Let $\triangle{ABC}$ be an acute triangle with orthocenter $H.$ Points $E$ and $F$ are on segments $\overline{AC}$ and $\overline{AB},$ respectively, such that $\angle{EHF}=90^\circ.$ Let $X$ be the foot of the perpendicular from $H$ to $\overline{EF}.$ Prove that $\angle{BXC}=90^\circ.$

The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

Let $\triangle{ABC}$ be a triangle with incenter $I.$ The incircle of triangle $\triangle{ABC}$ touches $\overline{BC}$ at $D.$ Let $M$ be the midpoint of $\overline{BC},$ and let line $AI$ meet the circumcircle of triangle $\triangle{ABC}$ again at $L \neq A.$ Let $\omega$ be the circle centered at $L$ tangent to $AB$ and $AC.$ If $\omega$ intersects $\overline{AD}$ at point $P,$ prove that $\angle{IPM}=90^\circ.$

Compute the remainder when $10^{2021}$ is divided by $10101$. [i]Proposed by Nathan Xiong[/i]

Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?

Solve for $x$ if $\sqrt{x + 1}+ \sqrt{x} = 5.$ [i]Proposed by Eric Oh[/i]

For some integer $n > 0$, a square paper of side length $2^{n}$ is repeatedly folded in half, right-to-left then bottom-to-top, until a square of side length 1 is formed. A hole is then drilled into the square at a point $\tfrac{3}{16}$ from the top and left edges, and then the paper is completely unfolded. The holes in the unfolded paper form a rectangular array of unevenly spaced points; when connected with horizontal and vertical line segments, these points form a grid of squares and rectangles. Let $P$ be a point chosen randomly from \textit{inside} this grid. Suppose the largest $L$ such that, for all $n$, the probability that the four segments $P$ is bounded by form a square is at least $L$ can be written in the form $\tfrac mn$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$.

Consider the polynomial \[P(x)=x^3+3x^2+6x+10.\] Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$. [i]Proposed by Nathan Xiong[/i]

George is taking a ten-question true-false exam, where the answer key has been selected uniformly at random; however, he doesn't know any of the answers! Luckily, a friend has helpfully hinted that no two consecutive questions have true as the correct answer. If George takes the exam and maximizes the expected number of questions he gets correct, how many of his answers are expected to be right?

Across all $x \in \mathbb{R}$, find the maximum value of the expression $$\sin x + \sin 3x + \sin 5x.$$

Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$. Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$. If $BC'=30$, find the sum of all possible side lengths of $\triangle ABC$. [i]Proposed by Connor Gordon[/i]

Suppose we have a regular $24$-gon labeled $A_1 \cdots A_{24}.$ We will draw $2$ similar $24$-gons within $A_1 \cdots A_{24}.$ For the sake of this problem, make $A_i=A_{i+24}.$ With our first configuration, we create $3$ stars by creating lines $\overline{A_iA_{i+9}}.$ A $24$-gon will be created in the center, which we denote as our first $24$-gon. With our second configuration, we create a start by creating lines $\overline{A_iA_{i+11}}.$ A $24$-gon will be created in the center, which we denote as our second $24$-gon. Find the ratio of the areas of the first $24$-gon to the second $24$-gon.

The binomial coefficient $\tbinom{n}{k}$ can be defined as the coefficient of $x^k$ in the expansion of $(1+x)^n.$ Similarly, define the trinomial coefficient $\tbinom{n}{k}_3$ as the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n.$ Determine the number of integers $k$ with $0 \le k \le 4048$ such that $\tbinom{2024}{k}_3 \equiv 1 \pmod{3}.$

Find the integer $n$ such that \[n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.\] Here, as usual, $\lfloor\cdot\rfloor$ denotes the floor function.

Find the product of all possible real values for $k$ such that the system of equations $$x^2+y^2= 80$$ $$x^2+y^2= k+2x-8y$$ has exactly one real solution $(x,y)$. [i]Proposed by Nathan Xiong[/i]

We have four registers, $R_1,R_2,R_3,R_4$, such that $R_i$ initially contains the number $i$ for $1\le i\le4$. We are allowed two operations: [list] [*] Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$. [*] Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. (For example if we do this once then $(R_1,R_2,R_3,R_4)=(1,4,2,3)$.) [/list] Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?

Let $ABC$ be a triangle. The incircle $\omega$ of $\triangle ABC$, which has radius $3$, is tangent to $\overline{BC}$ at $D$. Suppose the length of the altitude from $A$ to $\overline{BC}$ is $15$ and $BD^2 + CD^2 = 33$. What is $BC$?

In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?

Determine the maximum possible value of $$\sqrt{x}(2\sqrt{x}+\sqrt{1-x})(3\sqrt{x}+4\sqrt{1-x})$$ over all $x\in [0,1]$.

Given $10$ points arranged in a equilateral triangular grid of side length $4$, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?

Real numbers $x$ and $y$ satisfy \begin{align*} x^2 + y^2 &= 2023 \\ (x-2)(y-2) &= 3. \end{align*} Find the largest possible value of $|x-y|$. [i]Proposed by Howard Halim[/i]

Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.

Let $a,b,$ and $c$ be pairwise distinct positive integers such that $\tfrac{1}{a}, \tfrac{1}{b}, \tfrac{1}{c}$ is an increasing arithmetic sequence in that order. Prove that $\gcd(a,b)>1.$

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 $k$

Simplify $$\dbinom{2020}{1010}\dbinom{1010}{1010}+\dbinom{2019}{1010}\dbinom{1011}{1010}+\cdots+\dbinom{1011}{1010}\dbinom{2019}{1010} + \dbinom{1010}{1010}\dbinom{2020}{1010}.$$