If we let $g(n)$ denote the sum of the cubes of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 500$ and $g(i) = 1 + \sqrt{i^3} + i^3$?


Find the remainder when $7\cdot11^{24} + 2^{24}$ is divided by 12.


For which integer $a$ does $x^2 - x + a$ divide $x^{12} + x + 72$?


Let $\mathbf{v}_0$ be a vector. The vector $\mathbf{v}_0$ is projected onto $\begin{pmatrix} 2 \\ 2 \end{pmatrix}$, resulting in the vector $\mathbf{v}_1$. The vector $\mathbf{v}_1$ is then projected onto $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$, resulting in the vector $\mathbf{v}_2$. Find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2$.

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


What is the product of the numerator and the denominator when $0.\overline{0126}$ is expressed as a fraction in lowest terms?


What is the base six product of the numbers $132_{6}$ and $14_{6}$?


Let $O$ be the origin. Determine a scalar $m$ such that for any points $A,$ $B,$ $C,$ and $D,$ if the following condition holds:
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 7 \overrightarrow{OC} + m \overrightarrow{OD} = \mathbf{0},\]
then the four points $A,$ $B,$ $C,$ and $D$ are coplanar.


What is the sum of the squares of the lengths of the **medians** of a triangle whose side lengths are **13**, **14**, and **15**?


Evaluate $c$ if $x^2 - cx + 6 > 0$ only when $x \in (-\infty, -2) \cup (3,\infty)$.


If the quadratic $x^2 - 8mx + 15m$ has exactly one real root, find the positive value of $m$.


Consider sequences consisting entirely of $A$'s and $B$'s, where every run of consecutive $A$'s has an even length, and now every run of consecutive $B$'s must also have an even length. Determine the number of such sequences having a total length of 16.


Triangle $PQR$ has side lengths $PQ=6$, $QR=8$, and $PR=9$. Two bugs 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 $QS$?


A large sphere has a volume of $500\pi$ cubic units. A smaller sphere has a volume which is $25\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction.


Let $A = (-4, 0),$ $B=(-3,2),$ $C=(3,2),$ and $D=(4,0).$ Suppose that point $P$ satisfies \[PA + PD = PB + PC = 10.\]Find the $y-$coordinate of $P,$ and express it in its simplest form.


What is the coefficient of $x^2$ when $$3x^4 - 2x^3 + 4x^2 - 5x + 2$$ is multiplied by $$x^3 - 4x^2 + 3x + 6$$ and the like terms are combined?


Find the number of positive integers \( n \) that satisfy
\[
(n - 1)(n - 3)(n - 5) \dotsm (n - 99) < 0.
\]


The arithmetic progressions $\{7, 14, 21, 28, \ldots\}$ and $\{8, 20, 32, 44, \ldots \}$ have some common values. What is the largest value less than 500 that they have in common?


You have a picture with dimensions \(x\) and \(y\) (in inches), both of which are integers greater than one. You plan to place this picture in a frame of dimensions \(2x + 5\) by \(y + 4\). If the area of the frame (including both the picture and the region around it) equals 60 square inches, what is the area of the picture?


A binary operation $\diamondsuit$ is defined such that for any nonzero real numbers $a, b, c$, the operation satisfies $a \diamondsuit (b \diamondsuit c) = (a \diamondsuit b) \cdot c$ and $a \diamondsuit a = 1$. Find the value of $x$ for which $4050 \diamondsuit (9 \diamondsuit x) = 150.$


Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each having integer coefficients.


The mean (average) of 4, 16, and 20 is equal to the mean (average) of double 4 and $z$. What is the value of $z$?


A prism has 21 edges. How many faces does this prism have?


What is the arithmetic mean of the integers from -3 through 6, inclusive? Express your answer as a decimal to the nearest tenth.


A quadrilateral has vertices at $(0,0)$, $(2,5)$, $(5,5)$, and $(6,2)$. Find its perimeter and express it in the form of $c\sqrt{p}+d\sqrt{q}$, where $c$, $d$, $p$, and $q$ are integers. Determine $c + d$.


Solution: \boxed{4(x - 6)(x + 6)}


Within a cube of side length $a$, a regular tetrahedron is formed by connecting four vertices of the cube such that no two of these vertices belong to the same face. Determine the ratio of the volume of the tetrahedron to the volume of the cube.


Let $a$, $b$, $c$, $d$, $e$, and $f$ be positive integers with $a+b+c+d+e+f=3012$. Let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$, $d+e$, $e+f$, $a+c$, and $b+d$. What is the smallest possible value of $M$?


Determine the value of $\left(1 - \frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \dotsm \left(1-\frac{1}{100}\right)$. Express your answer as a common fraction.


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


Let $n$ be a positive integer. If $b\equiv (3^{2n+1}+5)^{-1}\pmod{9}$, what is the remainder when $b$ is divided by $9$?


I possess 8 shirts and 8 hats, available in the colors tan, black, blue, gray, white, yellow, red, and green. I own 4 pairs of pants in the colors tan, black, blue, and gray. I refuse to wear an outfit where the shirt and the pants are the same color, although the hat can match the color of either the shirt or pants. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?


Simplify the expression $\frac{3^{n+4} - 3(3^n)}{3(3^{n+3})}$ and express your answer as a common fraction.


In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The lengths of bases $BC$ and $AD$ are $10$ and $20$ respectively. The distance between the bases is $5$ units. Calculate the perimeter of trapezoid $ABCD$.

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


The Aquaflow system pumps water at a rate of 360 gallons per hour. How long will it take to pump a total of 540 gallons if after the first 30 minutes, the rate is increased to 480 gallons per hour?

Solution: \boxed{75 \text{ minutes}}


Compute \[ \left\lfloor \dfrac{2010^3}{2007 \cdot 2008^2} - \dfrac{2008^3}{2007^2 \cdot 2010} \right\rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.


Determine the value of $y$ that satisfies $\sqrt[3]{y\sqrt{y^5}} = 4$.


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


Mila constructs a sculpture by stacking four spherical clay balls of radii 1 inch, 4 inches, 6 inches, and 3 inches. Assuming that all clay balls are perfectly spherical, calculate the total volume of clay used in cubic inches. Express your answer in terms of $\pi$.


It is now 3:15:30 PM, as displayed on a 12-hour digital clock. In 174 hours, 58 minutes, and 16 seconds, the time will be $A:B:C$. What is the value of $A+B+C$?


Coach Grunt is preparing the 7-person starting lineup for his basketball team, the Grunters. The team has 15 players in total. Three players, Ace, Zeppo, and Chuck, are league All-Stars and must be in the starting lineup. How many different starting lineups can be made if at least 2 guards must be included, and there are 5 guards on the team?


I have a French exam with 600 vocabulary words and the exam grade is based on the percentage of these words that I recall correctly. I'm aware that I might misremember 10% of the words I learn. What is the least number of words I should learn to ensure I get at least 90% on the exam?


If the nine-digit number $8542m7618$ is divisible by $11$, what is $m$?


Evaluate the expression $\left(\frac{1}{3}\right)^{6} \cdot \left(\frac{5}{3}\right)^{-4}$.


A grocer creates a display of cans where the top row has one can, and each subsequent row has two more cans than the row above it. The display contains 225 cans. Additionally, each can weighs 5kg. How many rows does the display contain and what is the total weight supported by the display?

Solution: \boxed{1125 \text{ kg}}


Solve for $x:\ \log_2 x + \log_8 x = 9.$


Suppose $a$ and $b$ are positive integers such that $\gcd(a,b)$ is divisible by exactly $8$ distinct primes and $\mathop{\text{lcm}}[a,b]$ is divisible by exactly $32$ distinct primes.

If $a$ has fewer distinct prime factors than $b$, then $a$ has at most how many distinct prime factors?


A $3\times 4$ rectangle and a $4\times 5$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of this square?


The parabola $y=ax^2+bx+c$ has vertex $(q, q+1)$ and $y$-intercept $(0,-2q-1)$, where $q \neq -\frac{1}{2}$. What is $b$?

Solution: \boxed{6 + \frac{4}{q}}


In a triangle, two of the side lengths are 9 and 10, and the angle between them is $150^\circ$. Find the length of the third side.

Solution: \boxed{\sqrt{181 + 90\sqrt{3}}}


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

Solution: \boxed{\left(\frac{\pi}{12}, \frac{5\pi}{12}\right)}


How many integers from 1 to 9 are divisors of the five-digit number 91,837?


Consider the geometric sequence $4$, $\dfrac{12}{3}$, $\dfrac{36}{9}$, $\dfrac{108}{27}$, $\ldots$. Find the ninth term of the sequence. Express your answer as a common fraction.


It is currently $4\!:\!45\!:\!00 \text{ p.m.}$ What time will it be in $9999$ seconds?

Solution: \boxed{7\!:\!31\!:\!39 \text{ p.m.}}


Let $\mathbf{p}$ be the projection of vector $\mathbf{v}$ onto vector $\mathbf{w}$, and let $\mathbf{q}$ be the projection of $\mathbf{p}$ back onto $\mathbf{v}$. If the ratio of the magnitudes of $\mathbf{p}$ to $\mathbf{v}$ is $\frac{2}{3}$, determine the ratio $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}$.


A book with 73 pages numbered from 1 to 73 has its pages renumbered in reverse, from 73 to 1. For how many pages do the new page number and old page number share either the same units digit or consecutive units digits?


The side of a square has a length of $(x-5)$. A rectangle has a length of $(x - 4)$ and a width of $(x + 5)$. If the area of the rectangle is three times the area of the square, find the sum of the possible values of $x$.


Given are rows \(n-1\), \(n\), and \(n+1\) of Pascal's triangle, where \(n = 2021\). Compute the difference between the sum of the ratios of consecutive row elements:
\[
\sum_{i = 0}^{n} \frac{b_i}{c_i} - \sum_{i = 0}^{n-1} \frac{a_i}{b_i}.
\]


A decagon is inscribed in a square so that the vertices of the decagon divide each side of the square into five equal segments. The perimeter of the square is 150 centimeters. What is the number of square centimeters in the area of the decagon?


Find all solutions to the equation $\sqrt{5-4z+1} = 7$. Express your answer as a common fraction.


Suppose $543_7=cde_{10}$, where $c$, $d$, and $e$ represent base-10 digits. Find $\frac{c\cdot d\cdot e}{30}$.


An ellipse has foci at $(0, 3)$ and $(4, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair.

Solution: \boxed{\left(\frac{56}{11}, 0\right)}


Determine how many five-letter words can be formed such that they begin and end with the same letter and the third letter is always a vowel.


