \(\|\operatorname{proj}_{\mathbf{z}} \mathbf{u}\|\).


What is the inverse of the function \(f(x)=10-7x\)?

Solution: \boxed{g(x) = \frac{10 - x}{7}}


Triangles $XYZ$ and $XWV$ have areas $1200$ and $3600$, respectively, with $Y = (0,0)$, $Z = (150,0)$, $W = (500, 300)$, and $V = (510,290)$. Determine the sum of all possible $x$-coordinates of $X$.


Given that $-2 - 5\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx + 45 = 0\] and that $a$ and $b$ are rational numbers, compute $a$.


A circle with radius $r$ has chords $\overline{AB}$ of length $12$ and $\overline{CD}$ of length $9$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at point $P$, outside of the circle. Given that $\angle{APD}=60^\circ$ and $BP=10$, find the value of $r^2$.

Solution: \boxed{r^2 = 111}


Given that $9^2 = 81$, $99^2 = 9801$, and $999^2 = 998001$, which shows an increasing number of zeros in the square as the number of nines increases, calculate how many zeros are present in the expansion of $999,\!999^2$.


Two real numbers $a$ and $b$ satisfy $a+b=8$ and $a^3+b^3=172$. Compute the value of $ab$.


What is the smallest prime divisor of $3^{13} + 9^{11}$?


For \(-1 < r < 1\), let \( T(r) \) denote the sum of the geometric series \[20 + 20r + 20r^2 + 20r^3 + \cdots .\] Let \( b \) between \(-1\) and \(1\) satisfy \( T(b)T(-b)=5040 \). Find \( T(b) + T(-b) \).


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


Solve the following equation by completing the square: $$64x^2+96x-81 = 0.$$ Rewrite the equation in the form \((ax + b)^2 = c\), where \(a\), \(b\), and \(c\) are integers and \(a > 0\). What is the value of \(a + b + c\)?


Each square and each circle have a certain numerical value. Determine the sum of three circles, given:
\[\Box + \bigcirc + \Box + \bigcirc + \Box = 27,\]
\[\bigcirc + \Box + \bigcirc + \Box + \bigcirc = 25.\]


The line \( y = 2x + 5 \) is parameterized by the form
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r \\ -3 \end{pmatrix} + t \begin{pmatrix} 5 \\ k \end{pmatrix}.
\]
Determine the ordered pair \( (r, k) \).

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


Find the last three digits of $9^{107}.$


A girl has the following six coins in her wallet: $1$ penny, $2$ nickels, $2$ dimes, and $1$ half-dollar. She randomly selects two coins, calculates the sum of their values, and then returns them to her wallet. She repeats this process multiple times. What is the maximum number of different sums she can record?


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

Solution: \boxed{x^2 + 1}


If $\left\lfloor n^2/4 \right\rfloor - \lfloor n/2 \rfloor^2 = 5$, then find all integer values of $n$.


Let $u$, $v$, and $w$ be solutions of the equation $x^3 - 6x^2 + 11x - 6 = 0$.

Compute $\frac{uv}{w} + \frac{vw}{u} + \frac{wu}{v}$.


If 30% of 20% of a number is 24, what is 20% of 30% of the same number?


Determine the sum of all single-digit replacements for $z$ such that the number ${14{,}z48}$ is divisible by 9.


A segment with endpoints at $A(2, -2)$ and $B(14, 4)$ is extended through $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair.

Solution: \boxed{(20, 7)}


The measure of angle $ACB$ is 50 degrees. If ray $CA$ is rotated 540 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?

Solution: \boxed{50 \text{ degrees}}


Determine the smallest base-10 positive integer greater than 20 that is a palindrome when written in both base 2 and base 4.


Let \( a, b, c \) be the roots of the cubic polynomial \( x^3 - 3x - 1 = 0.\) Find
\[
a(b - c)^2 + b(c - a)^2 + c(a - b)^2.
\]


In how many ways can 10 people sit around a round table if one seat is permanently reserved for the guest of honor?


How many different positive, eight-digit integers can be formed using the digits 3, 3, 3, 5, 5, 7, 7, and 9?


Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 2500, \\
pr &=& qs &=& 1200.
\end{array}
\]
If \( T = p + q + r + s \), compute the value of \( \lfloor T \rfloor \).

Solution: \boxed{T = 140}


Find the greatest common divisor of 1237 and 1849.


Let \( g(n) \) denote the sum of all the positive divisors of the integer \( n \) including twice the greatest prime divisor of \( n \). How many integers \( j \) exist such that \( 1 \leq j \leq 5000 \) and \( g(j) = j + 2\sqrt{j} + 1 \)?


How many integers from 1 through 9999, inclusive, do not contain any of the digits 1, 5, 6, or 9?


What is the base 4 representation of the base 2 number \(1110010110_2\)?


Arc $AD$ is a semi-circle with center $B$. The shaded region $ABD$ is "rolled" along a straight board $PQ$ until it reaches its original orientation for the first time with point $B$ landing at point $B'$. If $BD = \frac{4}{\pi}$ cm, what is the length of the path that point $B$ travels?

Solution: \boxed{8\text{ cm}}


In a leap year, such as 2024, determine how many "prime dates" occur, where both the month and the day are prime numbers. Assume April is also considered as a prime month for this specific calculation aside from the usual prime months like February, March, May, July, and November.


A circle with center $C$ is tangent to both the positive $x$ and $y$-axes. It is also externally tangent to another circle centered at $(5,0)$ with radius $2$. Find the sum of all possible values of the radius of the circle with center $C$.


Calculate the area, in square units, of triangle $ABC$ with vertices $A(-1,2)$, $B(8,2)$, and $C(6,-1)$.

Solution: \boxed{13.5\text{ square units}}


How many of the positive divisors of 540 are multiples of 2?


How many different positive seven-digit integers can be formed using the digits 1, 1, 3, 3, 3, 8, and 8?


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


In a biology experiment, Thomas needs to prepare a saline solution by mixing 0.05 liters of salt solution with 0.03 liters of water, creating a base mixture. He aims to prepare a total of 0.6 liters of this saline solution. How many liters of water must Thomas use to achieve this?


