A circle is tangent to the lines $3x + 4y = 40$ and $3x + 4y = -20.$ The center of the circle lies on the line $x - 3y = 0.$ Find the center of the circle.

Solution: \boxed{\left(\frac{30}{13}, \frac{10}{13}\right)}


My friend Ana only likes numbers that are divisible by 6. How many different ones digits are possible in numbers that Ana likes?


Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^3+z^3|$.


Eight 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is exactly equal?


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


Let $x$ and $y$ be positive numbers such that $x^2 y$ is constant. Suppose $y = 8$ when $x = 3$. Find the value of $x$ when $y$ is $72$ and $x^2$ has increased by a factor of $z = 4$.


Let $p, q, r, s, t, u, v$ and $w$ be distinct elements in the set \[
\{-9, -8, -4, -1, 1, 5, 7, 10\}.
\] What is the minimum possible value of \[
(p+q+r+s)^{2} + (t+u+v+w)^{2}?
\]


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


How many positive four-digit integers are divisible by both 12 and 8?


Let $x$ and $y$ be positive real numbers such that $5x + 3y < 90.$ Find the maximum value of
\[xy(90 - 5x - 3y).\]


Let $x$, $y$, $z$ be positive real numbers where \( z = x^2 \). Find the set of all possible values of
\[ f(x,y,z) = \frac{x}{x+y} + \frac{y}{y+z} + \frac{z}{z+x} \]

Solution: \boxed{(1, 2)}


How many integers from 1 through 9999, inclusive, do not contain any of the digits 2, 3, 4, 5, or 9?


Three dice are rolled. Two of these dice are regular cubic dice with faces numbered from 1 to 6, and the third die has faces numbered with odd numbers only (1, 3, 5, 7, 9, 11). What is the probability that the sum of the numbers shown on the top faces of all three dice is even?


What integer $n$ satisfies $0 \leq n < 59$ and $$58n \equiv 20 \pmod {59}~?$$


From the new diagram below, find the positive difference in the $x$-coordinates when lines $l$ and $m$ reach $y = 20.$ Assume line $l$ passes through points $(0,10)$ and $(4,0)$, and line $m$ passes through points $(0,3)$ and $(9,0).$


The surface area of a sphere is given as $256\pi\text{ cm}^2$. Calculate the volume of the sphere, expressing your answer in terms of $\pi$.


In her science project, Emily finds that she can create a certain solution by mixing 0.08 liters of chemical B with 0.04 liters of water, yielding 0.12 liters of this solution. She needs to make a total of 1 liter of this new solution for her experiment. How many liters of chemical B will she need to use?


Linda sells three sandwiches and four cookies for $\$$4.20 and she sells four sandwiches and three cookies for $\$$4.50. Assuming a fixed price per item, what is the cost, in dollars, of four sandwiches and five cookies? Express your answer as a decimal to the nearest hundredth.


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

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


A point \((x,y)\) is randomly picked from inside the rectangle with vertices \((0,0)\), \((6,0)\), \((6,1)\), and \((0,1)\). What is the probability that \(x < 2y\)?


If 8 micks = 3 mocks and 5 mocks = 6 macks, how many micks are equal to 30 macks?

Solution: \boxed{\frac{200}{3} \text{ micks}}


Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to the two radii and the arc at three points. Determine the radius of the inscribed circle, expressed in simplest radical form.

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


How many prime numbers are there between 60 and 90?


If $\frac{7}{26}$ is expressed in decimal form, what digit is in the 105th place to the right of the decimal point?


Cara is sitting at a circular table with her six friends. How many different sets of two friends can Cara be directly sitting between?


If $x, y, z$ satisfy the system of equations
\begin{align*}
y + z &= 17 - 2x \\
x + z &= 1 - 2y \\
x + y &= 8 - 2z,
\end{align*}
what is $x + y + z$?


Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \epsilon$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \epsilon - \delta \epsilon^2.\]
Determine the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, 4, 5, 7\}$?


Find the remainder when \(x^4 + x^2 + 1\) is divided by \(x^2 - 4x + 7.\)

Solution: \boxed{12x - 69}


Given that $ABC$ is a triangle with coordinates $A=(0,0), B=(24,10)$, and both coordinates of $C$ are integers, determine the minimum area of $\triangle ABC$.


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


Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 40$, $b_1 = 75$, $b_n=0$, and $$ b_{k+1} = b_{k-1} - \frac{4}{b_k} $$ for $k = 1, 2, \dots, n-1$. Find $n$.


The equation $y = -18t^2 + 30t + 60$ describes the height (in feet) of a ball thrown upwards at $30$ feet per second from $60$ feet above the ground. Find the time (in seconds) when the ball will hit the ground. Express your answer as a common fraction.

Solution: \boxed{\frac{5 + \sqrt{145}}{6}}


Find the phase shift and the amplitude of the graph of $y = 3\sin(5x - \frac{\pi}{2})$.


What is the probability that Claudia gets at least 2 more heads than tails if she flips 12 coins?


Convert \(142_{10}\) to base 7. Represent any values normally represented by new symbols in decimal as \(A\) (if 10), \(B\) (if 11), and so on.


As \( n \) ranges over the positive integers, what is the maximum possible value for the greatest common divisor of \( 13n + 4 \) and \( 7n + 2 \)?


A large tank can be filled by pipes A, B, or C. Pipes A and B together take 5 hours to fill the tank. Pipes A and C together take 8 hours to fill the tank. Pipes B and C together would take 10 hours to fill the tank. The volume of the tank is 160 cubic meters. How many hours does it take pipes A, B, and C working together to fill the tank? Express your answer as a simplified fraction.


Let $T$ be the sum of all the real coefficients of the expansion of ${(1+ix)}^{2011}$.  What is $\log_{2}(T)$?


How many positive real solutions are there to \(x^{12} + 8x^{11} + 18x^{10} + 2048x^9 - 1638x^8 = 0\)?


Phil rolls 7 fair 6-sided dice. What is the probability that at least two dice show the same number?


In the diagram below, we have $\sin \angle RPQ = \frac{3}{5}$. Given that $\angle RPQ$ and $\angle RPS$ are supplementary angles, what is $\cos \angle RPS$?

[asy]
pair R,P,Q,S;
S = (-2,0);
P = (0,0);
Q = (2,0);
R = rotate(aSin(3/5))*(1.5,0);
dot("$S$",S,S);
dot("$Q$",Q,S);
dot("$R$",R,N);
dot("$P$",P,S);
draw(Q--S);
draw(P--R);
[/asy]


Find the remainder when $4x^5 - 3x^4 + 5x^3 - 7x^2 + 3x - 10$ is divided by $x - 2$.


How many integers between 1 and 500 are multiples of both 4 and 6 but not of either 5 or 9?


Five dice with faces numbered 1 through 6 are stacked in a similar manner to the original problem. Ten of the thirty faces are visible, leaving twenty faces hidden. The visible numbers are 1, 2, 2, 3, 3, 3, 4, 4, 5, and 6. What is the total number of dots NOT visible in this view?


Define a sequence of complex numbers by $z_1 = 0$ and
\[z_{n + 1} = z_n^2 - i\]
for all $n \ge 1$. In the complex plane, how far from the origin is $z_{101}$?


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


If the ratio of $b$ to $a$ is 4, then what is the value of $a$ when $b = 18 - 6a$?


A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, of which $30$ are triangular and $14$ are quadrilaterals. Determine the number of space diagonals this polyhedron has.


The greatest common divisor of two positive integers is $(x+5)$ and their least common multiple is $x(x+5)$, where $x$ is a positive integer. If one of the integers is 60, what is the smallest possible value of the other one?


Triangle $MNP$ has side lengths $MN = 10$, $NP = 24$, and $PM = 26$. Rectangle $WXYZ$ has vertex $W$ on $\overline{MN}$, vertex $X$ on $\overline{MP}$, and vertices $Y$ and $Z$ on $\overline{NP}$. In terms of the side length $WX = \delta$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[\text{Area}(WXYZ) = \gamma \delta - \delta \epsilon^2.\] Determine the coefficient $\epsilon = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.


In triangle $\triangle ABC$, where $\triangle ABC$ is a right triangle at $B$, $\tan A = \frac{4}{3}$. Given that $AB = 4$, find the length of side $AC$.


The circle $2x^2 + 2y^2 - 8x - 12y + 24 = 0$ is inscribed inside a square with sides parallel to the x-axis. Determine the area of the square.


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


What is $1254_6 + 3452_6$? Express your answer in base $6$.


A point has rectangular coordinates $(-3, -4, 5)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, -\theta, \phi)$.


Suppose you have 8 shirts and 7 ties. You also decide each day whether to wear a hat or not, and you have 2 hats to choose from. How many different outfits consisting of a shirt, a tie, and an optional hat can you create?


Expand $-(4-c)(c+2(4-c) + c^2)$. What is the sum of the coefficients of the expanded form?


There exist constants \(b_1, b_2, b_3, b_4, b_5\) such that
\[\sin^5 \theta = b_1 \sin \theta + b_2 \sin 2 \theta + b_3 \sin 3 \theta + b_4 \sin 4 \theta + b_5 \sin 5 \theta\]
for all angles \(\theta.\) Find \(b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2.\)


Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and
\[
|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|.
\]
What is the smallest possible value of $n$?


Let $x,$ $y,$ $z,$ $p,$ $q,$ $r$ be positive real numbers such that $x + y + z + p + q + r = 10.$  Find the minimum value of
\[\frac{1}{x} + \frac{9}{y} + \frac{4}{z} + \frac{25}{p} + \frac{16}{q} + \frac{36}{r}.\]


Thomas has twelve socks in his drawer: three of each color (red, blue, green, and yellow). He randomly selects five socks. What is the probability that he has exactly two pairs of socks with different colors?


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

Solution: \boxed{3x^3 - 14x^2 + 34x - 36}


Let $ABCDEFGH$ be a cube with each edge of length $s$. A right square pyramid is placed on top of the cube such that its base aligns perfectly with the top face $EFGH$ of the cube, and its apex $P$ is directly above $E$ at a height $s$. Calculate $\sin \angle FAP$.


The sum of 36 consecutive integers is $6^4$. What is their median?


A rectangular prism has six faces and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many total diagonals does a rectangular prism have if it measures differently along length, width, and height?


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


A rectangular mirror (the shaded region) is perfectly centered within a larger frame, which itself is within an even larger frame. The outermost perimeter of the largest frame measures 100 cm by 140 cm. Each side of this frame has a width of 15 cm. Inside this, there is another frame, whose sides are also 15 cm wide. What is the area of the mirror?