Solution: \boxed{1 - \frac{1}{\log_3 101}}


Altitudes \( \overline{AD} \) and \( \overline{BE} \) of an acute triangle \( \triangle ABC \) intersect at point \( H \). If \( HD = 8 \) and \( HE = 3 \), compute \( (BD)(DC) - (AE)(EC) \).


A $4$-foot by $16$-foot rectangular piece of plywood will be cut into $8$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece?


Find the last two digits of the following sum: $$3! + 6! + 9! + \cdots + 30!$$


In how many ways can the digits of 1225 be arranged to get a four-digit multiple of 5?


A circle with center \(A\) and radius 4 inches is tangent at \(C\) to a circle with center \(B\). Assume that point \(B\) is located on the smaller circle. What is the area of the shaded region formed by these circles? Express your answer in terms of \(\pi\).


Let $a,$ $b,$ $c,$ $d$ be nonzero integers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} 9 & 0 \\ 0 & 9 \end{pmatrix}.\]
Find the smallest possible value of $|a| + |b| + |c| + |d|.$


Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{8}$ and $\frac{\pi}{12}$ radians, respectively, with the positive $x$-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2$. Define $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$. Given that $l$ is the line $y=\frac{1}{3}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.


The amount of algae in a small lake doubles in size every day. By the end of the 20th day of a particular month, the lake is completely covered with algae. Determine on which day of that month was only $12.5\%$ of the lake covered by algae.


The orthocenter of triangle $ABC$ divides altitude $\overline{CF}$ into segments with lengths $HF = 8$ and $HC = 18.$  Calculate $\tan A \tan B.$


What is the greatest common divisor of $765432$ and $654321$?


Consider the sequence defined by $E_n = 3^{3^n} + 1$. What is the units digit of $E_{1000}$?


Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+15|$ and $y=x+2$.


A ball is dropped from 500 meters high and bounces back up to 60% of the height it fell. Find the number of bounces after which the ball first reaches a maximum height less than 5 meters. Assume the ball stops bouncing once it reaches a height less than 0.1 meter.


The integer $x$ has 18 positive factors. The numbers 18 and 21 are factors of $x$. What is the smallest possible value of $x$?


Solve the equation $9^{x+5} = 10^x$ and express $x$ in the form $x = \log_b 9^5$. What is $b$?


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


A sphere is inscribed in a cube. The volume of the cube is 343 cubic inches. Calculate the volume of the inscribed sphere. Express your answer in terms of $\pi$.


The square root of $3x + 1$ is greater than 4 and less than 5. How many integer values of $x$ satisfy this condition?


A square and a regular nonagon are coplanar and share a common side $\overline{AD}$, as illustrated. Calculate the degree measure of the exterior angle at vertex $BAC$, located at a vertex of the nonagon adjacent to the shared side $\overline{AD}$. Express your answer as a common fraction.


At constant temperature, the pressure of a sample of nitrogen is inversely proportional to its volume. A 12 liter container originally has nitrogen at 3 kPa. If this nitrogen is then transferred to a 30 liter container, what would be the new pressure in kPa? Additionally, if there is another sample of helium in a 15 liter container at 4 kPa, and it is also transferred to the 30 liter container, what is the total pressure inside the 30 liter container?

Solution: \boxed{3.2 \text{ kPa}}


Suppose \( x \), \( y \), and \( z \) are positive numbers satisfying:
\[
x^2 \cdot y = 2, \\
y^2 \cdot z = 4, \text{ and} \\
z^2 / x = 5.
\]
Find \( x \).


Jo adds up all the positive integers from 1 to 50. Kate does a similar thing with the first 50 positive integers; however, she first rounds every integer to its nearest multiple of 5 (rounding 2.5s up) and then adds the 50 values. What is the positive difference between Jo's sum and Kate's sum?


A right circular cone is 12 inches tall with a 1-inch radius at the opening. The interior of this cone is filled with ice cream. To add more appeal, a hemisphere of ice cream with a 1-inch radius exactly covers the opening of the cone. On top of the hemisphere, there is a smaller cone, with height 2 inches and the same radius that perfectly caps the hemisphere. What is the total volume of the ice cream? Express your answer in terms of $\pi$.


What is the smallest positive value of $n$ so that the equation $12x^2 - nx + 576 = 0$ has integral solutions?


Two of the vertices of a cube are to be chosen at random. What is the probability that they will be the endpoints of a space diagonal of the cube?

Solution: \boxed{\frac{4}{28} = \frac{1}{7}}


What is the remainder when $2024 \cdot 3047$ is divided by $800$?


Find the sum of the roots of the equation
\[(3x^3 - 6x^2 + 9x - 27)(4x^3 + 8x^2 - 16x + 32) = 0.\]


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


A function $f$ is defined by $f(z) = (5 + i)z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers. Suppose that $f(1)$ and $f(i)$ are both real. Find the smallest possible value of $|\alpha| + |\gamma|$.


Let $p$ and $q$ be the roots of the quadratic equation $x^2 - 6x + 8 = 0.$ Calculate
\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \]


Find the number of triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x &= 2023 - 2024 \operatorname{sign}(y + z), \\
y &= 2023 - 2024 \operatorname{sign}(x + z), \\
z &= 2023 - 2024 \operatorname{sign}(x + y).
\end{align*}


Amanda, Ben, Carlos, and Diana share a sum of money. Their portions are in the ratio of 1:2:7:3, respectively. If Amanda's portion is $\$$30, what is the total amount of money shared?


Calculate the value of the alternating sequence: $2 - 5 + 8 - 11 + 14 - \cdots - 59 + 62 - 65 + 68$.


Solve the inequality
\[\frac{x - 3}{(x - 1)^2} < 0.\]

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


Here are two functions:
$$\begin{array}{ccc}
f(x) & = & 4x^2 - 3x + 5 \\
g(x) & = & x^2 - mx - 8
\end{array}$$
If $f(5) - g(5) = 15$, what is the value of $m$?


A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $210$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$


Factor the expression \(5x^2(x-2) - 9(x-2)\).

Solution: \boxed{(x-2)(5x^2 - 9)}


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

Solution: \boxed{-8x + 1}


Jeff's six assignment scores include 89, 92, 88, 95, 91, and one unknown score. If Jeff wants to achieve an average score of 93, what must the unknown score be?


What is the greatest common divisor of 1549 and 1023?


Two historians, Alice and Bob, were both born in the last 400 years. Each lives to be 80 years old. Each historian is equally likely to be born at any point during those 400 years. What is the probability that they lived as contemporaries for any length of time?


A quadrilateral has vertices at $(1,2)$, $(5,6)$, $(6,5)$, and $(2,1)$. Its perimeter can be expressed in the form $c\sqrt2 + d\sqrt{10}$ with $c$ and $d$ integers. What is the sum of $c$ and $d$?


In a diagram, a square has a perimeter of 64 units and is adjoined on one side by a rectangle of width 8 units. A triangle with a height of 64 units also shares its base with the rectangle's length. If the square and the triangle have the same area, find the value of the rectangle's length, denoted as $y$.


In a closet, there are 6 shirts, 8 pairs of shorts, and 7 pairs of socks. If I randomly remove four articles of clothing, what is the probability that I get two shirts, one pair of shorts, and one pair of socks? Treat pairs of socks as one article of clothing.


Two circles are centered at the origin. The point $P(5,12)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=5$, what is the value of $k$?


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


In the arithmetic sequence $11, d, e, f, g, 53$, what is the value of $f$?


Emily and Eli each select a positive integer less than 250. Emily's number is a multiple of 20, and Eli's number is a multiple of 30. Additionally, both numbers must be multiples of 5. What is the probability that they selected the same number? Express your answer as a common fraction.


A circular sheet of paper with a radius of $12$ cm is cut into four congruent sectors. A cone is formed by rolling one of these sectors until the edges meet. Calculate both the height and the volume of this cone. Express the height in simplest radical form and the volume in terms of $\pi$.


Let $r$ and $s$ be the roots of $3x^2 + 4x - 18 = 0$. Calculate the value of $(3r^3 - 3s^3)(r - s)^{-1}$.


A $\textit{palindrome}$ is a number which reads the same forward as backward. What is the least natural number that can be added to 56,789 to create a palindrome?


Henry's Hamburger Heaven has expanded its menu to include an option for one, two, three, or four meat patties. Along with this, customers can select from the same eight condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. Each condiment can either be included or not. How many different kinds of hamburgers can now be ordered?


Let $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ -1 \\ 0 \end{pmatrix}$. Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}$.

Solution: \boxed{\begin{pmatrix} 5 \\ 0 \\ 1 \end{pmatrix}}


What is the remainder when 4672 is divided by 34?


The vertices of a $4 - 5 - 6$ right triangle are the centers of three mutually externally tangent circles. What is the sum of the areas of these circles?


Solution: \boxed{5 + \sqrt[3]{4}}


In a circle, segment $\overline{DE}$ is a diameter with $DE = 30$. Points $F$ and $G$ lie on the circle such that $\overline{DE}$, $\overline{DF}$, and $\overline{EF}$ form a triangle. What is the area of the circle traced by the centroid of $\triangle DEF$ as $F$ moves on the circumference of the circle excluding the points $D$ and $E$?


A traffic light runs through a repeated cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 40 seconds. Joseph picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching?


How many integers \( n \) satisfy the condition \( 200 < n < 300 \) and \( n \) has the same remainder whether it is divided by \( 5 \) or by \( 7 \)?


Consider the sum
\[
\text{cis } 40^\circ + \text{cis } 50^\circ + \text{cis } 60^\circ + \dots + \text{cis } 140^\circ
\]
expressed in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees.


The values of \(f\), \(g\), \(h\), and \(j\) are 3, 5, 7, and 9, but not necessarily in that order. What is the largest possible value of the sum of the four products \(fg\), \(gh\), \(hj\), and \(fj\)?


Let the first term of a geometric sequence be $2$, and let the second term be $70$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by five million?


Maya owns 16 pairs of shoes, consisting of 8 identical black pairs, 4 identical brown pairs, 3 identical grey pairs, and 1 pair of white shoes. If Maya randomly picks two shoes, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe?


If $x, y, z$ satisfy the system of equations:
\begin{align*}
y + z &= 20 - 4x \\
x + z &= 1 - 4y \\
x + y &= -12 - 4z,
\end{align*}
what is $3x + 3y + 3z$?


If the two roots of the quadratic \( 9x^2 + 5x + m \) are \( \frac{-5 \pm i\sqrt{391}}{18} \), what is \( m \)?


The numbers $a$ and $b$ are inversely proportional. When the sum of $a$ and $b$ is 60, $a$ is three times $b$. Find the value of $b$ when $a = -12$.