The following table is partially filled in with the results of a survey done by the radio station FM100. Determine how many of the males surveyed listen to this station.

\begin{tabular}{| l | c | r |}
\hline
& listen & don't listen \\
\hline
males &  & 98 \\ \hline
females & 76 &  \\ \hline
total & 160 & 180 \\
\hline
\end{tabular}


Laura is preparing for a triathlon, focusing only on biking and running. She bikes for 25 miles at a speed of $2x+1$ miles per hour, then takes five minutes to transition from biking to running, during which she parks her bike, switches shoes, and hydrates, followed by a run of 8 miles at a speed of $x$ miles per hour. The entire duration of Laura's training session is 140 minutes. Find the speed at which Laura ran, rounded to the nearest hundredth of a mile per hour. You may use a calculator for this problem.

Solution: \boxed{8.81 \text{ mph}}


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

Let $n$ be the number of possible values of $g(2),$ and let $s$ be the sum of all possible values of $g(2).$ Find $n \times s.$


Joe now eats four meals a day instead of three. At each meal, he randomly chooses with equal probabilities either an apple, an orange, or a banana to eat. On a given day, what is the probability that Joe will eat at least two different kinds of fruit?


A modified deck contains 60 cards, 30 of which are black and 30 are red. The cards are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both black?


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


I take variable $b$, triple it, add ten, subtract five times the square of $b$, and divide the resulting expression by five.  What is my final expression in simplest form?

Solution: \boxed{-b^2 + \frac{3b}{5} + 2}


Determine the total surface area of a cone with a diameter of 8 cm and a height of 12 cm. Express your answer in terms of \(\pi\).

Solution: \boxed{16\pi (\sqrt{10} + 1)}


How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{12}$?


A certain integer has $3$ digits when written in base $10$. The same integer has $d$ digits when written in base $16$. What is the sum of all possible values of $d$?


Let real numbers \( a \), \( b \), and \( c \) define the polynomial \( g(x) = x^3 + ax^2 + x + 5 \) which has three distinct roots. Each root of \( g(x) \) is also a root of the polynomial \( f(x) = x^4 + x^3 + bx^2 + 50x + c \). Find \( f(-1) \).


The stem and leaf plot represents the weights, in pounds, of the cats at a veterinary clinic. What is the mean weight of the cats? (Note: $10|4$ represents 104 pounds.)

Weight of the Cats at the Veterinary Clinic (pounds)

$8|7$

$9|0\;3\;5\;5\;8$

$10|4\;6\;6\;7\;9$

$11|0\;1\;2$

Solution: \boxed{101.64 \text{ pounds}}


Three friends, Ben, Tom, and Jerry, share a pack of 30 candies. Jerry eats a number of candies that is a positive multiple of the number of candies eaten by Ben. Tom eats twice the number of candies that Ben eats. If the friends finish all 30 candies, then what is the maximum number of candies that Ben could have eaten?


Consider the lines:
\[
y = 2x + 3, \quad 2y = 6x + 4, \quad 3y = 6x - 1, \quad 4y = 2x - 8, \quad 5y = 2x - 10.
\]
Determine how many pairs of these lines are either parallel or perpendicular to each other.


In right triangle $ABC$, $\sin A = \frac{3}{5}$ and $\sin B = 1$. Find $\sin C$.


A biased coin has a higher likelihood of landing tails than heads. When flipped six times, the probability of getting heads exactly two times is not equal to $0$ and equals that of getting heads exactly three times. Let $\frac{p}{q}$, in lowest terms, be the probability that the coin comes up heads in exactly $4$ out of $6$ flips. Find $p + q$.


The graph of the line $3x - y = c$ is a perpendicular bisector of the line segment from $(2,6)$ to $(8,10)$. What is the value of $c$?


How many integers from 1 through 9999, inclusive, do not contain any of the digits 1, 2, 5, or 6?


A function \( f(x) \) is defined for all real numbers \( x \). It satisfies the equation
\[ 2f(x) + f\left(\frac{1}{x}\right) = 6x + 3 \]
for all non-zero \( x \). Determine the integer nearest to the sum \( S \) of all values of \( x \) for which \( f(x) = 2023 \).


A line joining $(1,3)$ and $(4,0)$ divides a square into two parts. The square has vertices at $(1,0)$, $(4,0)$, $(4,3)$, and $(1,3)$. What fraction of the area of the square is above this line?


In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin.

Note: A point is a lattice point if all its coordinates are integers.


The volume of a box is 360 cubic units where $a, b,$ and $c$ are integers such that $1 < c < b < a$ with $c$ being a prime number. Determine the largest possible value of $b$.


An equilateral triangle has two vertices at $(2,3)$ and $(10,3)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form.

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


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


If the product $(2x + 3y + 2)(5x + 2y + 3)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?


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


If the least common multiple of two 7-digit integers has 11 digits, then their greatest common divisor has at most how many digits?


In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle?


Suppose that $p$ is prime and $1014_p + 502_p + 217_p + 231_p + 12_p = 254_p + 547_p + 675_p$. Determine how many possible values of $p$ there are.


In a club election, 10 officer positions are available. There are 20 candidates, of which 8 have previously served as officers. Determine how many different slates of officers include at least one of the past officers.


Find the area of the triangle with vertices $(2,1,-1),$ $(5,4,1),$ and $(11,10,5)$.


Matt's five cousins are coming to visit. There are four identical rooms in which they can stay. If any number of the cousins can stay in one room, how many different ways are there to put the cousins in the rooms?


Alice has been keen to write all the possible rearrangements of the letters in her name. She manages to write fifteen rearrangements every minute. Calculate how many hours it will take her to write all possible rearrangements of her name.

Solution: \boxed{\frac{2}{15} \text{ hours}}


Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations:
\begin{align*}
  11x +  by +  cz &= 0 \\
   ax + 24y +  cz &= 0 \\
   ax +  by + 41z &= 0.
\end{align*}
Suppose $ a \ne 11$ and $ x \ne 0$. What is the value of
\[ \frac{a}{a - 11} + \frac{b}{b - 24} + \frac{c}{c - 41} \, ? \]


A club has 15 members, 7 boys and 8 girls. Three of the members are chosen at random. What is the probability that all three chosen members are girls?