Solution: \boxed{3200 \text{ cm}^2}


Find the coefficient of the $x^2$ term in the expansion of the product $(2x^3 + 4x^2 - 3x + 5)(3x^2 - 9x + 1)$.


There are 7 blue chips, 6 red chips and 5 yellow chips in a bag. One chip is drawn and placed back into the bag, followed by a second and a third draw under the same conditions. What is the probability that the second and third chips are different in color?


Determine the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{d} & \frac{1}{c} \\ \frac{1}{b} & \frac{1}{a} \end{pmatrix} \renewcommand{\arraystretch}{1}.
\]


How many lattice points lie on the hyperbola \(x^2 - y^2 = 1800^2\)?


A car travels due east at 1 mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 60 miles, moves southeast at $\sqrt{2}$ miles per minute. At time $t=0$, the center of the storm is 150 miles due north of the car. Determine $\frac{1}{2}(t_1+t_2)$, where $t_1$ and $t_2$ are the times when the car enters and leaves the storm circle, respectively.


What is the number of degrees in the acute angle formed by the hands of a clock at 3:30?


Solution: \boxed{(-\infty, 1) \cup (2, 4) \cup (5, 6) \cup (7, \infty)}


The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ follows the recursive formula
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \geq 1$. If $a_7 = 210$, determine $a_8$.


Find the integer \(n\), \(0 \le n \le 360\), such that \(\cos n^\circ = \cos 340^\circ.\)


Find the quotient when \(x^5 - 21x^3 + 8x^2 - 17x + 12\) is divided by \(x-3\).

Solution: \boxed{x^4 + 3x^3 - 12x^2 - 28x - 101}


Determine the domain of the function $g(x) = \tan(\arcsin(x^3))$.

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


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


What is the total number of digits used when the first 1998 positive even integers are written?


What is the first year after 2000 for which the sum of the digits is 15?


What is the value of $34_{10}+27_{10}$ in base 5?


Let $b_n = \frac{8^n - 1}{7}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. Determine the maximum possible value that $e_n$ can take.


Consider triangle $XYZ$ with vertices at $X(-1,4)$, $Y(5,2)$, and $Z(3,10)$. Find the point $P$ with coordinates $(m,n)$ inside the triangle such that the three smaller triangles $XYP$, $XZP$, and $YZP$ all have equal areas. What is the value of $10m + n$?


If $\frac{720}{2^6 \cdot 3^5}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point?


Find the dot product of $\begin{pmatrix} 4 \\ -5 \\ 2 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} -6 \\ 3 \\ -4 \\ 2 \end{pmatrix}.$


Let $a$, $b$, $c$, and $d$ be real numbers such that $|a-b|=4$, $|b-c|=5$, and $|c-d|=6$. Additionally, let $a + b + d = 10$. Find the sum of all possible values of $|a-d|$.


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


Convex hexagon $ABCDEF$ has exactly two distinct side lengths. Side $AB$ measures 8 units, and side $BC$ measures 10 units. The perimeter of hexagon $ABCDEF$ is 58 units. How many sides of hexagon $ABCDEF$ have measure 10 units?


What is the greatest integer less than 150 for which the greatest common divisor of that integer and 16 is 8?


Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2?


In triangle \(XYZ,\) \(XY = 5,\) \(XZ = 7,\) \(YZ = 9,\) and \(W\) lies on \(\overline{YZ}\) such that \(\overline{XW}\) bisects \(\angle YXZ.\) Find \(\cos \angle YXW.\)


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


Find the last two digits of the following sum: $$3! + 6! + 9! + 12! + \cdots + 120!$$


Determine the number of solutions to the equation:
\[3\sin^3 x - 7 \sin^2 x + 4 \sin x = 0\]
in the range $0 \le x \le 2\pi$.


Tamika selects two different numbers at random from the set $\{7, 11, 14\}$ and adds them. Carlos takes two different numbers at random from the set $\{2, 5, 7\}$ and multiplies them. What is the probability that Tamika's result is greater than or equal to Carlos' result? Express your answer as a common fraction.

Solution: \boxed{\frac{6}{9} = \frac{2}{3}}


Let \(A = (1, 0), B = (4, 3),\) and \(C = (p, q)\) be three points on the parabola \(y = x^2 - 4x + 3,\) where \(1 \leq p \leq 4.\) Find the largest possible area of triangle \(ABC.\)


Two concentric circles are centered at point Q. The sides of a 60-degree angle at Q form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 48-degree angle at Q. Determine the ratio of the area of the smaller circle to the area of the larger circle. Express your answer as a common fraction.


There are 4 blue chips, 3 red chips, 2 yellow chips, 5 green chips, and 6 orange chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors?


The function $g$ is defined on the set of integers and satisfies \[g(m) = \begin{cases} m-3 & \mbox{if } m \ge 1010 \\ g(g(m+5)) & \mbox{if } m < 1010. \end{cases}\] Find $g(978)$.


Let \( p \) and \( q \) be real numbers so that the roots of
\[ z^2 + (12 + pi) z + (30 + qi) = 0 \]
are complex conjugates. Determine the ordered pair \( (p, q) \).


Let \(x\) be a real number. Consider the following five statements:

1. \(0 < x^2 < 4\)
2. \(x^2 > 1\)
3. \(-2 < x < 0\)
4. \(1 < x < 2\)
5. \(0 < x - x^2 < 2\)

What is the maximum number of these statements that can be true for any value of \(x\)?


Determine the sum of all integer solutions to $|n| < |n-4| < 6$.


Find the equation of the directrix of the parabola $x = -\frac{1}{8} y^2.$

Solution: \boxed{x = 2}


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


Determine $B^2$, where $B$ is the sum of the absolute values of all roots of the equation:
\[x = \sqrt{34} + \frac{57}{{\sqrt{34}+\frac{57}{{\sqrt{34}+\frac{57}{{\sqrt{34}+\frac{57}{{\sqrt{34}+\frac{57}{x}}}}}}}}}.\]


During a study session, Emily reviews a problem involving a quadratic $x^2 - tx + q$, with roots $\alpha$ and $\beta$. She discovers that \[\alpha + \beta = \alpha^2 + \beta^2 = \alpha^3 + \beta^3 = \cdots = \alpha^{2010} + \beta^{2010}.\] She is intrigued to determine the greatest possible value of \[\dfrac{1}{\alpha^{2012}} + \dfrac{1}{\beta^{2012}}.\]


The actual diameter of a circle is 30 cm, but Maria measures it with an error of up to 10%. What is the largest possible percent error, in percent, in Maria's computed area of the circle?


Let \( g(x) \) be a polynomial of degree 500 with real coefficients, and let its roots be \( s_1, s_2, \dots, s_{500} \). There are exactly 250 distinct values among
\[ |s_1|, |s_2|, \dots, |s_{500}|. \]
What is the minimum number of real roots that \( g(x) \) can have?


What is the inverse of the function \( h(x) = 6 - 3x \)?

Solution: \boxed{j(x) = \frac{6 - x}{3}}


John decides to draw a regular octagon in the sand. At each of the 8 vertices, he places one of eight different pebbles. How many ways can he place the pebbles, if reflections and rotations of an arrangement are considered equivalent?


What is the mean of the set $\{m, m + 6, m + 8, m + 11, m + 18, m + 20\}$ if the median of the set is 19?


The algae in a local lake expands to cover double its area each day. On the 24th day of a particular month, the lake is completely covered by algae. Determine the day of the month when only 12.5% of the lake was covered by algae.


How many perfect fourth powers are between 50 and 2000?


Suppose that all four of the numbers \[3 - \sqrt{8}, \;5 + \sqrt{11}, \;15 - \sqrt{28}, \;-2 - \sqrt{3}\] are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?


Let $Q$ be a quartic polynomial such that $Q(0) = k$, $Q(1) = 3k$, and $Q(-1) = 5k$. What is $Q(2) + Q(-2)$?


What is the sum of all the even integers between $200$ and $400$?


Three vertices of a cube are \(A=(3,7,4)\), \(B=(4,3,-5)\), and \(C=(7,-2,3)\). Determine the surface area of the cube.


Solve for $z$ in the equation $3 + 2iz = 5 - 3iz$ (where $i^2 = -1$).


Let \( S \) be the sum of all positive integers \( n \) such that \( n^2 + 12n - 1985 \) is a perfect square. Find the remainder when \( S \) is divided by \( 500 \).


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

Solution: \boxed{6(2y^6 + 7y^4 - 3)}


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


Let \( b \equiv (4^{-1} + 6^{-1} + 9^{-1})^{-1} \pmod{13} \). What is the remainder when \( b \) is divided by 13?


The sum of the positive divisors of a positive integer of the form $2^i 3^j 5^k$ is equal to $3600$. What is $i+j+k$?


The function $g$ is graphed below. Each small box on the grid has width and height 1.

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

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

import graph;

real i;

if(complexplane) {

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

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

} else {

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

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

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

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

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

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

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

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

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

}

if(useticks) {

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

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

} else {

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

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

}
};
rr_cartesian_axes(-1,9,-1,9);

dot((0,5),red+5bp);
dot((2,4),red+5bp);
dot((4,3),red+5bp);
dot((6,2),red+5bp);
dot((8,1),red+5bp);
dot((1,0),red+5bp);
dot((3,8),red+5bp);
dot((5,7),red+5bp);
dot((7,6),red+5bp);
dot((9,5),red+5bp);

[/asy]

Annie writes the number 4 on her thumb. She then applies $g$ to 4 and writes the output on her index finger. If Annie continues this process of applying $g$ and writing the output on a new finger, what number will Annie write on her tenth finger?


The parabola $y = ax^2 + bx + c$ has a vertex at $(h, k)$ and passes through the point $(0, -k)$, where $k \neq 0$. Determine the value of $b$.


A regular octagon $Q_1$ is plotted in the Cartesian plane. The sum of the $x$-coordinates of the $8$ vertices equals 512. The midpoints of the sides of $Q_1$ form a second octagon, $Q_2$. Again, the midpoints of the sides of $Q_2$ form a third octagon, $Q_3$. Determine the sum of the $x$-coordinates of the vertices of $Q_3$.


If I roll a fair eight-sided die five times, what is the probability that I will roll the number $2$ exactly two times?


Define $A \star B$ as $A \star B = \frac{(A+B)}{4}$. What is the value of $(3\star 8) \star 6$?


Let $x$ and $y$ be real numbers such that $3(x^3 + y^3) = x + y$ and $x + y = 1$. Find the maximum value of $x^3 - y^3$.