A rectangular picture frame is crafted from two-inch-wide pieces of wood. The total area of the frame itself is $30$ square inches, and the length of one of the outer edges of the frame is $7$ inches. What is the sum of the lengths of all four interior edges of the frame?


Matt's five cousins are coming to visit. There are four identical rooms where they can stay. Any number of cousins can stay in one room. Given these conditions, how many different ways can the cousins be assigned to the rooms?


Solution: \boxed{(2, 4)}


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


In the line $3x + 5y + c = 0$, the sum of the $x$- and $y$-intercepts is $30$. Find $c$.


Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 5,$ $\|\mathbf{w}\| = 8,$ and $\mathbf{v} \cdot \mathbf{w} = 20,$ then find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|$.


How many four-digit integers with a $5$ in the units place are divisible by 10?


Find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 1 \\ -2 \\ -4 \end{pmatrix} + t \begin{pmatrix} 5 \\ 4 \\ -2 \end{pmatrix}\]
is closest to
\[\bold{a} = \begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}.\]


The sum of the lengths of the twelve edges of a rectangular box is $240$, and the distance from one corner of the box to the farthest corner is $31$. What is the total surface area of the box?


A region $S$ in the complex plane is defined by:
\[
S = \{x + iy: -1 \leq x \leq 1, -1 \leq y \leq 1\}.
\]
A complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac{1}{2} + \frac{1}{2}i\right)z$ is also in $S$?


In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Cindy, and Dave, refuse to sit in four consecutive seats?


Suppose $f(x) = ax + b$ and $g(x) = 4x - 6$. If $h(x) = f(g(x))$ and $h^{-1}(x) = 2x + 14$, find the value of $a - 2b$.


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


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


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


Let $w$ be a nonreal complex number such that $|w| = 2.$ Find the real part of $\frac{1}{2 - w}.$


A right triangle has one leg of 24 meters and a hypotenuse of 25 meters. A second triangle, similar to the first, has a hypotenuse of 100 meters. What is the length of the other leg of the second triangle?

Solution: \boxed{28 \text{ meters}}


A rectangular box has a volume of 5184 cubic inches and a surface area of 1944 square inches. The sum of the lengths of its 12 edges totals 216 inches. Calculate the volume of the box if its length, width, and height were each increased by two inches.

Solution: \boxed{7352 \text{ cubic inches}}


What is the distance on the Cartesian coordinate plane from $(2, -3)$ to $(8, 5)$, and what is the midpoint of the line segment connecting these two points?


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


Find the smallest solution to the equation \[\lfloor x \rfloor = 7 + 75 \{ x \},\] where $\{x\} = x - \lfloor x \rfloor.$


What is the value of $\frac{3}{5} + \frac{2}{3} + 1\frac{1}{15}$?


A bag contains 7 blue chips, 5 yellow chips, and 2 red chips. A chip is drawn from the bag and then placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors?


The line $y = -\frac{5}{6}x + 10$ crosses the $x$-axis at point $P$ and the $y$-axis at point $Q$. Point $T(r,s)$ is on line segment $PQ$. If the area of $\triangle POQ$ is four times the area of $\triangle TOP$, what is the value of $r+s$?


I have seven apples and twelve oranges. If a fruit basket must contain at least two pieces of fruit, how many different fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it does not matter how the fruits are arranged in the basket.)


For how many digits $D$ is the positive four-digit number $10D4$ a multiple of both 3 and 4?


Three times the square of $B$ plus two more than three times $B$ is equal to 29. What is the value of $B$?

Solution: \boxed{\frac{-1 - \sqrt{37}}{2}}


In a modified city tax system, the rate of taxation is $x\%$ for an income of $2x$ thousand dollars. Determine the income, in dollars, that maximizes the take-home pay, where take-home pay is the income minus the tax on that income.


Solve the following: $\left(2\frac{1}{2} - 3\frac{1}{3}\right)^2 \div \left(4\frac{1}{4} + 1\frac{1}{6}\right)$. Express your answer as a mixed number.


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


Triangle $XYZ$ is an equilateral triangle. If the measure of $\angle X$ is artificially decreased by 5 degrees, what are the measures of the other interior angles of the triangle $XYZ$?


The expression $x^2 + 19x + 88$ can be written as $(x + a)(x + b)$, and the expression $x^2 - 23x + 132$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?


Thirty tiles are numbered 1 through 30 and are placed into box $C$. Thirty other tiles numbered 15 through 44 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 20 and the tile from box $D$ is either odd or greater than 40? Express your answer as a common fraction.


Let $D$ be the circle with equation $x^2 + 8x + 18y + 98 = -y^2 - 6x$. Find the values of $a$, $b$, and $r$ if $(a,b)$ is the center of $D$ and $r$ is its radius, and then compute $a + b + r$.

Solution: \boxed{-16 + 4\sqrt{2}}


How many positive perfect squares less than $2 \times 10^6$ are multiples of 36?


Find the difference between $3456_8$ and $1234_8$ in base $8$.


In a right isosceles triangle, one of the angles is $45^{\circ}$. What is the measure of the triangle's hypotenuse angle?


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


What integer $n$ satisfies $0 \leq n < 201$ and $$200n \equiv 144 \pmod {101}~?$$


A square is divided into four congruent rectangles. If the perimeter of each of these four rectangles is 32 inches, what is the perimeter of the square, in inches?

[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((1,0)--(1,4));
draw((2,0)--(2,4));
draw((3,0)--(3,4));
[/asy]

Solution: \boxed{51.2 \text{ inches}}


How many positive real solutions are there to $x^{12} + 8x^{11} + 15x^{10} + 2023x^9 - 1500x^8 = 0$?


A scalene triangle has side lengths which are prime numbers greater than 3, and the length of its perimeter is also prime. What is its smallest possible perimeter?


Let $g$ be a function from the positive integers to the positive integers, such that:

(i) $g$ is strictly increasing ($g(n+1) > g(n)$ for all $n$),
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m, n$,
(iii) if $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Find the sum of all possible values of $g(12)$.


A right cone has a base with a circumference of $12\pi$ meters and an original height of 20 meters. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is $108\pi$ cubic meters. What is the ratio of the shorter height to the original height? Express your answer as a common fraction.


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 $(18,0)$. 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.


In Pascal's Triangle, each entry is the sum of the two entries directly above it. Determine in which row of Pascal's Triangle do three consecutive entries occur that are in the ratio \(2:3:4\).


A rectangle measuring 15 cm by 10 cm has four quarter circles drawn with centers at the four corners. How many square centimeters are in the area of the shaded region? Express your answer in terms of $\pi$.

Solution: \boxed{150 - 25\pi}


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


In triangle $ABC$, where $\angle B = 90^\circ$, $AC = 13$, and $AB = 5$. Find the values of both $\tan A$ and $\sin A$.


Let $p, q, r, s, t, u, v$ and $w$ be distinct elements in the set
\[
\{-8, -6, -4, -1, 1, 3, 7, 12\}.
\]
What is the minimum possible value of
\[
(p+q+r+s)^2 + (t+u+v+w)^2?
\]


Given the equations $2009a + 2013b = 2021$ and $2011a + 2015b = 2023$, determine the value of $a - b$.


Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(2,2),$ $(4,6),$ $(6,12),$ and $(8,12),$ determine if there is an intersection point for the graphs of $y=h(x+2)$ and $y=j(2x)$, and find the sum of the coordinates of that point.


Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $P(z) = z^3 + sz + t,$ and $|p|^2 + |q|^2 + |r|^2 = 324.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane are the vertices of a right triangle with hypotenuse $k.$ Find $k^2.$


Determine the measure of angle \(Q\) in a hexagon where the other angles measure \(134^\circ, 98^\circ, 120^\circ, 110^\circ\), and \(96^\circ\).


Calculate the sum of the proper divisors of $540$.


What is the sum of the tens digit and the units digit in the decimal representation of \(8^{1002}\)?


Mrs. Crabapple teaches a British Literature class with 11 students and the class now meets five times per week. She decides to change her method and wants to pick a different student to receive a crabapple each day of the week. How many different sequences of crabapple recipients are possible under this week's arrangement?


Suppose a parabola has vertex $\left(\frac{3}{4},-\frac{16}{8}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. Find the smallest possible value of $a$.


Find the product of the divisors of $45$.


An organization originally consists of six leaders and the remaining members are regular members. Each year, the existing leaders are removed from the organization. Each regular member then recruits three additional members to join as regular members. Subsequently, six new people from outside the organization join as leaders. Initially, there are twelve people in the organization. How many total people will be in the organization after five years?


Find the smallest constant $D$ such that
\[x^4 + y^4 + 1 \ge D(x^2 + y^2)\] for all real numbers $x$ and $y$.


When three standard dice are tossed, and the numbers $a, b, c$ are obtained, find the probability that $abc = 144$.


A box contains eight cards. Four of the cards are black on both sides, two cards are black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is also red?


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


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots?


In triangle $ABC$, angle $A = 90^\circ$, $BC = 20$, and $\tan C = 3\cos B$. What is $AB$?

Solution: \boxed{AB = \frac{40\sqrt{2}}{3}}


Let $N$ be the sum of the divisors of $300$. What is the largest prime factor of $N$?


How many times do the digits 9 and 8 appear in the list of all integers from 1 to 700?


Find the value of parameter $b$ so that the lines described by
\[\begin{pmatrix} 3 \\ 0 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2b \\ 3 \\ -1 \end{pmatrix}\]
and
\[\begin{pmatrix} 0 \\ 1 \\ -4 \end{pmatrix} + u \begin{pmatrix} 2 \\ -6 \\ 0 \end{pmatrix}\]
are perpendicular.


In a basket, I have only white, purple, red, blue, and green marbles. The ratio of white to purple to red to blue to green marbles is $2:3:5:4:6$. There are 24 blue marbles in the basket. How many marbles are there in total?


Evaluate $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{35}\rceil$$Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$


A store owner reduces the price of a \$12 hat by 20% for an ongoing promotion. After a while, he decides to slash the price further by 25% off the already reduced price to clear inventory. What is the final price of the hat in dollars?


Find the distance between the points (0,12) and (16,0).


Define the function $g(x) = \frac{k}{3x - 4}$. If $g(3) = g^{-1}(k+2)$, find the product of all possible values of $k$.


Let $b_1, b_2, b_3, \dots$ be an increasing arithmetic sequence of integers. If $b_5b_6 = 21$, what is $b_4b_7$?


Given the region between the graph of $y = f(x)$ and the $x$-axis has an area of 15 square units, determine the area between the graph of $y = 4f(2(x - 1))$ and the $x$-axis.


Given that \( x \) is a multiple of \( 7200 \), what is the greatest common divisor of \( f(x)=(5x+6)(8x+3)(11x+9)(4x+12) \) and \( x \)?