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


Given \(\tan \theta = \frac{3}{4}\), find
\[
\frac{1 + \cos \theta}{\sin \theta} - \frac{\sin \theta}{1 - \cos \theta}.
\]


Let $\beta = bi$ where $b \neq 0$ and $b$ is a real number, and $\alpha$ be a complex number such that $\overline{\alpha} \beta \neq -1$. Find the maximum value of
\[
\left| \frac{\beta - \alpha}{1 + \overline{\alpha} \beta} \right|.
\]


Let \[p(x, y) =
\begin{cases}
xy &\quad \text{if } x \ge 0 \text{ and } y \ge 0, \\
x - 2y &\quad \text{if } x < 0 \text{ and } y < 0, \\
2x + 3y &\quad \text{if } x \ge 0 \text{ and } y < 0, \\
x + 3y &\quad \text{if } x < 0 \text{ and } y \ge 0 \\
3x + y &\quad \text{otherwise}.
\end{cases}
\]What is $p(p(2,-3),p(-1,4))$?


A rectangular array of books is an arrangement of the books in rows and columns such that each row contains the same number of books as every other row and each column contains the same number of books as every other column. If there must be at least three books in every row and column and all of the books in the room must be included, how many arrays are possible in a library room containing $48$ books? Note that $16$ rows of $3$ books is different from $3$ rows of $16$ books.


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


Given that $x = \frac{3}{4}$ is a solution to the equation $48x^2 + 25 = 77x + 4,$ find the other value of $x$ that will solve the equation. Express your answer as a common fraction.


The expression $\cos 2x + \cos 4x + \cos 8x + \cos 10x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$  Find $a + b + c + d.$


What is the sum of the last two digits of $15^{25} + 5^{25}$?


Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is along side $EG$ of the triangle, and side $AB$ is now one-third the length of side $AD$. The altitude from $F$ to side $EG$ is 12 inches, and side $EG$ is 15 inches. Determine the area of rectangle $ABCD$.


The number $0.425875$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?


Find $\tan A$ in the right triangle shown below.

[asy]
pair A,B,C;
A = (0,0);
B = (40,0);
C = (0,15);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,20));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$41$", (B+C)/2,NE);
label("$40$", B/2,S);
[/asy]


In isosceles triangle $ABC$, $AB=AC=10$, $BC=12$, and $\angle BAC = \angle BCA$. Points $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{AC}$ respectively; $\overline{CD}$ and $\overline{BE}$ intersect at point $X$. Compute the ratio of the area of quadrilateral $AEXD$ to the area of triangle $BXC$.


For what digit $d$ is the five-digit number $5617d$ a multiple of 9?


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


How many three-digit numbers are there such that the middle digit is the average of the first and last digits and the last digit is the difference between the first and middle digits?


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


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

Solution: \boxed{\begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}}


Let \[q(x,y) =
\begin{cases}
x - y &\quad \text{if } x \ge 0 \text{ and } y \le 0, \\
x + 3y &\quad \text{if } x < 0 \text{ and } y > 0, \\
4x - 2y &\quad \text{otherwise}.
\end{cases}
\]
What is $q(q(2,-3), q(-4,1))$?


Let $g(x)$ be a polynomial with real, nonnegative coefficients. If $g(8) = 32$ and $g(32) = 2048,$ find the largest possible value of $g(16).$


Let \( g \) be a function taking the integers to the integers such that
\[ g(m + n) + g(mn + 1) = g(m) g(n) + 1 \]
for all integers \( m \) and \( n \).

Let \( n \) be the number of possible values of \( g(2) \), and let \( s \) be the sum of all possible values of \( g(2) \). Find \( n \times s \).


For $\mathbf{v} = \begin{pmatrix} 2 \\ z \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} -1.5 \\ 2.5 \end{pmatrix}.\]Find $z$.


Each letter in the alphabet is assigned a distinct integer value. Let's say $F=23$. The point value of a word is the sum of its letters' values. Given that $FORMED=63$, $DEMO=30$, and $MODE=41$, what is the value of $R$?


What is the remainder when 1,493,825 is divided by 6?


Let $f(x) = \sin^4 x + \cos^2 x + \tan^2 x$. As $x$ ranges over all real numbers, find the range of $f(x)$.

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


Let $g(x) = x^2 - 2x + 2023$. What is the greatest common divisor of $g(102)$ and $g(103)$?


Determine the value of $b$ for which the equation $4(3x - b) = 3(4x + 16)$ has no solutions.


Mr. and Mrs. Singh have three children. When they travel in their family van, two people sit in the front seats, and three people sit in the back seats. Either Mr. Singh or Mrs. Singh must sit in the driver's seat. How many seating arrangements are possible?


A fair 8-sided die is rolled. If the roll is even and not a multiple of 3, you win that amount of dollars. If the roll is a multiple of 3, you win that amount minus $1 (for example, if you roll 6, you win $5). If the roll is odd and not a multiple of 3, you win nothing. What is the expected value of your winnings? Express your answer as a dollar value.


Determine the radius of the circle circumscribed around triangle $ABC$ if $AB = 13,$ $AC = 7,$ and $BC = 8$.


Find the sum of the coefficients in the polynomial $2(4x^8 - 3x^5 + 9) + 9(x^6 + 2x^3 - 8)$ when it is fully simplified.


Thirty tiles are numbered from 1 to 30 and placed into box $A$. Twenty-five other tiles numbered 10 through 34 are placed into box $B$. One tile is randomly drawn from each box. What is the probability that the tile from box $A$ has a number less than 20 and the tile from box $B$ is either a prime number or greater than 28? Express your answer as a common fraction.


Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} -2 \\ 4 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$.

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


What is the sum and the product of the distinct prime factors of 924?


In a bag, there are 7 blue chips, 5 yellow chips, and 4 red chips. One chip is drawn from the bag and then replaced. A second chip is then drawn. What is the probability that the two selected chips are of different colors?


I now have 7 shirts, 5 pairs of pants, and 7 hats. The pants come in tan, black, blue, gray, and red. The shirts and hats also come in those colors, plus white and yellow. I refuse to wear an outfit where the shirt, hat, and pants are all the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?


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


Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smallest one divides the larger two? Express your answer as a common fraction.


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


Evaluate the expression $\left(b^b - b(b-1)^b\right)^b$ when $b=4$.