$256_{10}$ in base \( b \) has exactly $6$ digits. What is the value of \( b \)?


Alexio now has 120 cards numbered from 1 to 120, inclusive, and places them in a box. He then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3, or 5? Express your answer as a common fraction.


If \( n = 2^{12} \cdot 3^{15} \cdot 5^{9} \), how many of the natural-number factors of \( n \) are multiples of 180?


In triangle $ABC,$ where $AB = 6,$ $BC = 8,$ and $CA = 4,$ find the coordinates of the incenter $I$ using vectors. Here, the ordered triple $(x, y, z)$ represents the vector $\overrightarrow{I} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C}$ where $x, y,$ and $z$ are constants such that $x + y + z = 1.$

Solution: \boxed{\left( \frac{2}{9}, \frac{1}{3}, \frac{4}{9} \right)}


Given two vectors, find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} + t \begin{pmatrix} -4 \\ 3 \\ 5 \end{pmatrix}\]
is closest to
\[\bold{a} = \begin{pmatrix} -2 \\ 7 \\ 0 \end{pmatrix}.\]

Solution: \boxed{t = \frac{17}{25}}


The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?


In a sequence of four positive integers, the first three terms form a geometric progression, the last three terms form an arithmetic progression, and the first and fourth terms differ by $40$. Find the sum of the four terms.


The stem and leaf plot represents the heights, in inches, of the players on the Hilltop High School boys' soccer team. What is the mean height of the players on the team? (Note: $5|4$ represents 54 inches.)

Height of the Players on the Soccer Team (inches)

$4|7\;8$

$5|0\;0\;4\;5\;7\;9$

$6|3\;3\;4\;5$

Solution: \boxed{56.25 \text{ inches}}


Solve
\[\arcsin x + \arcsin 3x = \frac{\pi}{2}.\]


Determine how many pairs of positive integers \( (m, n) \) satisfy \( m^2 + 3n < 50 \).


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 "MIST"? Express your answer as a common fraction.


A band has 90 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 4 and 15 students per row, in how many possible row-lengths can the band be arranged?


Bob rolls a fair six-sided die each morning. If Bob rolls an even number greater than 2, he eats sweetened cereal. If he rolls an odd number greater than 2, he eats unsweetened cereal. If Bob rolls a 1 or a 2, then he rolls again. In a leap year, what is the expected number of times Bob will roll his die?


Define a sequence of complex numbers by $z_1 = 1$ and
\[z_{n + 1} = z_n^2 - 1 + i\] for all $n \ge 1.$ In the complex plane, how far from the origin is $z_5$?


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


A pirate looted another ship called S.S. Fortune. He acquired $1432_5$ dollars worth of silver coins, $2134_5$ dollars worth of exotic spices, $3021_5$ dollars worth of silk fabrics, and $231_5$ dollars worth of rare books. What is the total dollar amount of his loot? Express your answer in base 10.


Sara used $\frac{2}{5}$ of a roll of wrapping paper to wrap five presents. If she used the same amount of wrapping paper on each present, what fraction of a roll of wrapping paper did she use on each present? Further, how much of the roll is left after wrapping all the presents?


Let $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the side lengths, and let $R$ denote the circumradius. Find $OH^2$ if $R = 8$ and $2a^2 + b^2 + c^2 = 50$.


A convex polyhedron $Q$ has $30$ vertices, $70$ edges, and $42$ faces, $30$ of which are triangular and $12$ of which are hexagons. Determine how many space diagonals $Q$ has.


Choose the largest of the following sums, and express it as a fraction in simplest form:
$$\frac{1}{4} + \frac{1}{5} \times \frac{1}{2}, \ \ \frac{1}{4} - \frac{1}{6}, \ \ \frac{1}{4} + \frac{1}{3} \times \frac{1}{2}, \ \ \frac{1}{4} - \frac{1}{8}, \ \ \frac{1}{4} + \frac{1}{7} \times \frac{1}{2}$$


Let \(0 \le x, y, z, w \le 1\). Find the possible values of the expression
\[
\sqrt{x^2 + (1 - y)^2} + \sqrt{y^2 + (1 - z)^2} + \sqrt{z^2 + (1 - w)^2} + \sqrt{w^2 + (1 - x)^2}.
\]

Solution: \boxed{[2 \sqrt{2}, 4]}


A circle with center $A$ has radius $10$ units and circle $B$ has radius $3$ units. The circles are externally tangent to each other at point $C$. Segment $XY$ is the common external tangent to circle $A$ and circle $B$ at points $X$ and $Y$, respectively. What is the length of segment $AY$? Express your answer in simplest radical form.


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


On a "prime date," both the month and the day are prime numbers. How many prime dates occurred in 2008?


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


What is the remainder when 5032 is divided by 28?


The letters of the alphabet are given numeric values based on the two conditions below.

$\bullet$  Only the numeric values of $-3,$ $-2,$ $-1,$ $0,$ $1,$ $2,$ $3$ are used.

$\bullet$  Starting with A and going through Z, a numeric value is assigned to each letter according to the following pattern: $$
2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 3
$$

Using this new cycle, what is the sum of the numeric values of the letters in the word "complex"?


Solution: \boxed{6 \times 10^9}


Let $a,$ $b,$ $c,$ $d,$ $e,$ $f$ be positive real numbers such that $a + b + c + d + e + f = 10.$  Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c} + \frac{16}{d} + \frac{25}{e} + \frac{36}{f}.\]


A rectangular picture frame is constructed from 1.5-inch-wide pieces of wood. The area of just the frame is \(27\) square inches, and the length of one of the interior edges of the frame is \(4.5\) inches. Determine the sum of the lengths of the four interior edges of the frame.


At Beaumont High School, there are 18 players on the basketball team. All 18 players are taking at least one of physics, chemistry, or biology. If 10 players are taking physics, 7 players are taking biology, 5 players are taking chemistry, 3 players are taking both physics and biology, 2 players are taking both biology and chemistry, and 1 player is taking both physics and chemistry, how many players are taking all three subjects?


There are 6 blue chips, 5 red chips, 4 yellow chips, and 3 green chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


Determine the mean of all solutions for the equation $x^3 + 3x^2 - 44x = 0$.


Solution: \boxed{8x^{12} + 3x^{11} + 7x^{10} + 9x^9 + 5x^7 + x^3}


A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 45 seconds. Jack picks a random five-second time interval to observe the light. What is the probability that he sees the color change during his observation?


Let $PQRS$ be a tetrahedron such that edges $PQ$, $PR$, and $PS$ are mutually perpendicular. Let the lengths of $PQ = a$, $PR = b$, and $PS = c$. If the areas of triangles $PQR$, $PRS$, and $PQS$ are denoted by $u$, $v$, and $w$ respectively, where $u = \frac{1}{2} ab$, $v = \frac{1}{2} bc$, and $w = \frac{1}{2} ca$, find the area of triangle $QRS$.


A rectangle has its length increased by $30\%$ and its width increased by $15\%$. What is the percentage increase in the area of the rectangle?


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

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


A bag contains three red, three blue, and three green marbles. If four marbles are randomly selected, without replacement, what is the probability that exactly one marble of each color is selected, with one color being chosen twice? Express your answer as a common fraction.


**Question**: Let $r_1, r_2,$ and $r_3$ be the roots of the polynomial equation
\[ x^3 - 4x^2 + 9 = 0. \]
Find the monic polynomial, in $x$, whose roots are $3r_1, 3r_2,$ and $3r_3$.

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


By partial fractions, express
\[\frac{1}{x(x + 1)(x + 2)(x + 3)(x + 4)(x+5)}\]
as
\[\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2} + \frac{D}{x + 3} + \frac{E}{x + 4} + \frac{F}{x + 5}\]
for some constants $A,$ $B,$ $C,$ $D,$ $E,$ and $F.$ Find $A + B + C + D + E + F.$


The base of a triangular piece of paper $ABC$ is $18\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the triangle. The area of the triangle that projects below the base is $25\%$ that of the area of triangle $ABC$. What is the length of $DE$, in cm?

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


**Find the value of $k$ such that the lines defined by the vectors**
\[ \begin{pmatrix} 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 6 \\ -9 \end{pmatrix} \]
**and**
\[ \begin{pmatrix} -1 \\ 0 \end{pmatrix} + s \begin{pmatrix} 3 \\ k \end{pmatrix} \]
**are parallel.**


Evaluate the expressions:
\[A = 1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 41 \times 42 + 43 \times 44\]
\[B = 1 + 2 \times 3 + 4 \times 5 + \cdots + 40 \times 41 + 42 \times 43\]
and find the positive difference between integers $A$ and $B$.


It is now 3:00:00 PM, as read on a 12-hour digital clock. In 137 hours, 58 minutes, and 59 seconds the time will be $X:Y:Z$. What is the value of $X + Y + Z$?


Let \( a, b, \) and \( c \) be the roots of the polynomial
\[ x^3 - 4x^2 + 6x - 3 = 0. \]
Find the monic polynomial in \( x \) whose roots are \( a + 3, b + 3, \) and \( c + 3 \).

Solution: \boxed{x^3 - 13x^2 + 57x - 84}


A cylindrical tower has a diameter of 25 feet and a height of 60 feet. A green stripe with a width of 2 feet wraps twice around the cylinder horizontally. Calculate the area of the stripe in square feet.

Solution: \boxed{100\pi \text{ square feet}}


What is the sum of all two-digit positive integers whose squares end with the digits 25?


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


What is the distance, in units, between the points $(1, 2)$ and $(4, 6)$? Express your answer in simplest form.


What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{2}{3}x - \frac{y}{3}\right)^8$?


A fair, twenty-faced die has $19$ of its faces numbered from $1$ through $19$ and has one blank face. Another fair, twenty-faced die has $19$ of its faces numbered from $1$ through $7$, $9$ through $19$ and has one blank face. When the two dice are rolled, what is the probability that the sum of the two numbers facing up will be $25?$ Express your answer as a common fraction.


Find the quotient when $x^5 - 17x^3 + 8x^2 - 9x + 12$ is divided by $x - 3$.

Solution: \boxed{x^4 + 3x^3 - 8x^2 - 16x - 57}


Let $M$ be the number of $8$-digit positive integers such that the digits are in both increasing order and even. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed.)


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


Twenty gremlins, twenty imps, and ten sprites are attending the Fantasy Creatures Gala. The imps only shake hands with gremlins and sprites. Gremlins are affable and shake hands with their fellow gremlins as well as the imps. Sprites, the shyest of all, only shake hands with imps. Calculate the total number of handshakes at the gala.