A segment with endpoints at $A(1, 3)$ and $B(9, -3)$ is extended through $B$ to point $C$. If $BC = \frac{1}{2} \cdot AB$, what are the coordinates of point $C$? Express your answer as an ordered pair.

Solution: \boxed{(13, -6)}


A grid is composed of 6 horizontal and 6 vertical lines, equally spaced, forming a rectangular arrangement. How many different squares can be traced using the lines in this larger grid?


Find the distance between the foci of the ellipse
\[\frac{x^2}{45} + \frac{y^2}{5} = 9.\]


Let $x$ and $y$ be integers such that $xy = 144.$  Find the minimum value of $x + y.$


The bottoms of two vertical poles are 8 feet apart on a flat surface. The first pole is 10 feet tall, and the second is 4 feet tall. How long, in feet, is a wire stretched from the top of the taller pole to the top of the shorter pole?


Let \( r_1, \) \( r_2, \) and \( r_3 \) be the roots of
\[ x^3 - x^2 + 3x - 7 = 0. \]
Find the monic polynomial, in \( x, \) whose roots are \( 3r_1, \) \( 3r_2, \) and \( 3r_3. \)

Solution: \boxed{x^3 - 3x^2 + 27x - 189}


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{2}{3}\] and $q$ is as small as possible. What is $q-p$?


A designer has access to 5 distinct fabric colors for a new clothing line: red, green, blue, yellow, and black. There are also 6 different patterns available: floral, stripe, polka dot, plaid, chevron, and solid. Unfortunately, due to current aesthetic trends, the designer decides not to use the plaid pattern with red or yellow fabric, nor the polka dot pattern with black fabric. How many different clothing designs is the designer still able to create?


Nyla constructs a snowman by placing snowballs of radius 4 inches, 6 inches, and 8 inches on top of a cylindrical stand with a radius of 3 inches and height of 5 inches. Assuming all snowballs and the stand are perfectly shaped, calculate the total volume of snow used in cubic inches, expressed in terms of $\pi$.


Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG=5$. If $EFGH$ has two sides with distinct integer lengths, then what is the area of $EFGH$? Express your answer in simplest radical form.


For each positive integer $n$, define $h(n) = n^4 - 500n^2 + 625$. What is the sum of all values of $h(n)$ that are prime numbers?


Find the largest integer $n$ for which $18^n$ divides $24!$.


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


Calculate the product:
\[
\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 10 \\ 0 & 1 \end{pmatrix} \dotsm \begin{pmatrix} 1 & 198 \\ 0 & 1 \end{pmatrix}.
\]

Solution: \boxed{\begin{pmatrix} 1 & 5000 \\ 0 & 1 \end{pmatrix}}


Annie bought 50 items each priced at 20 cents, 4 dollars, or 5 dollars. If her total purchase price was $\$$50.00, how many 20-cent items did she purchase?


To bake $24$ cookies, I use $4$ quarts of milk. There are $2$ pints in a quart and $2$ cups in a pint. How many cups of milk are required to bake $6$ cookies?

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


Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]
in the domain $\frac{1}{3} \le x \le \frac{3}{5}$ and $\frac{1}{4} \le y \le \frac{1}{2}.$


Compute without using a calculator: $\dfrac{11!}{8!3!}$.


An equilateral triangle $ABC$ shares a side with square $BCDE$. Another equilateral triangle $AEF$ is positioned such that $AF$ is parallel to $BC$. Determine the number of degrees in the measure of angle $FAE$.


Determine the value of $c + d^2$ for the solutions of the quadratic equation $5x^2 + 4x + 20 = 0$. The solutions are in the form $x = c \pm d i$ where $c$ and $d$ are real numbers.


At the mall's food court, Sarah wants to buy a meal consisting of one appetizer, one main course, one drink, and one dessert. The table below lists Sarah's options available in the food court, but she doesn't want to pick a soda if she chooses fries as an appetizer. How many distinct possible meals can she buy from these options?

\begin{tabular}{ |c | c | c | c|}
\hline \textbf{Appetizers} & \textbf{Main Courses} & \textbf{Drinks}& \textbf{Desserts} \\ \hline
Fries & Burger & Soda & Ice Cream \\ \hline
Salad & Chicken Wrap & Lemonade & Cake \\ \hline
Onion Rings & Pizza & Tea &  \\ \hline
\end{tabular}


What is the greatest possible number of digits in the product of a 4-digit whole number and a 5-digit whole number?


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


Suppose that the angles of triangle $DEF$ satisfy
\[\cos 4D + \cos 4E + \cos 4F = 1.\]
Two sides of the triangle have lengths 7 and 24. Find the maximum length of the third side.


The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 140^{\circ}$ and $\angle AOB = 130^{\circ}$. Determine the degree measure of $\angle ABC$.


Let $A$ be the set of four-digit odd numbers, and $B$ be the set of four-digit multiples of 5. Find the number of four-digit numbers that are either odd or multiples of 5 (i.e., calculate $|A \cup B|$).


Let $\mathbf{v}_0$ be a vector. The vector $\mathbf{v}_0$ is first projected onto $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$, resulting in the vector $\mathbf{v}_1$. The vector $\mathbf{v}_1$ is then projected onto $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$, resulting in the vector $\mathbf{v}_2$. Find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2$.

Solution: \boxed{\begin{pmatrix} \frac{24}{85} & \frac{6}{85} \\ \frac{48}{85} & \frac{12}{85} \end{pmatrix}}


Let $x$ and $y$ be two positive real numbers such that $x + y = 45$. Determine the ordered pair $(x, y)$ for which the value of $x^5 y^3$ is maximized.

Solution: \boxed{\left(\frac{225}{8}, \frac{135}{8}\right)}


If $x$ is real, compute the maximum integer value of
\[
\frac{3x^2 + 9x + 20}{3x^2 + 9x + 7}.
\]


Let $x = (3 + \sqrt{5})^{500},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$  Find
\[x(1 - f).\]


Triangle $PQR$ has side-lengths $PQ = 15, QR = 20,$ and $PR = 25.$ A line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y.$ What is the perimeter of $\triangle PXY?$


Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 5?


In triangle $DEF$, $DE = 8$, $EF = 6$, and $FD = 10$.

[asy]
defaultpen(1);

pair D=(0,0), E=(0,6), F=(8,0);
draw(D--E--F--cycle);

label("\(D\)",D,SW);
label("\(E\)",E,N);
label("\(F\)",F,SE);

[/asy]

Point $Q$ is arbitrarily placed inside triangle $DEF$. What is the probability that $Q$ lies closer to $D$ than to either $E$ or $F$?


What is the first year after 2010 for which the sum of the digits is 10?


For certain real values of \(a, b, c,\) and \(d,\) the equation \(x^4+ax^3+bx^2+cx+d=0\) has four non-real roots. The product of two of these roots is \(-7+4i\) and the sum of the other two roots is \(-2-6i.\) Find \(b.\)


Sam enjoys numbers that are divisible by 8. How many different units digits are possible in numbers that Sam likes?


Let $g(x, y)$ be the function for the set of ordered pairs of positive coprime integers such that:
\begin{align*}
g(x, x) &= x, \\
g(x, y) &= g(y, x), \quad \text{and} \\
(x + y) g(x, y) &= y g(x, x + y).
\end{align*}
Calculate $g(15, 33)$.


What is the smallest four-digit positive integer congruent to $1 \pmod{23}$?


Compute the sum of the series
\[\frac{7}{5^2 \cdot 9^2} + \frac{11}{9^2 \cdot 13^2} + \frac{15}{13^2 \cdot 17^2} + \dotsb.\]


An equilateral triangle has its two vertices at $(0,7)$ and $(10,7)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form.

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


Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \sin 670^\circ.$


Our school's volleyball team has expanded and now includes 16 players, among them a set of triplets (Alicia, Amanda, and Anna) and a set of twins (Beth and Brenda). In how many ways can we choose 7 starters, with the condition that both twins must either be included or excluded together?


A puppy, two smaller cats, and one larger cat together weigh 38 pounds. The puppy and the larger cat together weigh exactly three times as much as one smaller cat, and the puppy and the two smaller cats together weigh exactly the same as the larger cat. How many pounds does the puppy weigh?


Each of the twelve letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "SITE"? Express your answer as a common fraction.


For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$?


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


What is the sum of all of the positive even factors of $630$?


Calculate how many positive perfect square integers are factors of the product $\left(2^{12}\right)\left(3^{18}\right)\left(5^{20}\right)\left(7^8\right)$.


You have $7$ red shirts, $6$ green shirts, $5$ blue shirts, $6$ pairs of pants, $6$ green hats, $7$ red hats, and $5$ 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?


If
\[\frac{\sin^4 \alpha}{\sin^2 \beta} + \frac{\cos^4 \alpha}{\cos^2 \beta} = 1,\]
then find the sum of all possible values of
\[\frac{\tan^4 \beta}{\tan^2 \alpha} + \frac{\cot^4 \beta}{\cot^2 \alpha}.\]


Let $A$ equal the number of four-digit numbers that are odd and divisible by 3. Let $B$ equal the number of four-digit numbers divisible by 4. Find $A+B$.


If $a, b, c$ are non-negative integers less than $9$ such that:
\[
a + 2b + 3c \equiv 0 \pmod{9},
\]
\[
2a + 3b + c \equiv 5 \pmod{9},
\]
\[
3a + b + 2c \equiv 5 \pmod{9},
\]
then determine the remainder when $abc$ is divided by $9$.


Suppose $x,$ $y,$ and $z$ are positive real numbers such that $x + y + z = 1$ and $x=y$. If no one of these numbers is more than twice any other, find the maximum value of the product $xyz$.


An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,4)$ and the other focus is at $(3,12)$. Compute the x-coordinate of the center of the ellipse.


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


Determine the maximum integer value of the expression
\[\frac{3x^2 + 9x + 28}{3x^2 + 9x + 7}.\]


Add 53.463 to 12.98734 and 0.5697 then round to the nearest hundredth.


The sum of two positive integers $a$ and $b$ is 1729. What is the largest possible value of $\gcd(a, b)$?


In trapezoid $PQRS$, the sides $PQ$ and $RS$ are equal. Let the lengths of $QR=10$ (upper base) and $PS=20$ (lower base) with the vertical height from $PQ$ to $PS$ equal to $5$. Calculate the perimeter of $PQRS$.

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


Mary can mow a lawn in three hours and Tom can mow the same lawn in six hours. Mary starts mowing the lawn alone, and after one hour, Tom joins her. After Tom works for 2 more hours, what fractional part of the lawn remains to be mowed?


All the numbers 1, 3, 4, 6, 8, 9 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?