Find the smallest positive real number \( a \) such that there exists a positive real number \( b \) such that all the roots of the polynomial \( x^3 - 3ax^2 + bx - 2a \) are real. For this value of \( a \), determine the unique value of \( b \).


Rewrite the quadratic \( 4x^2 - 16x - 21 \) in the form of \( (gx + h)^2 + j, \) where \( g, h, \) and \( j \) are all integers. What is \( gh \)?


Bob is visiting Japan and decides to buy a souvenir for 340 yen. If one U.S. dollar is currently worth about 115 yen, and there is a service charge of 25 yen, how much does he need to spend in total, and what is the equivalent amount in USD? (You may use a calculator.)

Solution: \boxed{3.17\ \text{USD}}


Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are placed on the graph of $f(x) = e^x$. Let $x_1 = 0$ and $x_2 = 3$. The points $C$ and $D$ trisect $\overline{AB}$, with $AC < CB$. A horizontal line is drawn through $C$ to intersect the curve again at $E(x_3, y_3)$. Determine the value of $x_3$.

Solution: \boxed{\ln\left(\frac{2 + e^3}{3}\right)}


A fair 6-sided die is rolled. If I roll $n$, then I win $(n^3 - 2n)$ dollars if $n < 4$, and $n^3$ dollars if $n \geq 4$. What is the expected value of my winnings? Express your answer as a dollar value rounded to the nearest cent.


A square with vertices \((-a, 0), (a, 0), (a, 2a), (-a, 2a)\) is intersected by the line \(y = 2x\). Find the ratio of the perimeter of one of the resulting congruent quadrilaterals to \(a\) in simplest radical form.

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


How many integers, $x$, satisfy $|6x + 2| < 10$ and also $3x + 1 > 0$?


Solve the inequality
\[\frac{x^2 - 49}{x + 7} < 0.\]

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


A square region has quarter-circular arcs constructed at each of its corners. If the side of the square measures $4/\pi$, what is the perimeter of this region?


Four marbles are randomly selected, without replacement, from a bag containing three red, three blue, three green, and three yellow marbles. What is the probability that one marble of each color is selected? Express your answer as a common fraction.


Solve for $y$: $\log_3 y + \log_9 y = 5$.


Calculate the sum of the numbers in Row 10 of Pascal's Triangle.


What is the coefficient of \(x^4\) in the expansion of \((3x + 4)^8\)?


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


Find the integer $n$, $0 \le n \le 180,$ such that $\cos n^\circ = \cos 1124^\circ.$


How many license plates can be made if they consist of 2 vowels followed by 2 digits, where both digits are either both odd or both even?


Suppose that $a$ and $b$ are positive integers such that $a-b=4$ and $\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 4$. Find the smallest possible value of $b$.


Given Pascal’s triangle, calculate the sum of the fifth and sixth elements in Row 20.


Let $f(x) = \frac{2ax + 3b}{4cx - 5d}$ where $abcd \neq 0$ and $f(f(x)) = x$ for all $x$ in the domain of $f$. Determine the value of $2a - 5d$.


Find the largest value of $k$ such that $4$ is in the range of $g(x)=x^2-7x+k$.


At a conference with $30$ businessmen, fifteen businessmen drank coffee, twelve businessmen drank tea, and eight drank soda. Seven businessmen drank both coffee and tea, three drank both tea and soda, and two drank both coffee and soda. One businessman drank all three beverages. How many businessmen drank none of the beverages?


Kristen has to clear snow from a driveway that is 30 feet long and 3 feet wide. If the snow is initially 8 inches deep, and compacting the snow reduces its volume by 10%, how much snow (in cubic feet) must Kristen move?


The number of games won by five basketball teams are displayed in a graph, but the names of the teams are missing. Here are some clues to help identify the teams:

1. The Sharks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won at least 18 games.

How many games did the Warriors win? The number of games won by the teams are 20, 25, 30, 35, and 40.


Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, determine the total area in square centimeters of the football-shaped area of regions II and III combined. Express your answer as a decimal to the nearest tenth.


Suppose the roots of the cubic equation $x^3 + 5x^2 + 6x - 13 = 0$ are $a$, $b$, and $c$, and the roots of another cubic equation $x^3 + rx^2 + sx + t = 0$ are $a+1$, $b+1$, and $c+1$. Find the value of $t$.


What is the smallest positive integer $x$ that, when multiplied by $900$, produces a product that is a multiple of $1152$?


The vertical drops of five roller coasters at Mandelbrot Amusement Park after renovations and enhancements are shown in the updated table. Assume each roller coaster's drop has increased by 10%, except for The Tower of Power, which has been renovated to further enhance its drop by an additional 10%. The updated vertical drops are provided in the table below.
\begin{tabular}{|l|c|}
\hline
The Parabola & 181.5 feet \\ \hline
The G Force & 130.9 feet \\ \hline
The Mean Streak & 151.8 feet \\ \hline
The Tower of Power & 330 feet \\ \hline
The Maximum Ride & 217.8 feet \\ \hline
\end{tabular}
What is the positive difference between the mean and the median of these values?


Five people sit in a row, and each person will roll a standard six-sided die. What is the probability that no two adjacent people will roll the same number after they each roll the die once?


Compute $\begin{pmatrix} \sqrt{2} & -\sqrt{2} \\ \sqrt{2} & \sqrt{2} \end{pmatrix}^8.$

Solution: \boxed{\begin{pmatrix} 256 & 0 \\ 0 & 256 \end{pmatrix}}


A line segment begins at $(1, 3)$. It is 15 units long and ends at the point $(x, -4)$ where $x < 0$. What is the value of $x$?

Solution: \boxed{1 - 4\sqrt{11}}


Find $\tan \theta$ in the right triangle shown below.

[asy]
pair A,B,C;
A = (0,0);
C = (20,0);
B = (0,12);
draw(B--C--A--B);
draw(rightanglemark(B,A,C,20));
label("$A$",A,SW);
label("$C$",C,SE);
label("$B$",B,N);
label("$20$",C/2,S);
label("$25$",(B+C)/2,NE);
[/asy]


A car travels 60 kph for 30 kilometers, 70 kph for 35 kilometers, 80 kph for 1 hour, and 55 kph for 20 minutes. What is the average speed of the car, in kph?


For all composite integers $n$, what is the largest integer that always divides into the difference between $n^4 - n^2$?