The sum of five positive integers that form an arithmetic sequence is 75. Of all such possible sequences, what is the greatest possible fourth term?


Given that $\sin A + \sin B = 1$ and $\cos A + \cos B = \frac{5}{3}$, what is the value of $\cos (A-B)$?


What is the greatest integer less than 150 for which the greatest common divisor of that integer and 12 is 4?


What is the intersection point of the line $y = -3x + 4$ and the line perpendicular to it that passes through the point $(3, -1)$?

Solution: \boxed{(1.8, -1.4)}


Let $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \\ -4 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} \sqrt{3} \\ 5 \\ -2 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 8 \\ -5 \\ 12 \end{pmatrix}.$ Find the angle between the vectors $\mathbf{a}$ and $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c},$ in degrees.


In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. Determine the sum of all possible values for $k$.


In convex quadrilateral $PQRS$, side $\overline{RS}$ is perpendicular to diagonal $\overline{PQ}$, and side $\overline{PQ}$ is perpendicular to diagonal $\overline{SR}$. Given $RS = 52$ and $PQ = 34$, a line through $Q$ perpendicular to side $\overline{PS}$ intersects diagonal $\overline{PQ}$ at point $T$ where $PT = 9$. Find the length of $QT$.


Alexio now has 200 cards numbered from 1 to 200, inclusive, and places them in a box. He then selects one card at random from the box. What is the probability that the number on the card is a multiple of 4, 5, or 6? Express your answer as a common fraction.

Solution: \boxed{\frac{94}{200} = \frac{47}{100}}


If $\left\lfloor n^2/4 \right\rfloor - \lfloor n/2 \rfloor^2 = 3$, find all integer values of `n`.


Solve the system of equations $y = 5$ and $x^2 + y^2 + 2x - 4y = 80.$ What is the sum of the solutions for $x$?


If $BBB_5$ can be expressed as $44_c$, where $B$ is a digit in base 5 and $c$ is a base greater than 6, what is the smallest possible sum $B+c$?


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


When a polynomial is divided by $3x^3 - 2x^2 + x - 5,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


Jill has 8 red marbles and 4 blue marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are red? Express your answer as a decimal rounded to the nearest thousandth.


From the numbers in the set $\{1, 2, 3, 4, 5, 6, 7\}$, two distinct numbers are selected at random. What is the probability that their product is divisible by 5? Express your answer as a common fraction.


The side lengths of a triangle are 8 m, 15 m, and 17 m. How many square meters are in the area of this triangle, and what is its perimeter?

Solution: \boxed{\text{Area} = 60 \, \text{m}^2; \, \text{Perimeter} = 40 \, \text{m}}


Find the number of positive integers $n \le 1200$ such that $14n$ is a perfect square.


What is the degree of the polynomial $(7x^5 + 3e^x^2 + 4x^3 + 15 + 8\pi x^6 + 3\sqrt{2}x^2 + 2)$?


Find all values of $k$ such that the sum of cubes of the solutions of:
\[3x^2 + 6x + k = 0\]
equals the sum of these solutions itself.


The letters of the English alphabet are assigned numeric values based on the following conditions:

- Only the numeric values of $-2,$ $-1,$ $0,$ $1,$ and $2$ are used.
- Starting with A, a numeric value is assigned to each letter based on the pattern:
$$ 1, 2, 1, 0, -1, -2 $$
This cycle repeats throughout the alphabet.

If the letter A has a value of $1,$ B has a value of $2,$ and Z cycles back starting from the value of Y. Calculate the sum of the numeric values of the letters in the word "algebra".


How many integers $-15 \leq n \leq 15$ satisfy $(n-3)(n+5)(n + 9)<0$?


Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 9?


There are integers $d, e$ for which both roots of the polynomial $x^2 - 2x - 1$ are also roots of the polynomial $x^5 - dx - e$. Determine the product $de$.


Determine the number of six-letter words where the first and last letters are the same, and the second and fifth letters are also the same.


In triangle $XYZ$, we have $\angle Z = 90^\circ$, $XY = 13$, and $YZ = 5$. What are $\tan X$ and $\cos X$?

Solution: \boxed{\tan X = \frac{5}{12}, \cos X = \frac{12}{13}}


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


A state requires that all bicycle licenses consist of the letter B or C followed by any six digits, and the license must end with the digit 5. What is the number of groups of letters and numbers available for bicycle licenses?


Let \(x,\) \(y,\) and \(z\) be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]Find the minimum value of \(x^2 + y^2 + z^2.\)


Given that $y$ is a multiple of $4060$, what is the greatest common divisor of $g(y) = (3y + 5)(8y + 1)(11y + 3)(y + 15)$ and $y$?


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


$500_{10}$ in base $b$ has exactly $6$ digits. What is the value of $b$?


How many two-digit prime numbers can be formed by choosing digits from the set $\{1, 3, 7, 9\}$ for the tens and unit digits, where repetition of digits is allowed?


What is the value of $$\frac{1}{3}\times9\times\frac{1}{27}\times81\times\frac{1}{243}\times729\times\frac{1}{2187}\times6561?$$


Calculate the total number of entries from the $5$th row to the $25$th row in Pascal’s Triangle.


A right circular cone has base radius $r$ and height $h$. The cone rests on its side on a flat surface. When it rolls on the surface without slipping, the point where the cone's base meets the surface traces a circular arc centered at the vertex of the cone. The cone returns to its starting position after making $15$ complete rotations. Determine the value of $h/r$ in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m + n$.


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


Nathan has more than $\$1$ but less than $\$10$ worth of nickels. When he puts his nickels in stacks of 4, he has 2 left over. When he puts them in stacks of 5, he has 2 left over. When he puts them in stacks of 7, he also has 2 left over. How many nickels does Nathan have?

Solution: \boxed{n = 142}


The expression \(2y^2 + 3y - 35\) can be written as \((2y + a)(y + b)\), where \(a\) and \(b\) are integers. What is \(a - b\)?


Let $n$ be a positive integer. If $b \equiv (5^{3n} + 3)^{-1} \pmod{11}$, what is the remainder when $b$ is divided by 11?


The time right now is precisely 6:00 a.m. What time will it be in 2700 minutes?

Solution: \boxed{3\!:\!00 \text{ a.m.}}


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 150^\circ & -\sin 150^\circ \\ \sin 150^\circ & \cos 150^\circ \end{pmatrix}^n = \mathbf{I}.\]


For what value of $n$ is the four-digit number $514n$, with units digit $n$, divisible by 12?


The graph of $y = ax^2 + bx + c$ has a minimum value of 36, and passes through the points $(3,0)$ and $(7,0).$ Find $a + b + c.$


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


Determine the appropriate value of $h$ so that the following equation in base $h$ is accurate:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c}
&&8&6&7&4_h \\
&+&4&3&2&9_h \\
\cline{2-6}
&1&3&0&0&3_h.
\end{array}$$


In a triangle, two of the side lengths are 9 and 10, and the angle between them is $135^\circ$. Find the length of the third side.

Solution: \boxed{\sqrt{181 + 90\sqrt{2}}}


Marius is entering a wildlife photo contest with his nine snow leopards of different heights. If the three shortest leopards insist on being placed at the ends and the middle of the row, how many ways can he line up the leopards?

Solution: \boxed{4320\text{ ways.}}


If $\frac{120}{2^5 \cdot 5^{10}}$ is expressed as a decimal and then we subtract $.000001$ from the result, how many non-zero digits are to the right of the decimal point?


Suppose we wish to divide 12 dogs into three groups, one with 4 dogs, one with 6 dogs, and one with 2 dogs. We need to form these groups such that Rex is in the 4-dog group, Buddy is in the 6-dog group, and Bella and Duke are both in the 2-dog group. How many ways can we form these groups under these conditions?


Find $b^2$ if the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide.


Let us consider a similar problem but with a set of integers up to 15 instead of 12. Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy?


How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt[3]{500}$?


The functions \( g(x) = 4x + c \) and its inverse \( g^{-1}(x) \) intersect at the point \( (-2, d) \), where \( c \) and \( d \) are integers. Determine the value of \( d \).


In the following diagram, triangles $\triangle ABE$, $\triangle BCE$ contain right angles at B and C, with $\angle AEB = \angle BEC = 45^\circ$, and $AE=30$. [asy]
pair A, B, C, D, E;
A=(0,21.2132);
B=(0,0);
C=(15,0);
D=(20,0);
E=(30,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B", B, W);
label("C", C, SW);
label("D", D, SE);
label("E", E, NE);
[/asy] Calculate the area of triangle $ABE$.


For how many values of the digit $A$ is it true that $42$ is divisible by $A$ and $523{,}4A8$ is divisible by $4$?


Consider the polynomials
\[ f(x) = 2 - 6x + 4x^2 - 5x^3 + 7x^4 \]
and
\[ g(x) = 4 - 3x - 7x^3 + 11x^4. \]
Find \( c \) such that the polynomial \( f(x) + c g(x) \) has degree 3.


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ back onto $\mathbf{w}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{4}{5},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{w}\|}.$

Solution: \boxed{\frac{\|\mathbf{q}\|}{\|\mathbf{w}\|} = \frac{4}{5}}


When a number is divided by 11, the remainder is 5. What is the remainder when four times the number minus 9 is divided by 11?


Find the matrix $\mathbf{N}$ such that
\[\mathbf{N} \begin{pmatrix} 2 & -5 \\ 4 & -3 \end{pmatrix} = \begin{pmatrix} -19 & -7 \\ 10 & 3 \end{pmatrix}.\]

Solution: \boxed{\mathbf{N} = \begin{pmatrix} \frac{85}{14} & -\frac{109}{14} \\ -3 & 4 \end{pmatrix}}


Determine how many $x$-intercepts the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) has in the interval $(0.00005, 0.0005)$.


Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{2}{n}\] for all real numbers $x.$


What is the value of \( x \) if a cube's volume is \( 8x \) cubic units and its surface area is \( 4x \) square units?

Solution: \boxed{x = 216}


Let $x, y, z$ be real numbers such that $-1 < x, y, z < 0.$ Find the minimum value of
\[
\frac{1}{(1 - x)(1 - y)(1 - z)} + \frac{1}{(1 + x)(1 + y)(1 + z)}.
\]


Rationalize the denominator of $\frac{5}{4\sqrt{7} + 3\sqrt{9}}$ and express your answer in the form $\frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B<D$, the fraction is in lowest terms, and all radicals are in simplest radical form. What is $A+B+C+D+E$?


If 5 nicks = 3 nacks and 4 nacks = 5 nocks, how many nicks are equal to 40 nocks?


Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 3000$ is it true that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$?


In triangle $\triangle ABC$, where $CX$ bisects $\angle ACB$, the lengths are given as $BC = 33$ units and $AC = 45$ units. $CX$ extends from $C$ and meets line $AB$ at point $X$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Express your answer as a common fraction.


Points \(A\), \(B\), \(C\), and \(T\) are in space such that each of \(\overline{TA}\), \(\overline{TB}\), and \(\overline{TC}\) is perpendicular to the other two. If \(TA = TB = 12\) and \(TC = 10\), then what is the volume of pyramid \(TABC\)?

Solution: \boxed{240 \text{ cubic units}}


Find the sum of the first eight prime numbers that have a units digit of 3.


If 18 bahs are equal to 30 rahs, and 10 rahs are equal in value to 25 yahs, how many bahs are equal in value to 1250 yahs?


How many distinct products can you obtain by multiplying two or more distinct members of the set $\{2, 3, 5, 7, 11, 13\}$?


My five friends and I have a family gathering every month. Each month, three of us prepare the meal while the other three set up the dining area. How many different ways are there for us to choose who cooks and who sets up?


In triangle $ABC$, medians $AM$ and $BN$ are perpendicular. If $AM = 21$ and $BN = 28$, find the length of side $AB$.

Solution: \boxed{AB = \frac{70}{3}}


Let $x$, $y$, and $z$ be positive numbers such that $xyz = 1$, $x + \frac{1}{z} = 3$, and $y + \frac{1}{x} = 31$. Determine the value of $z + \frac{1}{y}$.


Each of four, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair, and the other two dice show different numbers from each other and from the pair?


Solution: \boxed{8xy + 10x - 4xz - 12y + 6z - 15}


For how many values of $c$ in the interval $[0, 1000]$ does the equation \[5 \lfloor x \rfloor + 3 \lceil x \rceil = c\]have a solution for $x$?


How many rows of Pascal's Triangle contain the number $37$?


Say that an integer $A$ is delicious if there exist several consecutive integers, including $A$, that add up to 2024. What is the smallest delicious integer?


Yan is in between his home and the stadium. He has two options to get to the stadium: either walk directly to the stadium, or walk back to his home and then ride his bicycle to the stadium. His bicycle speed is 5 times his walking speed, and both options take the same amount of time. What is the ratio of Yan’s distance from his home to his distance from the stadium?


Calculate
\[
\cos \left( 6 \arccos \frac{1}{4} \right).
\]


Let $a$ and $b$ be real numbers. One of the roots of $x^3 + ax + b = 0$ is $2 + i \sqrt{2}.$ Find $a + b.$


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


Sides $\overline{AB}$ and $\overline{GH}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$?


Let $g(x) = \left\lceil \dfrac{2}{x+3} \right\rceil$ for $x > -3$, and $g(x) = \left\lfloor \dfrac{2}{x+3} \right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Which integer is not in the range of $g(x)$?


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


Tamika chooses two different numbers at random from the set $\{11, 12, 13\}$ and adds them. Carlos selects two different numbers at random from the set $\{4, 6, 7\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction.


In a diagram, rectangle $ABCD$ is divided into four identical squares. If the perimeter of $ABCD$ is 160 cm, what is its area, in square centimeters?

Solution: \boxed{1600 \text{ square centimeters}}


There are four points that are $4$ units from the line $y = 13$ and $10$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?


Let
\[
\mathbf{C} = \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix}.
\]
Compute \(\mathbf{C}^{50}\).

Solution: \boxed{\begin{pmatrix} 101 & 50 \\ -200 & -99 \end{pmatrix}}


Let $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 3:5$. Then, describe $\overrightarrow{P}$ as a combination of $\overrightarrow{A}$ and $\overrightarrow{B}$:
\[\overrightarrow{P} = s \overrightarrow{A} + v \overrightarrow{B}\] for some constants $s$ and $v$. Enter the ordered pair $(s, v)$.

Solution: \boxed{\left(\frac{5}{8}, \frac{3}{8}\right)}


What is $1010101_2 + 1001001_2$? Write your answer in base $10$.


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


Let
\[f(x) =
\begin{cases}
3x + 4 & \text{if } x < 0, \\
x^2 - 2x + 1 & \text{if } x \geq 0.
\end{cases}
\]
Find $f(-1)$ and $f(2)$.


Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, 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 $S$. Determine the length of $QS$.


Without using a calculator, find the largest prime factor of $16^4 + 4 \times 16^2 + 4 - 15^4$.


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


On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$, meaning that if the length of the object was originally $x,$ then it is now $x + \frac{1}{3} x.$ On the second day he increased the object's longer length by $\frac{1}{4}$; on the third day he increased the object's new length by $\frac{1}{5}$; and so on. On the $n^{\text{th}}$ day of performing this trick, Barry will make the object's length exactly 50 times its original length. What is the value of $n$?


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


What is the smallest integer $b > 3$ for which the base $b$ number $23_b$ is a perfect cube?


One hundred cards, numbered 1-100, are placed in a box. One card is randomly selected. What is the probability that the number on the card is both a multiple of 13 and a prime number? Express your answer as a common fraction.


Express $0.\overline{2}+0.\overline{02}+0.\overline{0002}$ as a common fraction.


If altitude $CD$ is $1$ centimeter, what is the number of square centimeters in the area of $\Delta ABC$ where $\Delta ABC$ is a right triangle with a $45^\circ$ angle at $A$?

[asy] import olympiad; pair A,B,C,D; A = (0,1); B = (1,0);
C = foot(A,B,-B); D = foot(C,A,B); draw(A--B--C--A); draw(C--D,dashed);
label("$45^{\circ}$",A-(0.05,0.4),E);
label("$A$",A,N);label("$B$",B,E);label("$C$",C,W);label("$D$",D,NE);
draw((0,.1)--(.1,.1)--(.1,0)); draw(D + .1*dir(210)--D + sqrt(2)*.1*dir(165)--D+.1*dir(120));
[/asy]


In the triangular arrangement with points Q, R, P, and S, where $QRS$ is a straight line. Given $\angle PQS = 55^\circ$, $\angle PSQ = 40^\circ$, and $\angle QPR = 72^\circ$, determine the measure of $\angle RPS.$


Let \( p, q, r, \) and \( s \) be the roots of the polynomial \[ x^4 + 10x^3 + 20x^2 + 15x + 6 = 0. \] Calculate the value of \[ \frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}. \]


How many integers $-13 \leq n \leq 13$ satisfy $(n-3)(n+5)(n+7) < 0$?


Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7\}$. What is the probability that their product is an even number? Express your answer as a common fraction.


Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $39x + y = 1953$. Find the number of such distinct triangles whose area is a positive integer and where $x_1 \neq x_2$.


Define the operation \( a \star b = 3 + b^a \). What is the value of \( (2 \star 3) \star 2 \)?


Humanity has discovered 15 habitable planets, where 7 are "Earth-like" and 8 are "Mars-like". Colonizing an Earth-like planet requires 3 units of colonization, while a Mars-like planet requires 1 unit. If humanity has 21 units available for colonization, determine how many different combinations of planets can be occupied given that all planets are distinct.


Calculate the area of a parallelogram with vertices at (0,0), (4,0), (3,5), and (7,5).


How many two-digit positive integers are congruent to 3 (mod 5)?


Find the remainder when $8x^4 - 18x^3 + 6x^2 - 4x + 30$ is divided by $2x - 4$.


In triangle $PQR,$ the midpoint of $\overline{QR}$ is $(2,6,-2),$ the midpoint of $\overline{PR}$ is $(1,5,-3),$ and the midpoint of $\overline{PQ}$ is $(3,4,5).$ Find the coordinates of vertex $P.$

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


In $\triangle ABC$, where $\angle A = 90^\circ$ and $\tan C = 3$, if $BC = 30$, determine the length of side $AB$.

Solution: \boxed{AB = 9\sqrt{10}}


A room is 24 feet long and 14 feet wide. Find the ratio of the length to its perimeter and the ratio of the width to its perimeter. Express each ratio in the form $a:b$.


The radius of a sphere is $r$ units. A hemisphere is placed atop a cylinder, both having the same radius which is twice the radius of the sphere. Given the height of the cylinder as $2r$, calculate the ratio of the volume of the sphere to the combined volume of the hemisphere and the cylinder.


Find the maximum number of real roots of the polynomial
\[x^n - x^{n-1} + x^{n-2} - x^{n-3} + \ldots + (-1)^{n-1}x + (-1)^n = 0,\]
where \(n\) is a positive integer.


The graphs of $y=|x|$ and $y=-x^2-2x-1$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments.


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


How many positive integers, not exceeding 200, are multiples of 2 or 5 but not 6?


What is the domain of the function $f(x)=\log_2(\log_3(\log_5x))$?

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


Determine the value of the expression
\[
\log_3 (64 + \log_3 (64 + \log_3 (64 + \cdots))),
\]
assuming it is positive.


In triangle $XYZ,$ $XY = 5,$ $XZ = 7,$ $YZ = 9,$ and $W$ lies on $\overline{YZ}$ such that $\overline{XW}$ bisects $\angle YXZ.$ Find $\cos \angle YXW.$


How many total days were there in the years 2010 through 2014?


Determine the distance between the vertices of the parabolas described by the equation:
\[\sqrt{x^2 + y^2} + |y + 1| = 5.\]


Calculate the sum of the first 7 terms of the geometric series: $\frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \frac{1}{32} - \frac{1}{64} + \frac{1}{128}$.


Richard is planning to enclose a rectangular section of his yard using 500 feet of fencing. This enclosure is to cover three sides of the backyard, with his house forming the fourth side. Additionally, the length of the backyard must be twice the width. What is the maximum possible area for this backyard?


Five dice with faces numbered 1 through 6 are arranged in a configuration where 14 of the visible faces are showing. The visible numbers are 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 4, 5, 6. What is the total number of dots NOT visible in this view?


Consider a 5x5 grid of squares. How many different squares can be traced using the lines in this grid?


In a regular nonagon (a 9-sided polygon), two diagonals are chosen at random. What is the probability that these diagonals intersect inside the nonagon?


What is the smallest positive integer \( n \) for which \( n^2 \) is divisible by 50 and \( n^3 \) is divisible by 288?


A triangle is made of wood sticks of lengths 7, 24, and 25 inches joined end-to-end. Pieces of the same integral length, which must be an even number, are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. What is the length of the smallest even piece that can be cut from each of the three sticks to make this happen?