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


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


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

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


Determine the degree of the polynomial resulting from the expansion of $(3x^4+4x^3+2x-7)(3x^{10}-9x^7+9x^4+30)-(x^2+5)^8$.


Each of $a_1,$ $a_2,$ $\dots,$ $a_{200}$ is equal to $1$ or $-1.$  Find the minimum positive value of
\[\sum_{1 \le i < j \le 200} a_i a_j.\]


The orchestra has between 150 and 300 members. When lined up in rows of 6, there is one extra person; when lined up in rows of 8, there are three extra people; and when lined up in rows of 9, there are five extra people. How many members are in the orchestra?


Find the minimum value of $x$ such that $4x^2 + 8x + 3 = 5$.

Solution: \boxed{-1 - \frac{\sqrt{6}}{2}}


According to Moore's law, the number of transistors on a chip doubles every two years. If a processor in 1992 contained 2,000,000 transistors, how many transistors would it contain in the year 2002 and then in 2010?


Compute $4 \begin{pmatrix} 3 \\ -5 \end{pmatrix} - 3 \begin{pmatrix} 2 \\ -6 \end{pmatrix}$.

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


Calculate the value of \(725^2 - 675^2\) and then divide the result by 25.


In $\triangle ABC$, we have $AB = AC = 26$ and $BC = 24$. Point $D$ is the midpoint of $\overline{BC}$. Find $AD$.


Simplify: $$\dfrac{\sqrt{726}}{\sqrt{242}} + \dfrac{\sqrt{484}}{\sqrt{121}}.$$Express your answer as a common fraction.

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


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


A classroom has a total of $60$ chairs that need to be arranged in rows and columns. Each row and column must contain at least two chairs. How many different ways can the chairs be arranged such that all chairs are used, and every configuration is considered unique if the number of rows is different from the number of columns?


How many 6-digit numbers have exactly one zero?


Let $\alpha$ and $\beta$ be complex numbers such that $|\alpha| = 2$, $|\beta| = 1$, and $\overline{\alpha} \beta \neq 1$. Find the maximum value of
\[
\left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right|.
\]


The function \(g(x)\) satisfies \(g(1) = 1\) and
\[g(x + y) = 4^y g(x) + 3^x g(y)\] for all real numbers \(x\) and \(y.\) Find the function \(g(x).\)

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


The time right now is exactly noon. What time will it be in 1857 minutes?

Solution: \boxed{6\!:\!57 \text{ a.m.}}


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 $T(b)T(-b)=4800$. Find $T(b)+T(-b)$.


In $\triangle ABC, AB = 10, BC = 9, CA = 8$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$.


In the expression $c \cdot (a!)^b - d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?


Solution: \boxed{\frac{\log_2 5}{8}}


Determine for which values of $x$ the expression \[\frac{x-20x^2+100x^3}{16 - 2x^3}\] is nonnegative. Answer as an interval.

Solution: \boxed{[0, 2)}


Consider the reflection matrix over a line $\ell$ passing through the origin, given by
\[
\begin{pmatrix}
\frac{3}{5} & \frac{4}{5} \\
\frac{4}{5} & -\frac{3}{5}
\end{pmatrix}.
\]
Find the direction vector of line $\ell$. The direction vector should be in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ and $b$ are integers, $a > 0$, and $\gcd(|a|,|b|) = 1$.

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


The sides of a triangle have lengths \( 13, 17, \) and \( k, \) where \( k \) is a positive integer. For how many values of \( k \) is the triangle obtuse?


What is the least integer value of $x$ such that $\lvert2x+ 9\rvert\le 20$?


If \(a \div b = 3\) and \(b \div c = \frac{2}{5}\), what is the value of \(c \div a\)? Express your answer as a common fraction.


At Lila's Lily Shop, a bouquet containing 15 lilies costs \$30. The price per lily decreases by 10% for bouquets larger than 30 lilies due to a bulk discount. How much will a bouquet of 45 lilies cost?


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


How many of the natural numbers from 1 to 700, inclusive, contain the digit 4 at least once?


In $\triangle ABC$, points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$ respectively. Lines $\overline{AD}$ and $\overline{BE}$ intersect at point $T$ such that $AT/DT=2$ and $BT/ET=3$. Determine the ratio $CD/BD$.


Given a scalene triangle with side lengths $13$, $14$, and $15$, determine the sum of the squares of the lengths of the medians.


For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $8n$?


In a setting similar to the previous one, we now have $\cos \angle TUV = \frac{3}{5}$. What is $\cos \angle TUX$, where $X$ is the point at $180^\circ$ from $V$ through $U$?


Let \(a\), \(b\), and \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 4\). Find the maximum value of
\[6a + 3b + 10c.\]


"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x, y)$ where $0 \leq x, y < m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x, y)$ that satisfies the congruence. Consider the graph of $$5x \equiv 3y + 2 \pmod{17}$$ which has a single $x$-intercept $(x_0, 0)$ and a single $y$-intercept $(0, y_0)$, where $0 \leq x_0, y_0 < 17$.

What is the value of $x_0 + y_0$?


Find the remainder when \(x^{2023} + 1\) is divided by \(x^8 + x^6 + x^4 + x^2 + 1.\)

Solution: \boxed{x^3 + 1}


A particularly unusual four-sided die is designed such that:
- It lands on Side 1 with probability $\frac{1}{4}$, winning you $2 dollars.
- It lands on Side 2 with probability $\frac{1}{4}$, winning you $4 dollars.
- It lands on Side 3 with probability $\frac{1}{3}$, causing you to lose $6 dollars.
- It lands on Side 4 with probability $\frac{1}{6}$, resulting in no change in your money.

What is the expected winnings from rolling this die? Express your answer as a dollar value.


Arthur walks 8 blocks east, 15 blocks north, and then 3 blocks west. If each block is one-quarter of a mile, how many miles did he walk in total?


A standard deck of 52 cards has 13 ranks and 4 suits, with exactly one card of each rank and suit combination. Two suits are black and two are red. The deck is shuffled randomly. What is the probability that one of the top two cards is red and the other is black, regardless of order?


Trapezoid $ABCD$ (with bases $AB$ and $CD$) has an area of $48$ square units. Points $E$ and $F$ are the midpoints of the non-parallel sides $AD$ and $BC$, respectively. What is the area of triangle $BDF$?


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 \(\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.\)

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


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


A Senate committee has 6 Democrats and 5 Republicans. All politicians are distinguishable. The leader of the Democrats must sit at the head of the table. In how many ways can they sit around a circular table with this restriction?


Let $s_1,$ $s_2,$ and $s_3$ be the roots of
\[x^3 - 4x^2 + 9 = 0.\]Find the monic polynomial, in $x,$ whose roots are $3s_1,$ $3s_2,$ and $3s_3.$

Solution: \boxed{x^3 - 12x^2 + 243}


Sally's class has 360 students. At a school assembly, the students will sit in rows with the same number of students in each row. There must be at least 12 rows and at least 18 students in each row. Determine the sum of all possible values for the number of students \(x\) per row.


What is the smallest value of $x$ that satisfies the equation $\sqrt{3x} = 5x$? Express your answer in simplest fractional form.


On Wednesday, I worked $t+2$ hours and earned $4t-4$ dollars per hour. My friend Bob worked $4t-6$ hours but only earned $t+3$ dollars an hour. At the end of the day, I had earned three dollars more than Bob. What is the value of $t$?


Given that $x$ is a multiple of $12096$, what is the greatest common divisor of $g(x)=(3x+8)(5x+1)(11x+6)(2x+3)$ and $x$?


In triangle $PQR$, $\angle P = 90^\circ$, and $\cos Q = \frac{3}{5}$. Find $\sin R$.

Solution: \boxed{\sin R = \frac{3}{5}}


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


5 12-sided dice are rolled. What is the probability that the number of dice showing a two-digit number is equal to 3? Assume that the numbers on the 12 sides are the numbers from 1 to 12.


Let $x^2 + bx + c = 0$ be a quadratic equation whose roots are each three more than the roots of $4x^2 - 7x - 10$. Determine the value of $c$.


In triangle $\triangle DEF$, point $L$ lies on $EF$ such that $DL$ is an altitude of $\triangle DEF$. Given that $DE = 15$, $EL = 9$, and $EF = 24$, calculate the area of $\triangle DEF$.


My friend Ben only likes numbers that are divisible by 8, such as 24, or 8,016. How many different ones digits are possible in numbers that Ben likes?


In the following diagram, triangle $ABC$ has point $X$ on side $AB$ such that $X$ is the bisector of angle $ACB$. Given $AB=80$ units, $BC=90$ units, and $CA=40$ units, find the length of segment $AX$.


In a set of seven consecutive integers, the largest integer is less than twice the average of the seven integers. What is the smallest integer that could be in this set?


Express \(\sin (a + b) - \sin (a - c)\) as a product of trigonometric functions.

Solution: \boxed{2 \cos\left(a + \frac{b - c}{2}\right) \sin\left(\frac{b + c}{2}\right)}


What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by any of $19,$ $7,$ and $3?$


Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$'' where the digits $A$, $B$, $C$, and $D$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible?


Suppose the numbers \[3 - \sqrt{6}, \;5+\sqrt{15}, \;16 - 2\sqrt{10}, \;-\sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of this polynomial?


Mason is seeking to ascertain the Fermat point \(P\) of \(\triangle ABC\), where \(A\) is at the origin, \(B\) is at \( (12, 0) \), and \(C\) is at \( (4, 6) \). Thinking that \(P = (5,3)\) could be the Fermat point, he decides to compute the sum of the distances from \(P\) to the vertices. The final expression he achieves is in the form \(x\sqrt{a} + y\sqrt{b}\), where \(x\), \(y\), \(a\), and \(b\) are integers. Compute \(x + y\).


A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that fits inside the cylinder?


Ellie ran 4 miles in a third of the time it took Max to run 6 miles. If it took Max 36 minutes to run 6 miles on a flat road and assuming Ellie runs at the same rate on similar terrain, how many minutes would it take Ellie to run 8 miles?


A woman labels the squares of a very large chessboard from $1$ to $64$. On each square $k$, she places $3^k$ grains of rice. How many more grains of rice are placed on the $12^{th}$ square than on the first $10$ squares combined?


The number $0.84375$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?


Let $a$, $b$, $c$ be the sides of a triangle, and $\alpha$, $\beta$, $\gamma$ be the angles opposite them, respectively. If $a^2 + b^2 = 2023c^2$, find the value of
\[
\frac{\cot \gamma}{\cot \alpha + \cot \beta}.
\]


Calculate the value of $\frac{10! \cdot 6! \cdot 3!}{9! \cdot 7!}$.


