NASA is launching a spaceship at the south pole, but a sudden earthquake shock caused the spaceship to be launched at an angle of $\theta$ from vertical ($0 < \theta < 90^\circ$). The spaceship crashed back to Earth, and NASA found the debris floating in the ocean in the northern hemisphere. NASA engineers concluded that $\tan \theta > M$, where $M$ is maximal. Find $M$. Assume that the Earth is a sphere, and the trajectory of the spaceship (in the reference frame of Earth) is an ellipse with the center of the Earth one of the foci. [i]Proposed by Kevin You[/i]

Suppose that k and x are positive integers such that $$\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.$$ Find the sum of all possible values of $k$

A positive integer $N$ is a [i]triple-double[/i] if there exists non-negative integers $a$, $b$, $c$ such that $2^a + 2^b + 2^c = N$. How many three-digit numbers are triple-doubles? [i]Proposed by Giacomo Rizzo[/i]

A $3\times2\times2$ right rectangular prism has one of its edges with length $3$ replaced with an edge of length $5$ parallel to the original edge. The other $11$ edges remain the same length, and the $6$ vertices that are not endpoints of the replaced edge remain in place. The resulting convex solid has $8$ faces, as shown below. Find the volume of the solid. [i]Proposed by Justin Hsieh[/i]

Let $ABC$ be a triangle with $BC=30$, $AC=50$, and $AB=60$. Circle $\omega_B$ is the circle passing through $A$ and $B$ tangent to $BC$ at $B$; $\omega_C$ is defined similarly. Suppose the tangent to $\odot(ABC)$ at $A$ intersects $\omega_B$ and $\omega_C$ for the second time at $X$ and $Y$ respectively. Compute $XY$. Let $T = TNYWR$. For some positive integer $k$, a circle is drawn tangent to the coordinate axes such that the lines $x + y = k^2, x + y = (k+1)^2, \dots, x+y = (k+T)^2$ all pass through it. What is the minimum possible value of $k$?

Evaluate \[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\] [i]Proposed by Nathan Xiong[/i]

Alice and Bob have a fair coin with sides labeled $C$ and $M$, and they flip the coin repeatedly while recording the outcomes; for example, if they flip two $C$'s then an $M$, they have $CCM$ recorded. They play the following game: Alice chooses a four-character string $\mathcal A$, then Bob chooses two distinct three-character strings $\mathcal B_1$ and $\mathcal B_2$ such that neither is a substring of $\mathcal A$. Bob wins if $\mathcal A$ shows up in the running record before either $\mathcal B_1$ or $\mathcal B_2$ do, and otherwise Alice wins. Given that Alice chooses $\mathcal A = CMMC$ and Bob plays optimally, compute the probability that Bob wins.

In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.

Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$. [i]Proposed by Nathan Xiong[/i]

Let $ABC$ be points such that $AB=7, BC=5, AC=10$, and $M$ be the midpoint of $AC$. Let $\omega$, $\omega_1$ be the circumcircles of $ABC$ and $BMC$. $\Omega$, $\Omega_1$ are circles through $A$ and $M$ such that $\Omega$ is tangent to $\omega_1$ and $\Omega_1$ is tangent to the line through the centers of $\omega_1$ and $\Omega$. $D, E$ be the intersection of $\Omega$ with $\omega$ and $\Omega_1$ with $\omega_1$. If $F$ is the intersection of the circumcircle of $DME$ with $BM$, find $FB$. [i]Proposed by David Tang[/i]

Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this? [i]Proposed by Nathan Xiong[/i]

Vincent is playing a game with Evil Bill. The game uses an infinite number of red balls, an infinite number of green balls, and a very large bag. Vincent first picks two nonnegative integers $g$ and $k$ such that $g < k \le 2016$, and Evil Bill places $g$ green balls and $2016 - g$ red balls in the bag, so that there is a total of $2016$ balls in the bag. Vincent then picks a ball of either color and places it in the bag. Evil Bill then inspects the bag. If the ratio of green balls to total balls in the bag is ever exactly $\frac{k}{2016}$ , then Evil Bill wins. If the ratio of green balls to total balls is greater than $\frac{k}{2016}$ , then Vincent wins. Otherwise, Vincent and Evil Bill repeat the previous two actions (Vincent picks a ball and Evil Bill inspects the bag). If $S$ is the sum of all possible values of $k$ that Vincent could choose and be able to win, determine the largest prime factor of $S$.