Let $p,$ $q,$ and $r$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\overline{BC}$ is $(p,0,0),$ the midpoint of $\overline{AC}$ is $(0,q,0),$ and the midpoint of $\overline{AB}$ is $(0,0,r).$ Find
\[\frac{AB^2 + BC^2 + CA^2}{p^2 + q^2 + r^2}.\]


Determine the smallest positive real number $c$, such that for all nonnegative real numbers $x$ and $y$,
\[\sqrt{x^2y^2} + c |x^2 - y^2| \ge \frac{x^2 + y^2}{2}.\]


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


The product $N$ of three positive integers is $8$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.


Determine the number of digits $B$ such that the three-digit number $A7B$ is a multiple of 7, where $A=4$ is fixed and $B$ ranges from 0 to 9.


Find the product of the divisors of $45$.


Calculate how many zeros are in the decimal expansion of the square of the number $9,\!999$.


Cutting isosceles right triangle $BFC$ out of square $ABCD$ and translating it leftwards creates the shape $ABFCDE$. The perimeter of square $ABCD$ is 40 inches. What is the perimeter, in inches, of $ABFCDE$?

[asy]
import graph;
filldraw((0,0)--(sqrt(2)*2,2)--(2+2sqrt(2),2)--(2sqrt(2),0)--(2+2sqrt(2),-2)--(sqrt(2)*2,-2)--cycle,grey,linewidth(1));
draw((sqrt(2)*2,2)--(sqrt(2)*2,-2),linewidth(1));
draw((2+2sqrt(2),-2)--(2+2sqrt(2),2)--(2sqrt(2),0)--(2+2sqrt(2),-2)--cycle,dashed);

label("A",(sqrt(2)*2,2),N);
label("B",(2+2sqrt(2),2),N);
label("C",(2+2sqrt(2),-2),S);
label("D",(sqrt(2)*2,-2),S);
label("E",(0,0),W);
label("F",(2sqrt(2),0),N);
[/asy]

Solution: \boxed{20 + 20\sqrt{2} \text{ inches}}


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


Find $b^2$ where the foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{225} - \frac{y^2}{144} = \frac{1}{36}\]coincide.


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


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


The forecast predicts a 75 percent chance of rain for each of the five days of the extended Labor Day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri desire exactly two sunny days during this period for a special event. Calculate the probability of getting precisely two sunny days out of the five. Provide your answer as a fraction.


Find the degree measure of the least positive angle $\theta$ for which
\[\cos 10^\circ = \sin 20^\circ + \sin \theta.\]


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

Solution: \boxed{\cos^2 t}


Find the number of solutions to the equation
\[\sin x = \left(\frac{1}{3}\right)^x\]
on the interval \(0, 150\pi\).


Convert the point $(6, -6, 10)$ in rectangular coordinates to cylindrical coordinates. Enter your answer in the form $(r, \theta, z)$, where $r > 0$ and $0 \le \theta < 2\pi$.

Solution: \boxed{(6\sqrt{2}, \frac{7\pi}{4}, 10)}


For how many integers $n$ is it true that $\sqrt{n} \le \sqrt{5n - 8} < \sqrt{3n + 7}$?


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


In a different city, the rate of taxation is \(x\%\) for an income of \(x + 10\) thousand dollars. What income, in dollars, will yield the greatest take-home pay?


Find the point where the line passing through $(4,7,2)$ and $(6,3,7)$ intersects the $xz$-plane.

Solution: \boxed{\left(\frac{15}{2}, 0, \frac{43}{4}\right)}


A new type of license plate in a certain state consists of 5 digits and 3 letters. The 3 letters, not necessarily distinct, must appear next to each other, and these characters must begin in one of the first 6 slots of the license plate. How many distinct license plates are possible?


Jane is hiking in Utah. She walks northward three miles, then turns $45$ degrees eastward and walks eight miles. How far is she from her starting point? Express your answer in simplest radical form.

Solution: \boxed{\sqrt{73 + 24\sqrt{2}}}


For a positive integer $n > 3^2$, determine the greatest common divisor of $n^3 - 27$ and $n - 3$.


Wendy continues her math exploration and finds another torn piece of Michael's papers. On this piece, there is another monic polynomial of degree $n$, again with real coefficients. The first two terms following $x^n$ are $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$. Wendy notes that on this piece, $a_{n-1} = 2a_{n-2}$. Wendy aims to determine the sum of the squares of the roots of this polynomial knowing that neither the degree $n$ nor the coefficients $a_{n-1}$ and $a_{n-2}$ are directly specified, yet she's interested in finding a [greatest] lower bound for the sum of the squares of the roots. She wants to find the absolute value of this lower bound.


A four-inch wooden cube is painted blue on all its faces and then cut into one-inch cubes. How many of the one-inch cubes have blue paint on at least two faces?


Calculate $\sqrt{15p} \cdot \sqrt{10p^3} \cdot \sqrt{14p^5}$ and express your answer in the simplest radical form in terms of $p$.

Solution: \boxed{10p^4 \sqrt{21p}}


Alice experiences that the amount of time she studies for a test and her score on that test are inversely related. On her first test, she studied two hours and scored 60. How many hours does Alice believe she must study for her next test to average a 75 on the two tests?


Let $p$ and $q$ be real numbers such that the roots of
\[ z^2 + (6 + pi) z + (10 + qi) = 0 \] are complex conjugates. Determine the ordered pair $(p, q)$.

Solution: \boxed{(0, 0)}


The Gnollish language has expanded to include a fourth word, "kreeg". In a sentence, "splargh" cannot come directly before "glumph" or "kreeg". How many valid 3-word sentences are there now in this updated Gnollish language?


I have four distinct mystery novels, four distinct fantasy novels, three distinct biographies, and two distinct science fiction novels. I'm planning a reading marathon and want to select three books, ensuring that at least two of them belong to different genres. How many possible sets of three books can I choose?


Find the sum of the first eight terms in the geometric sequence starting with \( \frac{1}{3} \) and a common ratio of \( \frac{1}{3} \). Express your answer as a common fraction.


What is the 30th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{11}$ and $\frac{1}{13}$?


Solve for $x : \log_2 x + 2\log_4 x = 9.$


Let $\mathbf{p} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix}$. Find the vector $\mathbf{u}$ that satisfies $\mathbf{u} \times \mathbf{p} = \mathbf{q} \times \mathbf{p}$ and $\mathbf{u} \times \mathbf{q} = \mathbf{p} \times \mathbf{q}$.

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


Calculate the remainder when the sum \(1^3 + 2^3 + 3^3 + \dots + 150^3\) is divided by 6.


How many nonnegative integers can be written in the form \[ b_7\cdot4^7 + b_6\cdot4^6 + b_5\cdot4^5 + b_4\cdot4^4 + b_3\cdot4^3 + b_2\cdot4^2 + b_1\cdot4^1 + b_0\cdot4^0, \] where $b_i \in \{0, 1, 2\}$ for $0 \leq i \leq 7$?


How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?


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

[asy]
pair A,B,C;
A = (0,0);
B = (40,0);
C = (0,9);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,25));
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$41$",(B+C)/2,E);
label("$40$",B/2,S);
[/asy]


How many positive integers less than $201$ are multiples of either $8$ or $11$, but not both at once?


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


Simplify $24 \cdot \frac{8}{15} \cdot \frac{5}{18}$.


Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+1}$.


Calculate the total number of rectangles and the total number of squares formed by the lines on a $9 \times 9$ checkerboard. If the ratio of the number of squares to the number of rectangles can be expressed in its simplest form as $m/n$, where m and n are coprime integers, find the value of $m + n$.


Find the area of a triangle with side lengths 7, 7, and 10.


The notation $[x]$ is used for the greatest integer less than or equal to $x$, and $\lceil x \rceil$ is used for the smallest integer greater than or equal to $x$. Calculate $[3.7]$ and $\lceil 3.7 \rceil$.

Solution: \boxed{[3.7] = 3, \lceil 3.7 \rceil = 4}


Let $z_1$ and $z_2$ be the complex roots of the equation $z^2 + pz + q = 0$, where $p$ and $q$ are complex numbers. In the complex plane, 0, $z_1$, and $z_2$ are vertices that form a right triangle. Determine $\frac{p^2}{q}$.


Determine how many lattice points lie on the graph of the equation $x^2-y^2=65$ in the $x,y$-plane.


If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 294$ (changed from 168) and $\gcd(a,c) = 1155$ (changed from 693), what is the smallest possible value of $\gcd(b,c)$?


If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 119$, what is the value of $x + y$?


Alice walked 4 miles in a third of the time it took Bob to walk 6 miles. If it took Bob 36 minutes to walk 6 miles, using the same rates, how many minutes would it take Alice to walk 7 miles?


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


Two isosceles triangles are placed side-by-side with their bases on the same line. If $\triangle ABC$ and $\triangle DEF$ are such triangles with $AB = AC$, $DE = DF$, $\angle BAC = 25^\circ$, and $\angle EDF = 40^\circ$, find the sum $\angle DAC + \angle ADE$.


What is the area, in square units, of the interior region formed by the lines $y = 3x - 6$, $y = -4x + 24$, and the $y$-axis?


On a Cartesian coordinate plane, points \((4,-1)\) and \((-1, 3)\) are adjacent corners on a square. Calculate the area of this square.


I have 8 shirts, 4 pairs of pants, and 6 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which the shirt and hat are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do I have?


When the base-12 integer $2563_{12}$ is divided by $17$, what is the remainder?


5 12-sided dice are rolled. What is the probability that the number of dice showing a two-digit number is 2 and the number of dice showing a one-digit number is 3? Express your answer as a common fraction.


A hypothetical day is defined to last exactly $100,\!000$ seconds. In how many ways can this duration be divided into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers?


What is the sum of all the odd divisors of $450$?


In coordinate space, points \( A = (1, 2, 3) \), \( B = (5, 3, 1) \), \( C = (3, 4, 5) \), and \( D = (4, 5, 2) \) form a tetrahedron. Find the centroid of this tetrahedron.

Solution: \boxed{\left( \frac{13}{4}, \frac{7}{2}, \frac{11}{4} \right)}


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


Determine the number of square units enclosed by the following polygon formed by dots spaced one unit apart, horizontally and vertically.
[asy]
/* Modified polygon */
size(1inch,1inch);
pair a=(0,0), b=(20,0), c=(30,10), d=(20, 20), e=(10,10), f=(0,20);
dot(a);
dot(b);
dot(c);
dot(d);
dot(e);
dot(f);
draw(a--b--c--d--e--f--a);
[/asy]


A choir can be organized into rows of 9, 10, or 12 with no one left over. Additionally, the number of members is a multiple of a prime number larger than 10. What is the minimum number of members in the choir?


