Let $ABC$ and $DBC$ be triangles with incircles touching at a point $P$ on $BC.$ Points $A,D$ lie on the same side of $BC$ and $DB < AB < DC < AC.$ The bisector of $\angle BDC$ meets line $AP$ at $X,$ and the altitude from $A$ meets $DP$ at $Y.$ Point $Z$ lies on line $XY$ so $ZP \perp BC.$ Show the reflection of $A$ over $BC$ is on line $ZD.$ Proposed by squareman (Evan Chang), USA

Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$ Proposed by Henry Jiang

A sequence of moves is performed starting on the three letter string "$BAD.$'' A move consists of inserting an additional instance of the three letter string "$BAD$'' between any two consecutive letters of the current string, to achieve an elongated string. After $n$ moves, how many distinct possible strings of length $3n+3$ can be achieved? For example, after one move the strings "$BBADAD$'' and "$BABADD$'' are achievable. Proposed by squareman (Evan Chang), USA

Find all ordered pairs of integers $(a,b)$ such that there exists a function $f\colon \mathbb{N} \to \mathbb{N}$ satisfying $$f^{f(n)}(n)=an+b$$ For all $n\in \mathbb{N}$.

Let $f(x)=x+3x^{\frac 23}, g(x)=x+x^{\frac 13}$. Call a sequence $\{a_i\}_{i\ge 0}$ satisfactory if for all $i\ge 1, a_i\in \{f(a_{i-1}), g(a_{i-1})\}$. Find all pairs of real numbers $(x,y)$ such that there exist satisfactory sequences $(a_i)_{i\ge 0}, (b_i)_{i\ge 0}$ and positive integers $m$ and $n$, such that $a_0 =x$, $b_0 = y$, and $$|a_m-b_n|<1$$

Determine, with proof, if there exists an odd prime $p$ such that the following equation holds: $$\sum_{n = 1}^{\frac{p-1}{2}} \cot\left(\frac{\pi n^2}{p}\right) = 69\sqrt{p}$$ Proposed by Chris Bao

In terms of $p$ and $k$, compute the number of solutions in positive integers to the equation $ab+bc+ca=p^{2k}$ satisfying $a\leq b\leq c$ where $p$ is a fixed prime and $k$ is a fixed positive integer. Proposed by Alexander Wang

Let $m$ be a nonnegative integer. Show that the number of tilings of a $(2m + 2) \times (2m + 2)$ grid of squares by $1 \times 2$ or $2 \times 1$ rectangles is at least $$2 \cdot 2^{\frac{5}{2}m} \cdot 5120^{\frac{1}{8}m^2}.$$ Proposed by Milan Haiman