A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 8 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a common fraction.


How many perfect squares less than 5000 have a ones digit of 1, 5, or 6?


Cedric has deposited $12,000 into an account that pays $6\%$ interest compounded annually.

Daniel has deposited $12,000 into an account that pays $8\%$ simple annual interest.

After 20 years, Cedric and Daniel compare their respective balances. What is the positive difference between their balances to the nearest dollar?


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


I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $2pq - 5p - 3q = 4$? Express your answer as a common fraction.


What is the sum of the digits of the base-2 expression for $300_{10}$?


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


Find $y$ such that $\log_y 16 = \log_{125} 5$.


Let $n$ be the number of sides in a regular polygon. For $3\leq n < 15$, how many values of $n$ result in a regular polygon where the degree measure of the interior angles is not an integer?


A zoo has six pairs of different animals, each consisting of one male and one female. The zookeeper decides to feed the animals, ensuring that each time he feeds a single animal, the next one he feeds must be of the opposite gender. He begins by feeding the male lion. How many ways can he feed all the animals?


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

Determine a simplified expression for the sum of these seven consecutive integers.

Solution: \boxed{7n + 21}


Find $118^{-1} \pmod{119}$, as a residue modulo 119.


A choir has 90 members who need to be arranged in rows with the same number of members per row. The number of members per row must be between 4 and 15. Determine the number of possible row-lengths for arranging the choir.


Let $p$, $q$, $r$, and $s$ be real numbers such that $|p-q|=1$, $|q-r|=5$, and $|r-s|=6$. What is the sum of all possible values of $|p-s|$?


Given a triangle with two sides measuring 18 feet and 24 feet, and the included angle between them being 60 degrees, find the altitude to the longer side.


Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $100$, and $BC=12$. Find the volume of the tetrahedron.

Solution: \boxed{V = \frac{1250\sqrt{2}}{3}}


Solve for $z$ in the following equation: $2 - 3iz = -4 + 5iz$ (where $i^2 = -1$). Simplify your answer as much as possible.

Solution: \boxed{z = -\frac{3i}{4}}


A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$). What is the probability that the top card is a face card (Jack, Queen, King) and the second card is a number card (Ace through 10)?


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


Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 9.$  Find the minimum value of
\[\frac{a^2 + b^2}{a + b} + \frac{a^2 + c^2}{a + c} + \frac{b^2 + c^2}{b + c}.\]


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


Evaluate the determinant of the following matrix:
\[
\begin{vmatrix}
x & x & y \\
x & y & x \\
y & x & x
\end{vmatrix}.
\]

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


What is the base-10 integer 1357 when expressed in base 7?


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


Consider a regular nonagon (a 9-sided polygon). What is the probability that two randomly chosen diagonals intersect inside the nonagon?


We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules:
1. $g(b)=3b+1$ if $b$ is odd
2. $g(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $g$?


Rationalize the denominator of the expression \(\frac{3}{4\sqrt{6} + 3\sqrt{7}}\) and express your answer in the form \(\frac{A\sqrt{B} + C\sqrt{D}}{E}\) where \(B < D\), the fraction is in lowest terms, and all radicals are in simplest radical form. Calculate \(A+B+C+D+E\).


Two different numbers are selected simultaneously and randomly from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. What is the probability that the positive difference between the two numbers is exactly $3$?


A circle has an equilateral triangle and a rectangle inscribed in it. They share a common vertex. The rectangle has one angle of 60 degrees. Find the measure of the angle indicated by the question mark at the shared vertex.

[asy]
import markers; defaultpen(linewidth(0.8));

pair A,B,C,D,E,F,G;

draw(unitcircle);

A=(-1,0);
B=(0,-1);
C=(1,0);
D=(0,1);

draw(A--B--C--D--A);

E=(-0.9,-0.45);
F=(0.9,-0.45);

draw(D--E--F--D);

G=(-0.76,-0.23);

markangle(Label("?"),n=1,radius=10,D,G,A,marker(stickframe(n=0),true));

label("$A$",(-0.5,-0.34));
label("$B$",(0.5,-0.34));
label("$C$",B,S);

[/asy]


Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to color the dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.)


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


How many degrees are in the measure of the smaller angle formed by the hour-hand and minute-hand of a clock when it is 2:30?


A $10 \times 10$ checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 7 black squares, can be drawn on this checkerboard?


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


If $x = \frac{2}{3}$ and $y = \frac{9}{2}$, find the value of $\frac{1}{3}x^4y^5$.


A circle centered at $O$ is circumscribed about $\triangle ABC$. The angle $\angle AOB = 120^\circ$ and $\angle BOC = 90^\circ$. What is the measure of $\angle BAC$, in degrees?


You have $4$ red shirts, $4$ green shirts, $4$ blue shirts, $7$ pairs of pants, $6$ green hats, $6$ red hats, and $6$ 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?


Determine the positive integer $M$ for which $$ 12^2 \times 45^2 = 15^2 \times M^2 .$$


Let $a$, $b$, $c$, and $d$ be nonnegative numbers whose sum is 200. Find the largest possible value of
\[ab + ac + ad.\]


Pirate Pete shares his treasure with Pirate Paul in a similar way as before. Pete first says, ``One for me, one for you,'' giving himself one coin and starting Paul's pile with one coin. Then Pete says, ``Two for me, and two for you,'' giving himself two more coins but making Paul's pile two coins in total. This pattern continues until Pete gives himself $x$ more coins but makes Paul's pile $x$ coins in total. At this point all of the coins have been distributed, and Pirate Pete has exactly five times as many coins as Pirate Paul. How many gold coins do they have in total?


Let \(y_1, y_2, y_3\) be positive real numbers such that \(3y_1 + 2y_2 + y_3 = 30\). Find the smallest possible value of
\[ y_1^2 + y_2^2 + y_3^2. \]


What is the least five-digit positive integer which is congruent to 7 (mod 12)?


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


Let $x$ and $y$ be two positive real numbers such that $x + y = 50$. Find the ordered pair $(x, y)$ for which $x^7 y^3$ is maximized.

Solution: \boxed{(35, 15)}


The base of a triangular paper $ABC$ is $15\text{ cm}$ long. The paper is folded along a crease $DE$ parallel to the base $AB$. The area of the triangle that projects below the base is $25\%$ of the area of triangle $ABC$. Calculate the length of $DE$.