Keisha's dance team must decide on a new appearance for their competition. The juniors will pick the color of the pants (black, gold, or silver) and the seniors will pick the color of the shirt (black, white, or gold), but they won't communicate their choices to each other. Assuming each color choice is equally likely, what is the probability that the pants will be a different color than the shirt? Express your answer as a common fraction.


Solve the inequality
\[\frac{x - 5}{x^2 + x + 3} \ge 0.\]
Additionally, ensure $x \neq 2$.

Solution: \boxed{[5, \infty)}


The solutions of the equation $x(5x - 11) = -6$ can be expressed in the form $\frac{m + \sqrt{n}}{p}$ and $\frac{m - \sqrt{n}}{p}$, where $m$, $n$, and $p$ have the greatest common divisor of 1. Find $m+n+p$.


In the diagram, $\triangle ABC$ is right-angled at $A$, with $AB = 75$ and $AC = 100$. Point $D$ is on $BC$ such that $AD$ is perpendicular to $BC$. Determine the length of $BD$.
[asy]
pair A, B, C, D;
B=(0,0);
A=(45,60);
C=(125,0);
D=(45,0);
draw(A--B--C--A--D);
label("B", B, SW);
label("A", A, N);
label("D", D, S);
label("C", C, SE);
label("75", (A+B)/2, NW);
label("100", (A+C)/2, NE);
[/asy]


A square is circumscribed around a circle. The number of inches in the perimeter of the square is equal to the number of square inches in the area of the circle. What is the diameter of the circle in terms of pi and simplest radical form?


A list of five positive integers has the following properties:
- The only integer in the list that occurs more than once is $8$.
- Its median is $10$.
- Its average (mean) is $12$.
What is the largest possible integer that could appear in the list?


In a slightly different scenario, a teacher has modified the homework rules for Brian. For each of the first seven homework points Brian wants to earn, he needs to do one homework assignment; for each of the next seven points, he needs to do two homework assignments; and so on, so that to earn the $n^{\text{th}}$ homework point, he has to do $\lceil n\div7 \rceil$ (rounded up) homework assignments. What is the smallest number of homework assignments necessary for Brian to earn a total of 28 homework points?


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


A day can be evenly divided into 86,400 seconds; hence it can also be divided into various periods based on different lengths of time. How many ways are there to divide a day into $n$ periods of $m$ minutes, where $n$ and $m$ are positive integers and the product of $n$ and $m$ equals 1440 minutes?


Find the product of the roots of the equation \[(3x^4 + 2x^3 - 8x + 15)(4x^3 - 20x^2 + 25) = 0.\]


Simplify the product \[\frac{12}{6} \cdot \frac{18}{12} \cdot \frac{24}{18} \dotsm \frac{6n+6}{6n} \dotsm \frac{3036}{3030}.\]


Write the following expression as a polynomial:
$$(2x^2 + 5x + 8)(x+1) - (x+1)(x^2 - 2x + 50) + (3x - 7)(x+1)(x - 2).$$

Solution: \boxed{4x^3 - 2x^2 - 34x - 28}


When three standard dice are tossed, and the numbers $a, b, c$ are obtained, find the probability that $abc = 144$.


If the consecutive integers from $30$ to $1$ were written as $$3029282726...,$$ what would be the $39^{\text{th}}$ digit to be written?


Call a $3$-digit number arithmetic if it has $3$ distinct digits which, when read from left to right, form an arithmetic sequence. Find the difference between the largest and smallest arithmetic numbers.


Consider a square with side length $2a$. Inside the square, a smaller square with side length $b$ is placed in one corner, such that its sides are flush with two sides of the larger square. The remaining area of the large square is divided into four congruent rectangles. Find the perimeter, in units, of one of these four congruent rectangles. Express your answer in terms of $a$.


Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g$, and the $x$-coordinate of the vertex of $g$ is a root of $f$. Both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(50,-150)$, what is the value of $a + c$?

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


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


One hundred more than the square of a number is the same as the square of the quantity that is $20$ less than the number. What is the number?


Keisha's basketball team decides to involve their seventh-graders and eighth-graders in jersey design choices. The seventh-graders need to decide on the color of the shorts (from black, gold, or blue) and the eighth-graders will pick the color of the jersey (from black, white, or gold), and they do not confer together. Each color for both shorts and jerseys is equally likely to be chosen. What is the probability that the shorts will be a different color than the jersey? Express your answer as a common fraction.


Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-4)$, $R(-8,-4)$, and $S(-2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not below the $x$-axis? Express your answer as a common fraction.


The teacher asks Bill to calculate $a - b - c - d$, but Bill mistakenly calculates $(a - b) + c - d$ and gets an answer of 19. If the correct answer was 9, what is the value of $a - b$?


Determine in how many bases between three and ten inclusive the number \( 625_{10} \) has a final digit of 1 when represented in those bases.


It can be shown that for any positive integer $n,$
\[\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{pmatrix},\]where $F_n$ denotes the $n$th Fibonacci number.

Compute $F_{999} F_{1001} - F_{1000}^2.$


Math City plans to add more streets and now has 10 streets, but two pairs of these streets are parallel to each other. No other streets are parallel, and no street is parallel to more than one other street. What is the greatest number of police officers needed at intersections, assuming that each intersection has exactly one police officer stationed?


What is the smallest integer $n$ such that $n^2 - 14n + 45 > 0$?


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

Solution: \boxed{-1 - i}


Calculate the sum of the coefficients in the polynomial $3(x^8 - 2x^5 + x^3 - 6) - 5(x^4 + 3x^2 - 2) + 2(x^6 - 5)$.


Amelia drove from her office to a conference center and then to a coffee shop before heading back. Her car's odometer reading was 52,214 miles when she left her office and 52,235 miles when she returned home. Amelia's car has a fuel efficiency of 32 miles per gallon and the price of fuel is $3.89 per gallon. Determine the cost of the fuel used during her trip, rounding to the nearest cent.


A line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \end{pmatrix} + s \begin{pmatrix} 2 \\ 5 \end{pmatrix}.
\]
A second line is parameterized by
\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -7 \\ 15 \end{pmatrix} + v \begin{pmatrix} 4 \\ 1 \end{pmatrix}.
\]
If $\phi$ is the acute angle formed by the two lines, then find $\cos \phi.$


Find the number of integers $m$ such that $(m-1)(m-10) + 2 \leq 0$.


How many even abundant numbers are less than 50?


If $\frac{720}{2^5\cdot5^9}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


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


Twenty-five tiles are numbered 1 through 25 and are placed into box $A$. Thirty tiles, numbered 1 through 30, are placed into box $B$. One tile is randomly drawn from each box. Calculate the probability that the tile from box $A$ is 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.


How many sequences of 7 digits $x_1, x_2, \ldots, x_7$ can be formed where no two consecutive digits ($x_i$ and $x_{i+1}$) have different parity? Leading zeroes are allowed.


A ball is dropped from a height of 800 feet and bounces back to half the height of its previous bounce each time. Determine the minimum number of bounces required for the ball to first reach a maximum height less than 2 feet.

Solution: \boxed{k = 9}


Find the phase shift of the graph of $y = 5 \cos \left( x + \frac{\pi}{3} \right)$.


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


A relatively prime date is a date where the number of the month and the number of the day are relatively prime. How many relatively prime dates are in the month of April, which has 31 days?


Consider a rectangular region of 2x1 unit squares at the center of a large grid of unit squares. Each subsequent ring forms around this rectangle by one unit thickness. Determine the number of unit squares in the $50^{th}$ ring around this central rectangle.


A right circular cone is inscribed in a right rectangular prism as shown. The base of the prism has dimensions, where one side is exactly twice the length of the other ($a$ and $2a$). The cone's base fits perfectly into the base of the prism making one side of the rectangle the diameter of the cone's base. The height of the prism and the height of the cone are equal. Calculate the ratio of the volume of the cone to the volume of the prism, and express your answer as a common fraction in terms of $\pi$.


Consider a large square grid composed of unit squares. At the center there is a 1x1 unit square. Around this center square, a sequence of concentric rings of squares is formed. The first ring contains 8 unit squares, the second ring contains 16 unit squares, and so forth. Find the total number of unit squares in the first 100 rings.


In a 15-inch by 18-inch rectangle, two circles are drawn, each with a diameter of 7 inches. If the circles are positioned such that they do not extend beyond the rectangular boundaries, what is the greatest possible distance between the centers of the two circles?

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


Let $t$ be the positive real solution to $x^3 + \frac{3}{7}x - 1 = 0.$ Find the exact numerical value of
\[
t^3 + 2t^6 + 3t^9 + 4t^{12} + \dotsb.
\]

Solution: \boxed{\frac{49}{9} t}


A triangle has sides of lengths 8 and 15 units. Determine if it is possible to form a right triangle with these sides, and if so, calculate the length of the third side.


If Alex gives Bob one nickel, Bob will have four times as many nickels as Alex. If Bob gives Alex one nickel, then he will have three times as many nickels as Alex. How many nickels does Bob have initially?


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


For $\mathbf{v} = \begin{pmatrix} 2 \\ z \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 5 \\ 10 \end{pmatrix}$,
\[\text{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 5 \\ 10 \end{pmatrix}.\] Find $z$.


The function \( g(x) \) satisfies
\[ 4g(x) - 3g\left(\frac{1}{x}\right) = 2x \] for all \( x \neq 0 \). Find \( g(5) \).


Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a 10, and the third card is a Queen? Assume the dealing is done without replacement.


Solution: \boxed{\frac{\log_2 5}{4}}


Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$?


In a bag, there are 6 green chips, 5 blue chips, and 4 red chips. One chip is drawn from the bag and then replaced. This process is repeated for a second chip. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


Given a ten-sided regular polygon \(B_1B_2B_3B_4B_5B_6B_7B_8B_9B_{10}\), find how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set \(\{B_1, B_2, \ldots B_{10}\}\).


What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.1\overline{47}$?


Determine the value of $k$ such that the quadratic equation $kx^2 + 7x - 10 = 0$ has solutions $x = -2$ and $x = \frac{5}{2}$, and that the discriminant is positive.


If 500 is expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?


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


The lattice shown is continued for $12$ rows. What will be the fourth number in the $12$th row?
\begin{tabular}{rccccc}
Row 1: & 1 & 2 & 3 & 4 & 5 \\
Row 2: & 6 & 7 & 8 & 9 & 10 \\
Row 3: & 11 & 12 & 13 & 14 & 15 \\
Row 4: & 16 & 17 & 18 & 19 & 20
\end{tabular}


In a right triangle, one leg measures $5$, the hypotenuse measures $13$, and you want to find the ratio of the other leg to the hypotenuse. Present your answer in the form $x:y$, where $x$ and $y$ are relatively prime integers.