I have 16 distinguishable socks in my drawer: 5 white, 5 brown, 3 blue, and 3 black. In how many ways can I choose a pair of socks, provided that I get two socks of different colors?


What is the smallest positive integer $x$ that, when multiplied by $450$, produces a product that is a multiple of $648$?


What is the units digit of the sum of the first twelve terms of the sequence \(1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 11! + 11, \, 12! + 12\)?


A round cake with a diameter of $16\text{ cm}$ is sliced into four equally sized sector-shaped pieces. Let $m$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $m^2$?


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

Solution: \boxed{-c + 2}


Three dice with faces numbered 1 through 6 are placed vertically in a stack as described. The visible numbers on these dice are 2, 2, 3, 3, 4, 5, 6. Calculate the total number of dots that are NOT visible on these dice.


If the least common multiple of $A$ and $B$ is $200$, and the ratio of $A$ to $B$ is $2:5$, find their greatest common divisor.


Find the equation of the directrix of the parabola $y = -4x^2.$

Solution: \boxed{y = \frac{1}{16}}


The pages of a book are numbered starting from 2 up to $n$. When the page numbers were summed up, one of the page numbers was mistakenly added three times, resulting in an incorrect total sum of 4090. Determine the number of the page that was added thrice.


Find the sum of all angles \( x \) in \([0^\circ, 360^\circ]\) that satisfy
\[
\sin^3 x - \cos^3 x = \frac{1}{\cos x} - \frac{1}{\sin x}.
\]


A company purchases 300 markers from an online store at the price of $12.50$ per pack, with each pack containing 25 markers. There is an additional shipping charge of $7.85$. What is the average cost, in cents, for each marker? Express your answer rounded to the nearest whole number.


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


Determine the product of the roots of the equation \[(3x^4 - 9x^3 + 27x - 81)(4x^2 - 16x + 64) = 0.\]


Calculate the result of $180 \div \left(12 + 13 \cdot 3 - 5\right)$.


Find the integer $n$, $0 \le n \le 19$, that satisfies \[n \equiv -5678 \pmod{20}.\]


Determine the sum of all real numbers \( x \) satisfying
\[ (x^2 - 6x + 8)^{x^2 - 8x + 15} = 1. \]


There are 6 blue chips, 4 yellow chips, and 2 red chips in a bag. One chip is drawn from the bag. That chip is placed back into the bag. A second chip is then drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.


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


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


Let $\mathbf{p}$ be the projection of $\mathbf{v}$ onto $\mathbf{w},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{v}.$  If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{5},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{v}\|}.$


How many four-digit even numbers are there such that the sum of the tens and units digits is 12?


Solve for $x$, where $x>0$ and $4x^2 + 8x - 20 = 0$. Express your answer as a simplified common fraction.

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


The graphs of \( y = |x| \) and \( y = -x^2 - 5x - 4 \) are drawn. For every \( x \), a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments.


Sheila has been invited to a picnic tomorrow. The picnic will occur, rain or shine. However, she also needs to finish her homework to go. If it rains, there is a 40% probability that Sheila will decide to go, but if it is sunny, there is a 90% probability that she will decide to go. The forecast for tomorrow states that there is a 50% chance of rain. Additionally, irrespective of the weather, there's a 70% chance she finishes her homework, a condition required for her to attend. What is the overall probability that Sheila will attend the picnic? Express your answer as a percent.

Solution: \boxed{0.455 = 45.5\%}


Let \(s_1, s_2, s_3\) be the roots of the polynomial equation
\[x^3 - 4x^2 + 5x - 7 = 0.\]
Find the monic polynomial, in \(x\), whose roots are \(s_1+3, s_2+3, s_3+3\).

Solution: \boxed{x^3 - 13x^2 + 56x - 85}


What is the sum of the digits of the decimal representation of $2^{2010} \times 5^{2012} \times 7$?


Determine the value of $\frac{6x - 4y}{36xy}$, where $x = \frac{4}{7}$ and $y = \frac{5}{8}$.


In a school, there are 30 students who are enrolled in at least one of the offered foreign language classes: German or Italian. The information available indicates that 22 students are taking German and 26 students are taking Italian. Sarah, who is writing an article for the school magazine, needs to interview two students randomly chosen from this list. What is the probability that she will be able to gather information about both language classes after the interviews? Give your answer as a fraction in its simplest form.


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


The planet Zeta follows an elliptical orbit with its sun at one focus. At its closest approach (perigee), it is 3 astronomical units (AU) from the sun, and at its farthest point (apogee), it is 15 AU away. Calculate the distance from the sun to Zeta when Zeta is exactly halfway along its orbit from perigee to apogee.

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


The circular garden has an area of 616 square feet. Greg would like to install a decorative fence around the garden’s edge. To ensure he has enough fencing, he decides to buy 5 feet more than the garden’s circumference. Calculate how many feet of fencing Greg will need to purchase if he uses $\pi = \frac{22}{7}$ for his calculations.


In how many different ways can six students be arranged in a straight line if two pairs of students, say pairs (A, B) and (C, D), refuse to stand next to each other?


In right triangle $DEF$, $\cos{F}=\frac{5\sqrt{221}}{221}$. The hypotenuse $DE$ has a length of $\sqrt{221}$. Find the length of side $DF$.


Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. If $DP= 15$ and $EQ = 20$, then what is ${DE}$?


The sum
\[\text{cis } 80^\circ + \text{cis } 88^\circ + \text{cis } 96^\circ + \dots + \text{cis } 128^\circ\]
is expressed in the form \( r \, \text{cis } \theta \), where \( r > 0 \) and \( 0^\circ \le \theta < 360^\circ \). Find \( \theta \) in degrees.


If $(5000^{50}) \times 100^2$ is fully expanded, what is the result expressed in the form of '1 followed by how many zeros'?


The length of a rectangle is increased by \( 30\% \), and the width is decreased to keep the area of the rectangle unchanged. By what percent was the rectangle's width decreased?


Solution: \boxed{5 - 4i}


In trapezoid \(ABCD\), the lengths of the bases \(AB\) and \(CD\) are 7 and 15 respectively. The legs of the trapezoid are extended beyond \(A\) and \(B\) to meet at point \(E\), forming a triangle \(EAB\) with base \(AB\) and the height from \(E\) to \(AB\) being thrice the height of the trapezoid. What is the ratio of the area of triangle \(EAB\) to the area of trapezoid \(ABCD\)? Express your answer as a common fraction.


Find the smallest three-digit palindrome whose product with 102 is a five-digit number but not a palindrome.


A right pyramid has a square base with each side measuring 20 cm. The peak is 15 cm above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges, and express your answer to the nearest whole number.


The solution to the inequality
\[y = -2x^2 + cx + d \le 0\]
is $[\frac{-7}{2}, \infty).$ Find the vertex of the parabola $y=-2x^2 + cx + d.$

Solution: \boxed{(-\frac{7}{2}, 0)}


For all real numbers $x$ and $y$, define the mathematical operation $\#$ such that the following conditions apply: $x\ \#\ 0 = x, x\ \#\ y = y\ \#\ x$, and $(x + 1)\ \#\ y = (x\ \#\ y) + 2y + 1$. Calculate the value of $7\ \#\ 3$.


Express $\sqrt{p} \div \sqrt{q}$ as a common fraction, given:

$$\frac{ {\left( \frac{1}{3} \right)}^2 + {\left( \frac{1}{4} \right)}^2 }{ {\left( \frac{1}{5} \right)}^2 + {\left( \frac{1}{6} \right)}^2} = \frac{25p}{61q} $$


The Cookie Monster now encounters another cookie with a boundary described by the equation $x^2 + y^2 + 36 = 6x + 24y$. He is curious to compare this cookie's size to the previous one. What is the radius of this cookie?


If $m$ and $n$ are positive integers such that $\gcd(m,n) = 12$, then what is the smallest possible value of $\gcd(12m,18n)$?


If $\mathbf{v} \times \mathbf{u} = \begin{pmatrix} -3 \\ 6 \\ 2 \end{pmatrix}$ and $\mathbf{w} = 2\mathbf{v} + 3\mathbf{u}$, then find $(\mathbf{w} + \mathbf{u}) \times (\mathbf{w} - \mathbf{u}).$

Solution: \boxed{\begin{pmatrix} 12 \\ -24 \\ -8 \end{pmatrix}}


A sports league is divided into two divisions, one consisting of 5 teams and the other of 6 teams. In a season, every team must play every other team in its own division three times and every team in the opposing division twice. How many games are in a complete season for this league?


Let $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ.$ Find the area of the parallelogram whose diagonals correspond to $\mathbf{p} + 3 \mathbf{q}$ and $3 \mathbf{p} + \mathbf{q}.$


Circle $\omega$ has a radius of 8 and is centered at $O$. Point $A$ lies outside $\omega$ with $OA=17$. The two tangents to $\omega$ from $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ with $\omega$ lying outside triangle $ABC$. Compute $AB+AC$ given that $BC=12$.


A circular dartboard is divided into sectors. The probability of a dart randomly landing in one of the smaller sectors is $\frac{1}{8}$. There are 3 smaller sectors with equal probabilities and one larger sector. What is the central angle, in degrees, of one of the smaller sectors?


Determine the eighth term in the geometric sequence $12, 4, \frac{4}{3}, \ldots$.


Find the remainder when \( 8x^4 - 18x^3 - 6x^2 + 4x - 30 \) is divided by \( 2x - 8 \).


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 deck of cards now contains 54 cards, including two jokers, one being a red joker and the other a black joker, along with the standard 52 cards. In how many ways can we pick two different cards such that at least one of them is a joker? (Order matters.)


Find \(n\) if \(n \cdot n! + 2 \cdot n! = 5040\), where \(n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1\).


Determine the value of $x + y$ if the three points $(x,1,z)$, $(2,y,z)$, $(x,y,3)$ are collinear.


For any integer $n > 2$, the number of prime numbers greater than $n!+2$ and less than $n!+n+1$ is what?


The following line is parameterized, so that its direction vector is of the form $\begin{pmatrix} a \\ 2 \end{pmatrix}.$  Find $a.$

[asy]
unitsize(0.4 cm);

pair A, B, L, R;
int i, n;

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

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

A = (-3,4);
B = (2,-1);
L = extension(A, B, (0,8), (1,8));
R = extension(A, B, (0,-8), (1,-8));

draw(L--R, red);

label("$x$", (8,0), E);
label("$y$", (0,8), N);
[/asy]


What is the sum of all two-digit primes greater than 30 but less than 99, which are still prime when their two digits are interchanged?