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


What is the smallest positive integer $n$ such that $5n$ is a perfect fourth power and $4n$ is a perfect cube?


A large square quilt block is constructed from sixteen smaller unit squares arranged in a 4x4 grid. Eight of these unit squares are divided into diagonal halves, with one of the two halves shaded. Four whole unit squares are completely shaded, and four are divided into quarters, with opposite quarters shaded. What fraction of the total quilt is shaded? Express your answer as a common fraction.


Find the distance between the planes $3x - y + z - 3 = 0$ and $6x - 2y + 2z + 4 = 0$.


Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of $4000\text{ km}$ from $A$ and with $B$ a distance of $4500\text{ km}$ from $A$. To take a bus, it costs Piravena $\$0.20$ per kilometer. To take an airplane, it costs her a $\$150$ booking fee, plus $\$0.15$ per kilometer. To begin her trip, she took a bus from $A$ to $B$. Determine the cost to travel by bus from $A$ to $B$.


Consider the line $y = 2x + 5$ parameterized in the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} s \\ -2 \end{pmatrix} + t \begin{pmatrix} 3 \\ m \end{pmatrix}.\] Determine the ordered pair $(s,m).$

Solution: \boxed{\left(-\frac{7}{2}, 6\right)}


What is the value of $$\frac{1}{2}\times4\times\frac{1}{8}\times16\times\frac{1}{32}\times64\times\frac{1}{128}\times256\times\frac{1}{512}\times1024\times\frac{1}{2048}\times4096?$$


What is the smallest five-digit positive integer which is congruent to 11 (mod 14)?


Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 8.$ Find the maximum value of
\[\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.\]


In the diagram below, $BC$ is 12 cm, and the angle $C$ is $30^\circ$. In square centimeters, what is the area of triangle $ABC$?

[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(12,0)--(12, 6)--cycle);
draw((12,0)--(12,6),dashed);
label("12 cm",(6,0),S);
label("30^{\circ}",(12,1),NE);
label("$A$",(12,6),N);
label("$B$",(0,0),SW);
label("$C$",(12,0),SE);
[/asy]


Find all real numbers $x$ such that \[2 \le \frac{x}{2x-1} < 5.\] (Give your answer in interval notation.)

Solution: \boxed{(\frac{5}{9}, \frac{2}{3}]}


In how many ways can 8 people be seated in a row of chairs if two of the people, Alice and Bob, must not sit next to each other, and Charlie has to sit at one end of the row?


The curve \(y = \sin x\) intersects with the line \(y = \sin 60^\circ\). Determine the ordered pair \((p, q)\) of relatively prime positive integers representing the ratios of the segment lengths formed by these intersections on one period of \(y = \sin x\).

---

Solution: \boxed{(1, 5)}


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


A cylinder has a height of `15` and a radius of `5`. If only 75% of the lateral surface area of the cylinder is painted, determine the painted area. Also include the areas of both ends in your total calculation.


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


Solution: \boxed{98x^7 - 266x^{13} = 14x^7(7 - 19x^6)}


How many different counting numbers will each leave a remainder of 3 when divided into 73?


On the complex plane, the parallelogram formed by the points 0, \( z \), \( \frac{1}{z} \), and \( z + \frac{1}{z} \) has area \( \frac{12}{13} \). If the real part of \( z \) is positive, find the smallest possible value of \( \left| z + \frac{1}{z} \right| \) and compute the square of this value.


What is the sum of all the integers between -15 and 5?


Tom has recorded times, in seconds, of 92, 86, 101, 95, and 90 for running 100 meters. After his sixth attempt, he managed to bring his median time down to 93 seconds. What was his time, in seconds, for his sixth attempt?


Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p  +q + r + s = 10$ and
\[pq + pr + ps + qr + qs + rs = 20.\]Find the largest possible value of $s.$

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


Mark has $\frac{5}{8}$ of a dollar and Carolyn has $\frac{2}{5}$ of a dollar. How many dollars do they have altogether? (Give your answer in fraction and as a decimal.)


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, and $k$ is a power of 2. How many total divisors does $g(1005)$ have?


Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{10}{x - 10} = x.
\]


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


An equilateral triangle has two vertices at $(0,5)$ and $(8,5)$. If the third vertex is in the third quadrant, what is the y-coordinate? Express your answer in simplest radical form.

Solution: \boxed{5 - 4\sqrt{3}}


Fully factor the following expression: $3x^2 - 75$.


In triangle $ABC$, angle $ACB$ is 40 degrees, and angle $CBA$ is 60 degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of the diameter which goes through $A$. Find the angle $DAE$, in degrees.


The Gauss family has four boys aged $8,$ a girl aged $16,$ and a boy aged $17.$ What is the mean (average) of the ages of the children?


The Lopez family has grown and now includes three children. They still travel in their family car that has a driver's seat, a front passenger seat, and three seats in the back. Either Mr. Lopez, Mrs. Lopez, or their eldest child must sit in the driver's seat. How many different seating arrangements are possible now?


Each pack of hot dogs contains 12 hot dogs (no buns), but each pack of hot dog buns contains 10 buns. After a barbecue, Phil finds that he has 6 hot dogs left over. What is the second smallest number of packs of hot dogs he could have bought?


The number $1057$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q+r$?


In convex quadrilateral $PQRS$, side $\overline{RS}$ is perpendicular to diagonal $\overline{PQ}$, and side $\overline{PQ}$ is perpendicular to diagonal $\overline{RS}$. Side lengths are $RS = 75$, and $PQ = 36$. A line through $Q$ perpendicular to side $\overline{PR}$ intersects diagonal $\overline{PQ}$ at a point $T$ such that $PT = 12$. Find the length of segment $QT$.


A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 35 seconds. Find the probability that the color changes during a random four-second interval observed by Leah.


Consider the polynomial $Q(x) = 3x^3 + dx^2 + ex + f$. The polynomial has the property that the reciprocal of the sum of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. Find the value of $e$.


A square sheet of paper has each of its four corners folded into smaller isosceles right triangles which are then cut out. If the side length $AB$ of the original square is $16$ units, and each corner triangle is made from folds such that the legs of the triangles are equal to half the side length of smaller squares formed inside the original square, calculate the total area of the four triangles removed.