Calculate the value of the following expression: $3 - 8 + 13 - 18 + 23 - \cdots - 58 + 63 - 68 + 73$.


Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g((x - y)^2) = g(x)g(y) - xy\] for all real numbers $x$ and $y$.
Let $m$ be the number of possible values of $g(1)$, and let $t$ be the sum of all possible values of $g(1)$. Find $m \times t.$


If eight people decide to come to a basketball game, but four of them are only 1/3 sure that they will stay for the entire time (the other four are sure they'll stay the whole time), what is the probability that at the end, at least 7 people stayed the entire time?


When the base-12 integer $1732_{12}$ is divided by $9$, what is the remainder?


Given the equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$, reduce it to an equivalent form \[\cos ax \cos bx \cos cx = 0,\] for some integers $a, b, c$. Find the sum $a+b+c$.


Calculate the value of the sequence $\frac{1}{3}\cdot\frac{9}{1}\cdot\frac{1}{27}\cdot\frac{81}{1} \dotsm \frac{1}{2187}\cdot\frac{6561}{1}$.


In a certain sequence, the first term is $a_1 = 2010$ and the second term is $a_2 = 2011$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = n + 3$ for all $n \ge 1$. Determine $a_{1000}$.


Suppose a number $n$ has $4$ divisors. How many divisors does $n^2$ have?


An equilateral triangle has vertices at $(0,3)$ and $(10,3)$. Determine the y-coordinate of the third vertex if it is in the first quadrant.

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


Let \(x, y, z\) be real numbers such that \(x^2 + y^2 + z^2 = 9\). Find the maximum value of \(x^3 + y^3 + z^3\).


A large domino set contains tiles where each integer from 0 to 12 is paired with every integer from 0 to 12 exactly once. A $\textit{double}$ in this set is a domino that has the same integer on both squares. What is the probability that a domino randomly selected from this set is a $\textit{double}$? Express your answer as a common fraction.


Determine the area of triangle $ABC$ given that $AC$ is the hypotenuse of a right triangle $ABC$ with $AC = 40$ units and $\angle BAC = 45^\circ$.


Suppose $\{a_n\}$ is an arithmetic sequence with
$$
a_1 + a_2 + \cdots + a_{50} = 50 \quad \text{and} \quad
a_{51} + a_{52} + \cdots + a_{100} = 150.
$$
What is the value of $a_2 - a_1$? Express your answer as a common fraction.


A girl has three different types of coins in her wallet: $3$ nickels, $3$ dimes, and $2$ quarters. She also adds $2$ half-dollars. She randomly picks two coins, calculates the total value, and puts them back. She repeats this process multiple times. What is the maximum number of different sums she can record?


The graphs of four functions, labelled (2) through (5), are shown. Note that the domain of function (3) is $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ Function (3) is represented by distinct points on the graph. Determine the product of the labels of the functions which are invertible. Assume the following descriptions:
- Function (2): Graph of a quadratic function opening upwards.
- Function (3): Graph represented by distinct points with no two points having the same y-coordinate.
- Function (4): Graph of the arctangent function.
- Function (5): Graph of the function $y = \frac{5}{x}$.


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


On a $9 \times 9$ chessboard, determine the total number of rectangles and the number of those rectangles that are squares. Then, find the simplified fraction of the ratio of the number of squares to the total number of rectangles and compute $m + n$ where the fraction is expressed as $\frac{m}{n}$ with $m$ and $n$ as coprime integers.


A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$.


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


If $x$ and $y$ are positive real numbers such that $3x^2 + 18xy = x^3 + 3x^2y + 6x,$ what is the value of $x$?


A math seminar plans to schedule a series of eight different lectures. Dr. Brown must lecture before Dr. Green, and similarly, Dr. Clark must go before Dr. May. How many ways can the eight lecturers be scheduled given these constraints?


A community has 4 schools, each with 6 members. The community wants to arrange a leadership meeting with the following conditions:
i. They must choose one of the 4 schools to host the meeting.
ii. The host school sends 2 representatives to the meeting, and each of the other three schools sends 2 representatives as well.


Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $-3 - i \sqrt{8}$ as a root.

Solution: \boxed{x^2 + 6x + 17}


Calculate the number of increasing sequences of positive integers $b_1 \le b_2 \le b_3 \le \cdots \le b_7 \le 2010$ such that $b_i - i$ is odd for $1 \le i \le 7$. The result can be expressed as ${p \choose q}$ for some integers $p > q$. Determine the remainder when $p$ is divided by 1000.


In a revised shooting match, nine clay targets are arranged in three hanging columns; the first two columns each contain three targets, and the third column has three targets. A marksman must break all the targets following the rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest remaining target in the chosen column.
How many different orders can the nine targets be broken, following these rules?


Given three points on a coordinate plane, $\{(1,3), (7,3), (-2,-4)\}$. Calculate the perimeter of the triangle formed by these points.

Solution: \boxed{6 + \sqrt{58} + \sqrt{130}}


Two pairs of real numbers $(x, y)$ satisfy the equation $x + y = 3xy = 6$. Given that the solutions for $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$?


Let \( a_0, a_1, a_2, \ldots \) be an infinite sequence of real numbers where \( a_0 = \frac{7}{17} \) and for every positive integer \( n \),
\[ a_n = 2a_{n-1}^2 - 1. \]
Let \( c \) be the smallest number such that for every positive integer \( n \), the product of the first \( n \) terms satisfies
\[ |a_0 a_1 \ldots a_{n-1}| \le \frac{c}{2^n}. \]
What is the value of \( 100c \), rounded to the nearest integer?


Let $n$ be a positive integer. If $b \equiv (3^{2n+1} + 5)^{-1} \pmod{9}$, what is the remainder when $b$ is divided by 9?


The number $m$ is a three-digit positive integer and is the product of three distinct prime factors: $x$, $y$, and $10x - y$, where $x$ and $y$ are distinct primes each less than 10. What is the largest possible value of $m$?


The Science club consists of 24 members: 12 boys and 12 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?


The blue parabola displayed is the graph of the equation $x = ay^2 + by + c$. The vertex of this parabola is at $(5,3)$, and the graph passes through the point $(3,5)$. If $a = -1$, find the value of $c$.


Below is a portion of the graph of a function, $y=F(x)$:

[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.5,xmax=4.5,ymin=-0.99,ymax=8.5;

pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);

Label laxis; laxis.p=fontsize(10);

xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f2(real x){return sqrt(abs(x+1))+(9/pi)*atan(sqrt(abs(x+1)));}
draw(graph(f2,xmin,xmax),linewidth(1));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
label("$y=F(x)$",(xmax+0.25,f2(xmax)),E);
[/asy]

The value of $F(-1)$ is an integer. What is it?


Let \(a\) and \(b\) be real numbers. One of the roots of
\[x^3 + ax^2 + 2x + b = 0\]is \(2 - 3i.\) Determine the ordered pair \((a, b)\).

Solution: \boxed{\left(-\frac{5}{4}, \frac{143}{4}\right)}


Tamika selects two different numbers at random from the set $\{10, 11, 12\}$ and adds them. Carlos takes two different numbers at random from the set $\{4, 6, 7\}$ and adds them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.


What is the remainder when $3079$ is divided by $67$?


The fraction $\frac{a}{a+36}$, where $a$ is a positive integer, equals $0.775$. What is the value of $a$?


Points $A$, $B$, $C$, $D$, and $E$ are located on $\overline{AB}$ such that $AB = 4AD = 8BC = 2BE$. If a point is selected at random on $\overline{AB}$, what is the probability that it is between $C$ and $E$?


Three identical circles touch one another, with each circle's circumference being 30. What is the perimeter of the triangular shaded region formed where all three circles touch? Each shaded arc corresponds to the area between the touch points of each circle with its adjacent circles.


For how many values of the digit $A$ is it true that $72$ is divisible by $A$ and $273{,}1A2$ is divisible by $4$?


How many positive three-digit integers less than 600 have at least two digits that are the same?


How many positive integers less than $500$ are multiples of $5$, but not multiples of $10$ or $15$?


Find the projection of the vector $\begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix}$ onto the line:
\[\frac{x}{3} = \frac{y}{-1} = \frac{z}{2}.\]

Solution: \boxed{\begin{pmatrix} \frac{57}{14} \\ -\frac{19}{14} \\ \frac{19}{7} \end{pmatrix}}


$ABCDEFGH$ is a cube. Find $\sin \angle HAD$.


Calculate $\sqrt{\sqrt[3]{0.008}} \times \sqrt[4]{16}$. Express your answer as a decimal to the nearest tenth.


In how many ways can the digits of $47,\!770$ be arranged to form a 5-digit number, ensuring the number does not begin with 0?


Find the radius of the circle with the equation $2x^2 - 8x + 2y^2 - 4y + 10 = 0$.


How many integers are between $(9.5)^3$ and $(9.7)^3$?


Cara is sitting at a circular table with her seven friends. Determine the different possible pairs of people Cara could be sitting between on this table.


Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $q-p$?


In a geometric sequence, the first term is $125$, the second term is $b$, and the third term is $\frac{60}{49}$. If $b$ is positive, what is the value of $b$?


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


What is the largest integer less than $\log_2 \frac{2^2}{1^2} + \log_2 \frac{3^2}{2^2} + \cdots + \log_2 \frac{2010^2}{2009^2}$?


Consider a modified set $\{ 6, 14, 18, 27, 35, 42, 54 \}$. If two distinct members are randomly selected and multiplied, what is the probability that the product is a multiple of 63?


A convex polyhedron $Q$ has $30$ vertices, $70$ edges, and $42$ faces, of which $30$ are triangular and $12$ are pentagonal. Determine the number of space diagonals that $Q$ has.


How many prime numbers between 50 and 90 have a prime remainder when divided by 10?


A standard die is rolled eight times. What is the probability that the product of all eight rolls is divisible by 3? Express your answer as a common fraction.


Let $AB$ be a diameter of a circle centered at $O$. Let $F$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $F$ and $AF$ at $G$ and $H$, respectively. If $\angle BAF = 30^\circ$, find $\angle HFG$, in degrees.


If
\[\tan x = \frac{3ab}{a^2 - b^2},\]where $a > b > 0$ and $0^\circ < x < 90^\circ,$ then find $\sin x$ in terms of $a$ and $b.$

Solution: \boxed{\frac{3ab}{\sqrt{a^4 + 7a^2b^2 + b^4}}}


A penny, a nickel, a half-dollar, and a dollar coin are flipped. If the probability of each coin landing heads is $\frac{1}{2}$, what is the expected value in cents of the total amount that comes up heads? Express your answer in cents, rounded as necessary.