Prove that $a^3 + b^3 + c^3 + abc +a^{3}b^{2}c^{-1}+a^{3}c^{2}b^{-1}+b^{3}a^{2}c^{-1}+b^{3}c^{2}a^{-1}+c^{3}a^{2}b^{-1}+c^{3}b^{2}a^{-1}+a^{5}b^{3}c^{-3}+ abc^{14} + a^{5}c^{3}b^{-3}+b^{5}a^{3}c^{-3}+b^{5}c^{3}a^{-3}+c^{5}a^{3}b^{-3}+c^{5}b^{3}a^{-3}+a^{6}b^{1}c^{-1}+a^{6}c^{1}b^{-1}+b^{6}a^{1}c^{-1}+b^{6}c^{1}a^{-1}+c^{6}a^{1}b^{-1}+c^{6}b^{1}a^{-1}+ a^{6}b^{4}c^{-3}+a^{6}c^{4}b^{-3}+b^{6}a^{4}c^{-3}+b^{6}c^{4}a^{-3}+c^{6}a^{4}b^{-3}+c^{6}b^{4}a^{-3}+a^{7}b^{2}c^{-1}+a^{7}c^{2}b^{-1}+b^{7}a^{2}c^{-1}+b^{7}c^{2}a^{-1}+c^{7}a^{2}b^{-1}+ abc + a^{14}bc + c^{7}b^{2}a^{-1}+a^{4}b^{1}c^{4}+a^{4}c^{1}b^{4}+b^{4}a^{1}c^{4}+b^{4}c^{1}a^{4}+c^{4}a^{1}b^{4}+c^{4}b^{1}a^{4}+a^{6}c^{4}+a^{6}b^{4}+b^{6}c^{4}+b^{6}a^{4}+c^{6}b^{4}+c^{6}a^{4}+a^{9}b^{6}c^{-4}+a^{9}c^{6}b^{-4}+ ab^{14}c + b^{9}a^{6}c^{-4}+b^{9}c^{6}a^{-4}+c^{9}a^{6}b^{-4}+ abc + c^{9}b^{6}a^{-4}+a^{12}b^{1}c^{-1}+a^{12}c^{1}b^{-1}+b^{12}a^{1}c^{-1}+b^{12}c^{1}a^{-1}+c^{12}a^{1}b^{-1}+ c^5 b^5 a^5 - c^5 b^5 a^2 + 3 c^5 b^5 - c^5 b^2 a^5 + c^5 b^2 a^2 - 3 c^5 b^2 + 3 c^5 a^5 - 3 c^5 a^2 + 9 c^5 - c^2 b^5 a^5 + c^2 b^5 a^2 - 3 c^2 b^5 + c^2 b^2 a^5 - c^2 b^2 a^2 + 3 c^2 b^2 - 3 c^2 a^5 + 3 c^2 a^2 - 9 c^2 + 3 b^5 a^5 - 3 b^5 a^2 + 9 b^5 - 3 b^2 a^5 + 3 b^2 a^2 - 9 b^2 + 9 a^5 - 9 a^2 + 27 + c^{12}b^{1}a^{-1}+a^{13}b^{9}c^{-9}+a^{13}c^{9}b^{-9}+b^{13}a^{9}c^{-9}+b^{13}c^{9}a^{-9}+c^{13}a^{9}b^{-9}+c^{13}b^{9}a^{-9}+a^{12}b^{11}c^{-9}+a^{12}c^{11}b^{-9}+b^{12}a^{11}c^{-9}+b^{12}c^{11}a^{-9}+c^{12}a^{11}b^{-9}+c^{12}b^{11}a^{-9}+a^{8}b^{7}+a^{8}c^{7}+b^{8}a^{7}+b^{8}c^{7}+c^{8}a^{7}+c^{8}b^{7} + a^{16} + b^{16} + c^{16} + a^{16} + b^{16} + c^{16} + a^{16} + b^{16} + c^{16}\ge c^3 + 3 c^2 a + 3 c b^2 + 6 c b a + b^3 + 3 b^2 a + a^3 + a^{1}c^{2}+a^{1}b^{2}+4b^{1}c^{2}+4b^{1}a^{2}+c^{1}b^{2}+4c^{1}a^{2}+a^{1}c^{3}+a^{1}b^{3}+b^{1}c^{3}+b^{1}a^{3}+c^{1}b^{3}+c^{1}a^{3}+a^{3}b^{2}+a^{3}c^{2}+b^{3}a^{2}+b^{3}c^{2}+c^{3}a^{2}+c^{3}b^{2}+a^{5}c^{1}+a^{5}b^{1}+b^{5}c^{1}+b^{5}a^{1}+c^{5}b^{1}+c^{5}a^{1}+a^{2}b^{1}c^{4}+a^{2}c^{1}b^{4}+b^{2}a^{1}c^{4}+b^{2}c^{1}a^{4}+c^{2}a^{1}b^{4}+c^{2}b^{1}a^{4}+a^{1}c^{7}+a^{1}b^{7}+b^{1}c^{7}+b^{1}a^{7}+c^{1}b^{7}+c^{1}a^{7}+a^{1}c^{8}+a^{1}b^{8}+b^{1}c^{8}+b^{1}a^{8}+c^{1}b^{8}+c^{1}a^{8}+a^{5}b^{1}c^{4}+a^{5}c^{1}b^{4}+b^{5}a^{1}c^{4}+b^{5}c^{1}a^{4}+c^{5}a^{1}b^{4}+c^{5}b^{1}a^{4}+a^{2}b^{1}c^{8}+a^{2}c^{1}b^{8}+b^{2}a^{1}c^{8}+b^{2}c^{1}a^{8}+c^{2}a^{1}b^{8}+c^{2}b^{1}a^{8}+a^{1}c^{11}+a^{1}b^{11}+b^{1}c^{11}+b^{1}a^{11}+c^{1}b^{11}+c^{1}a^{11}+a^{6}b^{2}c^{5}+a^{6}c^{2}b^{5}+b^{6}a^{2}c^{5}+b^{6}c^{2}a^{5}+c^{6}a^{2}b^{5}+c^{6}b^{2}a^{5}+a^{3}b^{2}c^{9}+a^{3}c^{2}b^{9}+b^{3}a^{2}c^{9}+b^{3}c^{2}a^{9}+c^{3}a^{2}b^{9}+c^{3}b^{2}a^{9}+a^{3}b^{1}c^{11}+a^{3}c^{1}b^{11}+b^{3}a^{1}c^{11}+b^{3}c^{1}a^{11}+c^{3}a^{1}b^{11}+c^{3}b^{1}a^{11} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15}+c^{2}a^{1}b^{4}+c^{2}b^{1}a^{4}+a^{1}c^{7}+a^{1}b^{7}+b^{1}c^{7}+b^{1}a^{7}+c^{1}b^{7}+c^{1}a^{7}+a^{1}c^{8}+a^{1}b^{8}+b^{1}c^{8}+b^{1}a^{8}+c^{1}b^{8}+c^{1}a^{8}+a^{5}b^{1}c^{4}+a^{5}c^{1}b^{4}+b^{5}a^{1}c^{4}+b^{5}c^{1}a^{4}+c^{5}a^{1}b^{4}+c^{5}b^{1}a^{4}+a^{2}b^{1}c^{8}+a^{2}c^{1}b^{8}+b^{2}a^{1}c^{8}+b^{2}c^{1}a^{8}+c^{2}a^{1}b^{8}+c^{2}b^{1}a^{8}+a^{1}c^{11}+a^{1}b^{11}+b^{1}c^{11}+b^{1}a^{11}+c^{1}b^{11}+c^{1}a^{11}+a^{6}b^{2}c^{5}+a^{6}c^{2}b^{5}+b^{6}a^{2}c^{5}+b^{6}c^{2}a^{5}+c^{6}a^{2}b^{5}+c^{6}b^{2}a^{5}+a^{3}b^{2}c^{9}+a^{3}c^{2}b^{9}+b^{3}a^{2}c^{9}+b^{3}c^{2}a^{9}+c^{3}a^{2}b^{9}+c^{3}b^{2}a^{9}+a^{3}b^{1}c^{11}+a^{3}c^{1}b^{11}+b^{3}a^{1}c^{11}+b^{3}c^{1}a^{11}+c^{3}a^{1}b^{11}+c^{3}b^{1}a^{11} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15} + a^{15}b + ab^{15} + a^{15}c + ac^{15} + b^{15}c + bc^{15}$ for all $a,b,c\in\mathbb R^+$. Proposed by Henry Jiang and C++