Let $ABCD$ be a cyclic quadrilateral with $\angle BAD \lt \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD$, $PM$, $BC$ are concurrent.
Let $ABC$ be an acute-angled triangle with $AB \lt AC$. Denote its circumcircle by $\Omega$ and denote the midpoint of $\operatorname{arc} CAB$ by $S$. Let the perpendicular from $A$ to $BC$ meet $BS$ and $\Omega$ at $D$ and $E \neq A$ respectively. Let the line through $D$ parallel to $BC$ meet line $BE$ at $L$ and denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$.
Prove that the line tangent to $\omega$ at $P$, and line $BS$ intersect on the internal bisector of $\angle BAC$.
Let $ABC$ be an equilateral triangle. Points $A_1$, $B_1$, $C_1$ lie inside triangle $ABC$ such that triangle $A_1B_1C_1$ is scalene, $BA_1 = A_1C$, $CB_1 = B_1A$, $AC_1 = C_1B$ and
$$
\angle BA_1C + \angle CB_1A + \angle AC_1B = 480^\circ.
$$
Lines $BC_1$ and $CB_1$ intersect at $A_2$; lines $CA_1$ and $AC_1$ intersect at $B_2$; and lines $AB_1$ and $BA_1$ intersect at $C_2$.
Prove that the circumcircles of triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ have two common points.
Determine all positive, composite integers $n$ that satisfy the following property: if the positive divisors of $n$ are $1=d_{1} \lt d_{2} \lt \cdots \lt d_{k}=n$, then $d_{i}$ divides $d_{i+1}+d_{i+2}$ for every $1 \leqslant i \leqslant k-2$.
Let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be $2n$ positive integers such that the $n+1$ products
$$
\begin{gathered}
a_{1} a_{2} a_{3} \cdots a_{n}, \\
b_{1} a_{2} a_{3} \cdots a_{n}, \\
b_{1} b_{2} a_{3} \cdots a_{n}, \\
\vdots \\
b_{1} b_{2} b_{3} \cdots b_{n}
\end{gathered}
$$
form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.
Let $a_{1} \lt a_{2} \lt a_{3} \lt \cdots$ be positive integers such that $a_{k+1}$ divides $2(a_{1} + a_{2} + \cdots + a_{k})$ for every $k \geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_{k}$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_{k}$.
A sequence of integers $a_{0}, a_{1}, a_{2}, \ldots$ is called kawaii, if $a_{0}=0, a_{1}=1$, and, for any positive integer $n$, we have
$$
\left(a_{n+1}-3 a_{n}+2 a_{n-1}\right)\left(a_{n+1}-4 a_{n}+3 a_{n-1}\right)=0 .
$$
An integer is called kawaii if it belongs to a kawaii sequence.
Suppose that two consecutive positive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that 3 divides $m$, and that $m / 3$ is kawaii.
Let $a, b, c, d$ be positive integers satisfying
$$
\frac{a b}{a+b}+\frac{c d}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}
$$
Determine all possible values of $a+b+c+d$.
Let $\mathbb{Z}_{ \gt 0}$ be the set of positive integers. Determine all functions $f: \mathbb{Z}_{ \gt 0} \rightarrow \mathbb{Z}_{ \gt 0}$ such that
$$
f^{b f(a)}(a+1) = (a+1) f(b)
$$
holds for all $a, b \in \mathbb{Z}_{ \gt 0}$, where $f^{k}(n) = f(f(\cdots f(n) \cdots))$ denotes the composition of $f$ with itself $k$ times.
Let $r \gt 0$ be a real number. All the interior points of the disc $D(r)$ of radius $r$ are colored with one of two colors, red or blue.
1. If $r \gt \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that the distance $AB = \pi$ and $A$ and $B$ have the same color.
2. Does the conclusion in (a) hold if $r \gt \frac{\pi}{2}$?
Alice has an integer $N \gt 1$ written on a blackboard. Each minute, she deletes the current number $x$ on the blackboard and writes $2x + 1$ if $x$ is not the cube of an integer, or the cube root of $x$ otherwise. Prove that at some point in time, she writes a number larger than $10^{100}$.
Let $N \ge 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are each given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.
Let $f, g$ be functions $\mathbb{R} \to \mathbb{R}$ such that
$$
f(g(x) + y) = g(x + y) \quad \forall x, y \in \mathbb{R}
$$
then either $f$ is the identity¹ function or $g$ is periodic².
Let $k$ be a positive integer. A sequence of integers $\langle a_i \rangle_{i=1}^\infty$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \dots, a_{n+k}\}$. Determine, as a function of $k$, how many $k$-pop sequences there are.
Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P, Q, R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90 - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$.
Find all $f: \mathbb{N} \to \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of 1's in their binary representation for any $x, y \in \mathbb{N}$.
Suppose an acute scalene triangle $ABC$ has incentre $I$ and incircle touching $BC$ at $D$. Let $Z$ be the antipode of $A$ in the circumcircle of $ABC$. Point $L$ is chosen on the internal angle bisector of $\angle BZC$ such that $AL = LI$. Let $M$ be midpoint of arc $BZC$, and let $V$ be the midpoint of $ID$. Prove that $\angle IML = \angle DVM$.
In triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to line $BC$. Point $K$ lies inside triangle $ABC$ such that $\angle KAB = \angle KCA$ and $\angle KAC = \angle KBA$. The line through $K$ perpendicular to line $DK$ meets the circle with diameter $BC$ at points $X$ and $Y$. Prove that $AX \cdot DY = DX \cdot AY$.
In a school, every pair of students are either friends or strangers. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called *mischievous* if
* Total number of friends of $A_1$ is odd.
* $A_i$ and $A_{i+1}$ are friends for $i = 1, 2, \dots, 2022$.
* Total number of friends of $A_{2023}$ is even.
Prove that the total number of *mischievous* sequences is even.
Prove that for all integers $k \gt 2$, there exists $k$ distinct positive integers $a_1, \dots, a_k$ such that
$$
\sum_{1 \le i \lt j \le k} \frac{1}{a_i a_j} = 1.
$$