A school has between 180 and 250 students enrolled. Every afternoon, all the students come together to participate in gym class. The students are separated into eight distinct sections of students. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


Among the eight numbers 1234, 1468, 1520, 1672, 1854, 2010, 2256, and 2409, six numbers have a mean (average) of 1800. What is the mean (average) of the other two numbers?


If $g(x) = \sqrt[3]{\frac{2x+6}{4}}$, for what value of $x$ will $g(3x) = 3(g(x))$? Express your answer in simplest form.


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


If $f(x) = ax^2 + bx + c$ and $f^{-1}(x) = cx^2 + bx + a$ with $a, b, c$ real numbers, find the value of $a + c$.


Compute $\arctan(\tan 75^\circ - 2\tan 30^\circ)$. (Express your answer in degrees as an angle between $0^\circ$ and $180^\circ$.)


A right circular cone with base radius $r$ and height $h$ rolls on its side on a horizontal surface. The cone completes $20$ full rotations before returning to its starting orientation. The point where the cone's base meets the surface traces a circular arc centered at the vertex of the cone. Find the ratio $h/r$ in the form $p\sqrt{q}$, where $p$ and $q$ are integers, and $q$ is not divisible by the square of any prime. Determine $p + q$.


A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $7.5$ cm, $9.5$ cm, and $12$ cm. What is the area of the square in square centimeters?


A square is inscribed in the ellipse
\[\frac{x^2}{4} + \frac{y^2}{8} = 1,\]
with its sides parallel to the coordinate axes. Find the area of the square.


Suppose $x$, $y$, $z$, and $w$ are integers satisfying: $x-y+z=10$, $y-z+w=15$, $z-w+x=9$, and $w-x+y=4$. What is the value of $x+y+z+w$?


Jonathan purchases a rectangular mirror and places it within a wooden frame. The overall dimensions of the framed mirror are 100 cm by 50 cm. The frame itself has a uniform width of 8 cm around the mirror. What is the area of the mirror?

Solution: \boxed{2856 \text{ cm}^2}


A theorem states that for any five points in the plane, with no three collinear, there exists a unique conic section that passes through all of them. Consider the five points \[(0, 0),\; (0, 4),\; (6, 0),\; (6, 4),\; (-3, 2)\]. Determine the type of conic section and find the length of its minor axis assuming it is an ellipse with axes parallel to the coordinate axes.


Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=8$ and $HQ=3$, then calculate $(BP)(PC)-(AQ)(QC)$.


The values of \( p, q, r, \) and \( s \) are 2, 3, 4, and 5, but not necessarily in that order. What is the largest possible value of the sum of the four products \( pq, qr, rs, \) and \( sp \)?


The average of 12 numbers is 90. If the numbers 80 and 85 are removed from the set of numbers and two numbers, each 75, are then added, what is the average of the remaining numbers?


Suppose $\cos R = \frac{5}{13}$ in the triangle below. What is $QS$?

[asy]
pair Q,R,S;
S = (0,0);
Q = (sqrt(144),0);
R = (sqrt(144),-5);
draw(S--Q--R--S);
draw(rightanglemark(S,Q,R,13));
label("$S$",S,NW);
label("$Q$",Q,NE);
label("$R$",R,SE);
label("$13$",(R+S)/2,SW);
[/asy]


A function $g$ from the integers to the integers is defined as follows:
\[ g(n) = \left\{
\begin{array}{cl}
n + 5 & \text{if $n$ is even}, \\
\frac{n+1}{2} & \text{if $n$ is odd}.
\end{array}
\right. \]
Suppose $k$ is even and $g(g(g(k))) = 61$. Find $k$.

Solution: \boxed{k = 236}


In triangle $ABC$, where $\angle B = 90^\circ$, $AC = 25$, and $\tan C = 3\sin C$. What is $BC$?


A rectangle has integer length and width, and a perimeter of 200 units. Calculate the number of square units in the least possible area.


On a 12-hour clock, an elapsed time of three hours looks the same as an elapsed time of 15 hours. Because of this, we can say that three hours is "clock equivalent" to its square number of hours. What is the least whole number of hours that is greater than 3 hours and is "clock equivalent" to its square number of hours?


In the diagram shown, $\overrightarrow{OA} \perp \overrightarrow{OC}$ and $\overrightarrow{OB} \perp \overrightarrow{OD}$. If $\angle AOD$ is 4 times $\angle BOC$, what is $\angle AOD$? The diagram remains unchanged from the original setup.


Given the set $\{1, 2, 3, \ldots, 19, 20\},$ determine how many different positive integers can be represented as:
1. The difference of two distinct members of the set.
2. The sum of two distinct members of the set.


David continues to manage his savings account for one more month after April. His balances over these five months, from January through May, are as follows: $100, $200, $150, $150, and $180. What is the average monthly balance of David's savings account over this five-month period?


Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.00001, 0.0001)$.


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


Find the matrix $\mathbf{N}$ such that
\[\mathbf{N} \mathbf{v} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix} \times \mathbf{v}\] for all vectors $\mathbf{v}.$

Solution: \boxed{\begin{pmatrix} 0 & -4 & -1 \\ 4 & 0 & -3 \\ 1 & 3 & 0 \end{pmatrix}}


Consider a year where each month has a unique number of days ranging from 28 to 31. A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, March 4 is not a relatively prime date because the greatest common factor of 3 and 4 is greater than 1. Given that September has 30 days and is represented by the number 9, calculate how many relatively prime dates are in September.


A line segment begins at $(2, y)$ and ends at point $(6, 10)$. The length of the line segment is 10 units. What are the coordinates of the starting point $(2, y)$ where $y > 0$?

Solution: \boxed{(2, 10 + 2\sqrt{21})}


Solution: \boxed{3x^3 - 4x^2 - 62x + 116}


The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 110^{\circ}$ and $\angle AOB = 150^{\circ}$. Determine the degree measure of $\angle ABC$.


Let set $\mathcal{C}$ be a 70-element subset of $\{1,2,3,\ldots,120\}$, and let $U$ be the sum of the elements of $\mathcal{C}$. Find the number of possible values of $U$.


Points $P$, $Q$, $R$, and $S$ lie in the plane of the square $EFGH$ such that $EPF$, $FQG$, $GRH$, and $HSE$ are equilateral triangles. If $EFGH$ has an area of 36, find the area of $PQRS$. Express your answer in simplest radical form.

Solution: \boxed{72 + 36\sqrt{3}}


Find the minimum value of the expression
\[ x^2 + xy + y^2 - 3y \]
over all real numbers $x$ and $y$.


Let $z$ be a nonreal complex number such that $|z| = 2.$ Find the real part of $\frac{1}{2 - z}$.


The graph of \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] has its foci at $(0,\pm 5),$ while the graph of \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] has its foci at $(\pm 7,0).$ Compute the value of $|ab|$.

Solution: \boxed{2 \sqrt{111}}


Rachel places 480 marbles into $m$ total boxes such that each box contains an equal number of marbles. Each box must contain more than one marble, and there must be more than one box. For how many values of $m$ can this be done?


In a right triangle $ABC$, where $\angle B = 90^\circ$, $\angle A = 40^\circ$, and $BC = 7$. Find the length of $AB$ to the nearest tenth. Use a calculator for this problem.


Consider points $P = (1, -2, 3)$, $Q = (3, -6, 6)$, $R = (2, -1, 1)$, and $S = (4, -5, 4)$ in space. Determine whether these points form a parallelogram and find the area of this parallelogram if they do.


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


Find the smallest positive integer $b$ for which $x^2 + bx + 1800$ factors into a product of two polynomials, each having integer coefficients.


Convert the point $(-2,1)$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2\pi$.

Solution: \boxed{\left( \sqrt{5}, \pi - \tan^{-1}\left(\frac{1}{2}\right) \right)}


On a $12$-hour clock, find the least whole number of hours that is greater than $6$ hours and is "clock equivalent" to both its square number of hours and its cube number of hours.


Two sides of an isosceles triangle are 8 inches and 16 inches. Given that the shortest side of a similar triangle is 40 inches, determine the perimeter of this larger triangle.

Solution: \boxed{200 \text{ inches}}


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


Let $y_1, y_2, \ldots, y_n$ be real numbers which satisfy $|y_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|y_1| + |y_2| + \dots + |y_n| = 23 + |y_1 + y_2 + \dots + y_n|.\] What is the smallest possible value of $n$?


Evaluate
\[
\begin{vmatrix} x + 2 & x & x \\ x & x + 3 & x \\ x & x & x + 4 \end{vmatrix}.
\]

Solution: \boxed{26x + 24}


A collection of four positive integers has a mean of 6.5, a unique mode of 6, and a median of 7. If a 10 is added to the collection, what is the new median?


Translating the graph of $y = 3x^2 + 2x - 5$ three units to the left yields the new graph $y = ax^2 + bx + c$. Calculate the value of $a + b + c$.


Determine the number of intersection points for the lines represented by the equations $2y - 3x = 2$, $5x + 3y = 6$, and $x - 4y = 8$ in the plane.


Ken owns 15 pairs of shoes, consisting of eight identical black pairs, four identical blue pairs, and three pairs of green shoes which are all distinct. If he picks two shoes at random, what is the probability that they are from the same color set with one being a left shoe and the other a right shoe? Express your answer as a common fraction.


Find the number of ordered pairs \((x,y)\) of real numbers such that
\[
9^{x^2 + y} + 9^{x + y^2} = \frac{1}{3}.
\]


How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6?


Let $g(x) = x^2 - 3x + 2$. What is the value of $g(g(g(g(g(g(2))))))$?


Find the quotient when $x^5 + 5$ is divided by $x - 1.$

Solution: \boxed{x^4 + x^3 + x^2 + x + 1}


In how many ways can 6 people be seated around a round table if person A and person B must always sit next to each other?


A special coin is designed so that the probability of flipping heads is $\frac{2}{5}$ and the probability of flipping tails is $\frac{3}{5}$. If you flip a heads, you win $\$5$, but if you flip tails, you lose $\$4$. What is the expected value, in dollars, of your winnings after one flip?


A line passes through the points \(\begin{pmatrix} -3 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\). Determine the value of \(b\) if the line's direction vector can be expressed in the form \(\begin{pmatrix} b \\ -1 \end{pmatrix}\).


In triangle $PQR$, points $M$ and $N$ are midpoints of sides $\overline{PQ}$ and $\overline{PR}$, respectively. The area of $\triangle PQR$ is 50 square units. Determine the area of $\triangle QMN$.


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

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


Given that $x$ is a positive integer less than 150, how many solutions does the congruence $x + 20 \equiv 75 \pmod{45}$ have?