A rectangular prism has dimensions of 12 inches, 16 inches, and 21 inches. Calculate the length of the diagonal connecting opposite corners of the prism and find the total surface area of the prism. Express the diagonal's length in simplest radical form.

Solution: \boxed{1560 \text{ square inches}}


A rectangle has an area of 100 square inches, and its sides (length and width) must add up to 24 inches. Find the number of distinct rectangles that meet these conditions.


Peter has between $\$2$ and $\$20$ worth of nickels. When he organizes them in stacks of 4, 2 are left; in stacks of 5, 2 are left; and in stacks of 7, 2 are left. How many nickels does Peter have?


Let $M = 123456789101112\dots4950$ be the $95$-digit number formed by writing integers from $1$ to $50$ in order, one after the other. What is the remainder when $M$ is divided by $45$?


Find the value of $x$ such that $\log_x 8 = \log_{64} 4$.


Let $x^2 + bx + c = 0$ be a quadratic equation whose roots are each three less than the roots of $5x^2 - 8x + 2$. What is the value of $c$?


Express $\frac{0.\overline{142857}}{0.\overline{285714}}$ as a common fraction.


When a biased coin is flipped six times, the probability of getting heads exactly twice is the same as that of getting heads exactly three times. Let $\frac{p}{q}$, in lowest terms, be the probability that the coin comes up heads in exactly $4$ out of $6$ flips. Find $p+q$.


On "modulo \(9\) graph paper", two graphs are drawn from the equations:
- \(y \equiv 3x + 6 \pmod{9}\)
- \(y \equiv 7x + 3 \pmod{9}\)

What is the sum of the \(x\)-coordinates of points that both graphs have in common?


Find the integer $n$, $0 \le n \le 6$, such that
\[n \equiv 12345 \pmod{7}.\]


Assume we have two spinners, where one is divided into fifths and the other into sixths. Each segment of the first spinner contains numbers: $2, 4, 5, 7, 9$, and each segment of the second spinner contains numbers: $3, 4, 6, 8, 10, 12$. If each spinner is spun once, what is the probability that the product of the results of the two spins is an even number?


Every week, Judy buys the following from the supermarket: $6$ carrots at $\$1.50$ each, $4$ bottles of milk at $\$3.50$ each, $3$ pineapples at $\$5$ each, $3$ bags of flour at $\$6$ each, and a $\$8$ tub of ice cream. This week, both pineapples and flour have a discount: pineapples are offered at 25% off and flour has a $10%$ discount. Judy also has a coupon for $\$10$ off any order of $\$50$ or more. Calculate Judy's total spending this shopping trip.


Cara is sitting at a circular table with her six friends. How many different possible pairs of people could Cara be sitting between?


A club has 18 members divided into three departments: 6 in Department A, 7 in Department B, and 5 in Department C. The club needs to choose 3 members to form a new committee, ensuring that each member on the committee is from a different department. In how many ways can the club form this committee?


How many positive divisors does 8! have?


The line $y = -x + 3$ is parameterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ v \end{pmatrix} + u \begin{pmatrix} m \\ 8 \end{pmatrix}.\]
Find the ordered pair $(v, m)$.

Solution: \boxed{(1, -8)}


What is the greatest common factor (GCF) of 300 and 125, and what is their least common multiple (LCM)?


Find the modular inverse of \( 35 \), modulo \( 36 \).


Determine the base \( b \) such that the number \( 256_{10} \) has exactly \( 4 \) digits in base \( b \).


How many integers between 100 and 300 have the sum of their digits equal to 11?


Given the function $y = E(x)$ illustrated on the graph, where $E(x) = \sqrt{\lvert x + 1 \rvert} + \frac{9}{\pi} \arctan(\sqrt{\lvert x \rvert})$, find the integer value of $E(-2)$.


Determine the value of \( y \) in the equation \( 8^3 + 8^3 + 8^3 + 8^3 = 4^y \).


Find the distance between the foci of the ellipse
\[
\frac{x^2}{36} + \frac{y^2}{9} = 5.
\]


Find the sum of all $x$ that satisfy the equation $\frac{-15x}{x^2-1} = \frac{3x}{x+1} - \frac{9}{x-1}$.


The stem-and-leaf plot shows the duration of one lap in minutes and seconds for each of the $15$ fastest cars in a race. In the stem-and-leaf plot, $1 \ 15$ represents $1$ minute, $15$ seconds, which is equivalent to $75$ seconds. Determine the median duration of these laps. Express your answer in seconds.

\begin{tabular}{c|ccccc}
0&45&48&50&&\\
1&10&15&20&30&45\\
2&05&10&15&20&25\\
3&10&15&&&\\
\end{tabular}


For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (nx + \frac{\pi}{4})$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$?


A unit cube is cut once along a diagonal of one face and then along the diagonal of the opposite parallel face creating four triangular prisms. One of these prisms is cut again along a plane passing through one of its longest edges and the midpoint of its opposite edge. This results in six pieces in total. What is the volume of the triangular prism that includes vertex $W$ on the original cube?


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


Find the product of all positive integer values of $c$ such that $10x^2 + 24x + c = 0$ has two real roots.


Point $Q$ lies on the line $x= -3$ and is 13 units from the point $(3,-1)$. Find the product of all possible $y$-coordinates that satisfy the given conditions.


"Modulo $m$ graph paper" consists of a grid of $13^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <13$. To graph a congruence on modulo $13$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. Consider the graph of $$4x \equiv 3y + 1 \pmod{13}.$$ Find the sum of the $x$-intercept and the $y$-intercept, where the intercepts are represented as $(x_0,0)$ and $(0,y_0)$ with $0\le x_0,y_0<13$.


In the diagram, each circle is divided into two equal areas and $O$ is the center of the larger circle. The area of the larger circle is $81\pi.$ What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),3,180,360)--cycle,mediumgray);fill(Arc((0,1.5),1.5,0,180)--cycle,mediumgray);
draw(Circle((0,0),3));
draw(Circle((0,1.5),1.5));

dot((0,0)); label("$O$",(0,0),N);
draw((-3,0)--(3,0)); draw((-1.5,1.5)--(1.5,1.5));
[/asy]


Our school's track and field team consists of 16 athletes, including a set of triplets: Ben, Bill, and Bob and a set of twins: Tom and Tim. In how many ways can we choose 7 athletes for a relay team if exactly one of the triplets is in the team and at least one of the twins must be in the team?


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


In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 700$ and $AD = 1400$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$.


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


In Pascal's Triangle, each number is the result of adding the number directly above and to the left with the number directly above and to the right. Calculate the sum of the numbers in Row 10 of Pascal's Triangle.


In quadrilateral $ABCD$, $\angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD}$, $AB=15$, $BC=20$, and $CD=9$. Calculate the perimeter of $ABCD$.

Solution: \boxed{44 + \sqrt{706}}


Captain Charlie and First Mate Fred divide their bounty using the following scheme: Charlie says, "One for me, one for you," giving himself one coin and starting Fred's pile with one coin. They continue, "Two for me, and two for you," giving himself two more coins but making Fred's pile two coins in total. They pronounce, "Three for me, three for you," and so on, incrementally until Charlie eventually gives himself $x$ more coins while adjusting Fred's pile to just precisely $x$ coins in total. At the end of the division, Charlie has five times as many coins as Fred. Determine how many gold coins they had initially.


If $a$ and $b$ are odd integers, how many terms in the expansion of $(a+b)^8$ are odd?


Mrs. Johnson recorded the scores for a test taken by her 180 students. The data provided is shown below. What was the average percent score for these students?

\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
95&12\\\hline
85&30\\\hline
75&50\\\hline
65&45\\\hline
55&30\\\hline
45&13\\\hline
\end{tabular}


Find the remainder when $x^4 + x^3 + 1$ is divided by $x^2 - 2x + 3.$

Solution: \boxed{-3x - 8}


Two dice are tossed. What is the probability that the sum of the numbers on the two dice is greater than 8?


On "Modulo 20 graph paper," which illustrates all pairs (x,y) for integer residues where $0 \leq x < 20$, we graph the equations $$y \equiv 7x + 3 \pmod{20}$$ and $$y \equiv 13x + 17 \pmod{20}$$. Determine the sum of the x-coordinates for all points that appear in both graphs.


Find the smallest value of $c$ such that $1$ is in the range of the function $f(x) = x^2 + 5x + c$.


A printer has two printing modes: fast mode and normal mode. In fast mode, it prints 23 pages per minute, while in normal mode, it prints 15 pages per minute. If an operator needs to print a document of 280 pages, of which the first 150 pages must be printed in fast mode and the remaining in normal mode, how many total minutes will it take to print the whole document? Round your answer to the nearest whole number.


If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?


A wooden block is 6 inches long, 4 inches wide, and 2 inches high. The block is painted red on all six sides and then cut into forty-eight 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?


Let \(a, b, c, d, e, f, g,\) and \(h\) be distinct elements in the set
\[
\{-9, -6, -3, 0, 1, 3, 6, 10\}.
\]
What is the minimum possible value of \[
(a+b+c+d)^2 + (e+f+g+h)^2?
\]


A regular $50$-gon $Q_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $50$ vertices equals 1010. The midpoints of the sides of $Q_1$ form a second $50$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $50$-gon, $Q_3$. Find the sum of the $x$-coordinates of the vertices of $Q_3$.


Let $g(x)$ be a polynomial with real, nonnegative coefficients. If $g(4) = 16$ and $g(16) = 256$, find the largest possible value of $g(8)$.


Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$.


Determine the number of positive integers $a$ less than $30$ such that the congruence $ax \equiv 1 \pmod{20}$ has a solution in $x$.


Lucy designs a cylindrical vase for her botany project, which is 15 inches tall with a base diameter of 8 inches. Lucy needs to find the smallest cube-shaped box in which she can place her vase upright for transportation to her school. What is the volume of this box, in cubic inches?

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


A sphere is inscribed in a cube. The edge length of the cube is 8 inches. Calculate the volume of the inscribed sphere in terms of \(\pi\).

Solution: \boxed{\frac{256}{3} \pi}


In a revised shooting match setup, ten clay targets are arranged in four hanging columns with the configuration of three targets in the first and second column, two targets in the third column, and two targets in the fourth column. The rules for the marksman remain the same:
1) The marksman must choose one of the columns to shoot from.
2) The marksman must shoot the lowest remaining target in the chosen column.
Determine the number of different sequences in which all ten targets can be broken, while respecting the above-stated rules.