Chloe wants to buy a pair of sneakers that costs $ \$45.50$. She checks her wallet and finds she has four $ \$10$ bills, ten quarters, and some dimes and nickels. What is the minimum number of dimes and nickels combined that must be in her pile so she can pay for the sneakers?


For all composite integers $n$ that are multiples of $4$, what is the largest integer that always divides into the difference between $n^3$ and $n^2$?


Last year, Isabella took 8 math tests and received 8 different scores, each an integer between 91 and 100, inclusive. After each test, she noted that the average of her test scores was an integer. Her score on the seventh test was 97. What was her score on the eighth test?


Fifty more than the square of a number is the same as the square of the quantity that is 10 less than the number. What is the number?


The arithmetic mean of eleven numbers is 32. If three numbers $x$, $y$, and $z$ are added to this list, the mean of the fourteen-member list becomes 45. What is the mean of $x$, $y$, and $z$?


Find the $2023^{\text{th}}$ digit past the decimal point in the decimal expansion of $\dfrac{7}{22}$.


In a specific hyperbola, the center is at $(1, 1),$ one focus is at $(1, 7),$ and one vertex is at $(1, -2).$ The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\]
Find $h + k + a + b.$

Solution: \boxed{5 + 3\sqrt{3}}


Determine the area enclosed by the region defined by the equation $x^2 + y^2 + 10x + 20y - 75 = 0$.


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 other bears a number $b$ that is not equal to $a$. Calculate the value of $q/p$.


A circular cake with a diameter of $18\text{ cm}$ is cut into four equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$?


A line is parameterized by a parameter $t$, so that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$, and the vector on the line at $t = 4$ is $\begin{pmatrix} 5 \\ -7 \end{pmatrix}.$ Find the vector on the line at $t = -3$.

Solution: \boxed{\begin{pmatrix} -2 \\ 21 \end{pmatrix}}


If $x - 4$ is a factor of $dx^4 + 11x^3 + 5dx^2 - 28x + 72$, find the constant $d$.


A square is divided into four congruent rectangles. If the perimeter of each rectangle is 40 inches, what is the perimeter of the square?

Solution: \boxed{64 \text{ inches}}


Calculate $\begin{pmatrix} 4 \\ -9 \end{pmatrix} \times 2 + \begin{pmatrix} -7 \\ 15 \end{pmatrix}.$

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


For how many values of $c$ in the interval $[0, 500]$ does the equation \[5 \lfloor x \rfloor + 3 \lceil x \rceil = c\] have a solution for $x$?


Jimmy finds out that 9 pears weigh the same as 6 peaches. If Jimmy has 36 pears, how many peaches would Jimmy need to equal the weight of his 36 pears?


The legs of a right triangle have lengths $\log_9 512$ and $\log_3 64$. If the length of the hypotenuse is $h$, compute $9^h$.


At a different university, the department of natural sciences consists of the physics, chemistry, and biology departments. Each department has three male and three female professors. A committee is to be formed with four men and four women. Additionally, the committee must include:
- At least one professor from each gender in each department,
- Exactly three professors from any one department.
Find the number of possible committees that can be formed following these conditions.


Solution: \boxed{-1 + 7i}


A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. Determine how many relatively prime dates are in July, considering July has 31 days.


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


Seventy percent of a cruise ship's passengers are women, and fifteen percent of those women are in luxury suites. What is the number of women in luxury suites if the cruise ship is carrying 250 passengers?


A circle with the equation $x^2 + y^2 - 8x - 6y + 24 = 0$ is inscribed inside a square that has a pair of sides parallel to the x-axis. What is the area of this square?


Find the maximum value of
\[\frac{\sin^4 x + \cos^4 x + 2}{\sin^2 x + \cos^2 x + 2}\]
for all real values $x$.


A bag contains 4 blue, 3 green, and 5 red chips. If the 12 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 4 blue chips are drawn consecutively, the 3 green chips are drawn consecutively, and the 5 red chips are drawn consecutively, but not necessarily in the blue-green-red order? Express your answer as a common fraction.


Find the number of positive integers less than or equal to $1200$ that are neither $5$-nice nor $6$-nice.


Divide the product of the first four positive composite integers by the product of the next four composite integers. Express your answer as a common fraction.


What is the value of $x$ if a cube's volume is $8x$ cubic units and its surface area is $2x$ square units?


The members of a choir are arranged in a rectangular formation. When they are arranged in 10 rows, there are 4 positions unoccupied in the formation. When they are arranged in 11 rows, there are 5 positions unoccupied. How many members are in the choir if the membership is between 200 and 300?


If
\[\tan y = \frac{3cd}{c^2 - d^2},\]where \( c > d > 0 \) and \( 0^\circ < y < 90^\circ,\) find \(\sin y\) in terms of \(c\) and \(d\).

Solution: \boxed{\sin y = \frac{3cd}{\sqrt{c^4 + 7c^2d^2 + d^4}}}


Suppose $f(x) = \sqrt[4]{\frac{2x+6}{5}}$. Determine the value of $x$ where $f(3x) = 3(f(x))$.


A special deck of 60 cards has 15 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Prince, Princess) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. The deck is randomly arranged. What is the probability that the top card is a $\diamondsuit$?


In a right triangle $LMN$, with $\angle M = 90^\circ$ and $\sin N = \frac{3}{5}$. The length of side LM is 15 units. Find the length of side LN.


In square $ABCD$, two right-angled isosceles triangles are cut off from two opposite corners, $A$ and $C$, each with legs of length $x$. This leaves an irregular octagon. The combined area of the triangles cut off is $100 \text{ cm}^2$. Find the length of the shortest side of octagon, assuming $AB = 20$ cm.

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


Let $b_1, b_2, \dots$ be a sequence defined by $b_1 = b_2 = 1$ and $b_{n+2} = b_{n+1} + b_n$ for $n \geq 1$. Find \[
\sum_{n=1}^\infty \frac{b_n}{3^{n+1}}.
\]


What is the modulo $7$ remainder of the sum $1+2+3+4+5+6+7+\ldots+132+133+134+135?$


What is the sum of all the positive divisors of $117$ and how many of these divisors are prime numbers?


The 6-digit number $142\,85\square$ is a multiple of 6. Which digit is represented by $\square$?