Jason starts in a cell of the grid below. Every second he moves to an adjacent cell (i.e., two cells that share a side) that he has not visited yet. Find the maximum possible number of cells that Jason can visit. [asy] size(3cm); draw((1,0)--(4,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((1,5)--(4,5)); draw((0,1)--(0,4)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,1)--(5,4)); [/asy]

Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)

Given $n=2020$, sort the $6$ values $$n^{n^2},\,\, 2^{2^{2^n}},\,\, n^{2^n},\,\, 2^{2^{n^2}},\,\, 2^{n^n},\,\,\text{and}\,\, 2^{n^{2^2}}$$ from [b]least[/b] to [b]greatest[/b]. Give your answer as a 6 digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.

A fair coin is flipped $6$ times. The probability that the coin lands on the same side $3$ flips in a row at some point can be expressed as a common fraction $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?

Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$? Let $T = TNYWR$, and let $T = 10X + Y$ for an integer $X$ and a digit $Y$. Suppose that $a$ and $b$ are real numbers satisfying $a+\frac1b=Y$ and $\frac{b}a=X$. Compute $(ab)^4+\frac1{(ab)^4}$.

The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Currently, Selena’s analog clock says $4{:}00$. Suddenly her clock breaks, so the hour hand moves $12$ times as fast as it normally does, but the minute hand stays the same speed. Find the degree measure of the smaller angle formed by the minute and the hour hand $2024$ minutes from now.

Given a triangle, $AB=78, BC=50, AC=112,$ construct squares $ABXY, BCPQ, ACMN$ outside the triangle. Let $L_1, L_2, L_3$ be the midpoints of $\overline{MP}, \overline{QX}, \overline{NY},$ respectively. Find the area of $L_1L_2L_3.$

Evaluate \[2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.\] [i]Proposed by Nathan Xiong[/i]

Let $\omega_1$ and $\omega_2$ be two circles intersecting at distinct points $A$ and $B.$ Point $X$ varies along $\omega_1,$ and point $Y$ is chosen on $\omega_2$ such that $AB$ bisects angle $\angle{XAY}.$ Prove that as $X$ varies along $\omega_1,$ the circumcenter of $\triangle{AXY}$ (if it exists) varies along a fixed line.

On Misha's new phone, a passlock consists of six circles arranged in a $2\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle on the way between two others (e.g. the top left and right circles cannot be adjacent). For example, the left path shown below is allowed but the right path is not. (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$. However, in the diagrams below, the paths are valid/invalid regardless of orientation.) How many passlocks are there consisting of all six circles? [asy] size(270); defaultpen(linewidth(0.8)); real r = 0.3, rad = 0.1, shift = 3.7; pen th = linewidth(5)+gray(0.2); for(int i=0; i<= 2;i=i+1) { for(int j=0; j<= 1;j=j+1) { fill(circle((i,j),r),gray(0.8)); fill(circle((i+shift,j),r),gray(0.8)); } draw((0,1)--(2-rad,1)^^(2,1-rad)--(2,rad)^^(2-rad,0)--(0,0),th); draw(arc((2-rad,1-rad),rad,0,90)^^arc((2-rad,rad),rad,270,360),th); draw((shift+1,0)--(shift+1,1-2*rad)^^(shift+1-rad,1-rad)--(shift+rad,1-rad)^^(shift+rad,1+rad)--(shift+2,1+rad),th); draw(arc((shift+1-rad,1-2*rad),rad,0,90)^^arc((shift+rad,1),rad,90,270),th); } [/asy]

$2$ identical red tokens and $2$ identical black tokens are placed on distinct cells of a $5\times5$ grid. Suppose it is impossible to color some additional cells of the grid red or black such that there exists a red path between the red tokens and a black path between the black tokens. Find the number of possible arrangements of the tokens on the grid. (A red path is a path of edge adjacent red cells, and same for a black path.)