For how many integer values of $n$ between 1 and 569 inclusive does the decimal representation of $\frac{n}{570}$ terminate?


In the diagram, lines $l\|k$. Determine the number of degrees in $\angle SRQ$. The diagram setup includes:
- Lines $l$ and $k$ are parallel.
- Line $RQ$ is perpendicular to both $l$ and $k$.
- The angle $\angle RSQ$ measures $120^\circ$.
[asy]
draw((-.4,-.4)--(2,2)--(2,-.4));
draw((-.5,0)--(3,0),Arrows);
draw((-.5,1)--(3,1),Arrows);
draw((1.9,0)--(1.9,.1)--(2,.1));
label("$S$",(1,1),NNW);
label("$R$",(2,2),N);
label("$Q$",(2,1),NE);
label("$l$",(3,1),E);
label("$k$",(3,0),E);
label("$120^{\circ}$",(1,1),SSE);
[/asy]


From a set of the first 12 positive integers, $\{1,2,3,\ldots,12\}$, eight distinct integers are picked at random. What is the probability that, among those selected, the third smallest is $5$?


Consider the ellipse defined by the equation
\[
\sqrt{(x-4)^2 + (y-5)^2} + \sqrt{(x+6)^2 + (y+9)^2} = 22.
\]
Determine the distance between its foci.


Laura is training for a marathon and decides to include biking in her workout. She bikes 25 miles at \(3x + 2\) miles per hour, then spends five minutes transitioning to her running gear before she runs 8 miles at \(x\) miles per hour. Her entire workout, including the transition, lasts 140 minutes. How fast did Laura run, to the nearest hundredth of a mile per hour?


Five of the seven numbers 1234, 1567, 1890, 2023, 2147, 2255, and 2401 have a mean (average) of 2020. What is the mean (average) of the other two numbers?


There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are from Spain. The gold, silver, and bronze medals are awarded to the top three finishers. In how many ways can the medals be awarded if at most two Spaniards get medals?


A cylinder with a height of `15` and a radius of `5` is topped with a cone that has the same radius and a height of `8`. Calculate the total surface area of this combined shape.

Solution: \boxed{175\pi + 5\pi \sqrt{89}}


In rectangle $ABCD$, $AB = 10$ cm, $BC = 14$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fifth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form.


Let $p, q, r, s, t, u$ be positive real numbers such that $p+q+r+s+t+u = 11$. Find the minimum value of
\[
\frac{1}{p} + \frac{9}{q} + \frac{25}{r} + \frac{49}{s} + \frac{81}{t} + \frac{121}{u}.
\]


Quantities \( a \) and \( b \) vary inversely. Initially, when \( a \) is 800, \( b \) is 0.5. Later, due to some external changes, the product of \( a \) and \( b \) increases by 50%. What is the value of \( b \) when \( a \) is 1600?


Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:
\[x = \sqrt{31} + \frac{105}{{\sqrt{31}+\frac{105}{{\sqrt{31}+\frac{105}{{\sqrt{31}+\frac{105}{{\sqrt{31}+\frac{105}{x}}}}}}}}}.\]


Below is the graph of \( y = a \sin(bx + c) + d \) for some positive constants \( a, b, c, \) and \( d \). Find \( a \).

[asy]
import TrigMacros;

size(400);

real f(real x)
{
    return 4*sin(3*x + pi) + 3;
}

draw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-3,7,pi/2,1);
layer();
rm_trig_labels(-5,5, 2);

label("$1$", (0,1), E);
label("$3$", (0,3), E);
label("$5$", (0,5), E);
label("$7$", (0,7), E);
label("$-1$", (0,-1), E);
label("$-3$", (0,-3), E);
[/asy]


Let \(D\) be the circle with equation \(x^2 + 2x - 8y - 7 = -y^2 - 6x\). Find the center \((a, b)\) and radius \(r\) of \(D\), and compute the value of \(a + b + r\).


A club has between 300 and 400 members. The members gather every weekend and are divided into eight distinct groups. If two members are absent, the groups can all have the same number of members. What is the sum of all possible numbers of members in the club?


Let \[f(x) = \left\{
\begin{array}{cl}
x + 1 & \text{if } x \le 0, \\
3x - 5 & \text{if } x > 0.
\end{array}
\right.\] How many solutions does the equation $f(f(x)) = 7$ have?


A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 4-primable positive integers are there that are less than 1000?


Given the volume of a cylinder is given by the formula $V=\pi r^2h$, increasing the radius by $3$ units and the height by $3$ units increased the volume by $z$ cubic units. If increasing either the radius by $3$ units or the height by $3$ units increases the volume by the same amount, $z$, find the original radius if the original height is $3$.

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


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

Solution: \boxed{-1 + i}


I have two 20-sided dice. The first die has 5 maroon sides, 6 teal sides, 7 cyan sides, 1 sparkly side, and 1 silver side. The second die has 4 maroon sides, 6 teal sides, 7 cyan sides, 1 sparkly side, and 2 silver sides. If I roll both dice, what is the probability they come up showing the same color?


Determine the number of ways to arrange the letters of the word BANANA where no two 'A's are adjacent.


Lisa has 12 friends and 50 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least 2 marbles and no two friends receive the same number of marbles?


Express \( 7.\overline{036} \) as a common fraction in lowest terms.


At constant temperature, the pressure of a gas is inversely proportional to its volume. A sample of helium is placed in a 3-liter container at a pressure of 6 kPa. If the gas is then transferred to a 4.5-liter container at the same temperature, what will be the new pressure of the gas in kPa?

Solution: \boxed{4 \text{ kPa}}


Find \( \cot G \) in the right triangle shown below.

[asy]

pair H,F,G;

H = (0,0);

G = (12,0);

F = (0,5);

draw(F--G--H--F);

draw(rightanglemark(F,H,G,20));

label("$H$",H,SW);

label("$G$",G,SE);

label("$F$",F,N);

label("$13$",(F+G)/2,NE);

label("$12$",G/2,S);

[/asy]


If $\sum_{n = 0}^{\infty} \cos^{2n} \theta = 12$, what is the value of $\cos 2\theta$?


Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\sin^4 x - \cos^4 x = \frac{1}{\cos x} - \frac{1}{\sin x}.\]