Suppose $ H$ and $ O$ are the orthocenter and the circumcenter of acute triangle $ ABC$; $ AA_1$, $ BB_1$ and $ CC_1$ are the altitudes of the triangle. Point $ C_2$ is the reflection of $ C$ in $ A_1B_1$. Prove that $ H$, $ O$, $ C_1$ and $ C_2$ are concyclic.

For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$ where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?

Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$. Note: $N = \{0,1,2,...\}$

Determine the smallest positive integer $m$ such that $529^n+m\cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.

Call a positive integer $m$ $\textit{good}$ if there exist integers $a, b, c$ satisfying $m=a^3+2b^3+4c^3-6abc$. Show that there exists a positive integer $n<2024$, such that for infinitely many primes $p$, the number $np$ is $\textit{good}$.

For positive real numbers $x,y,z$ that satisfy $ xy + yz + zx + xyz=4$, prove that $$\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)$$

Consider a unit circle with center $O$. Let $P$ be a point outside the circle such that the two line segments passing through $P$ and tangent to the circle form an angle of $60^\circ$. Compute the length of $OP$.

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that \[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\] Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.

Five numbers from a list of nine integers are $7,8,3,5,$ and $9$. The largest possible value of the median of all nine numbers in this list is $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

A 16-step path is to go from $ ( \minus{} 4, \minus{}4)$ to $ (4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $ \minus{} 2 \le x \le 2$, $ \minus{} 2 \le y \le 2$ at each step? $ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,\!800$

Consider all 1000-element subsets of the set $\{1,2,3,\dots,2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? $\textbf{(A) } 32 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 48 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 64 $

On a board, Ana writes $a$ different integers, while Ben writes $b$ different integers. Then, Ana adds each of her numbers with with each of Ben’s numbers and she obtains $c$ different integers. On the other hand, Ben substracts each of his numbers from each of Ana’s numbers and he gets $d$ different integers. For each integer $n$ , let $f(n)$ be the number of ways that $n$ may be written as sum of one number of Ana and one number of Ben. [i]a)[/i] Show that there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{c}.$$ [i]b)[/i] Does there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{d}?$$ [i]Proposed by Besfort Shala, Kosovo[/i]

The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is: $\textbf{(A)}\ 33\qquad \textbf{(B)}\ 34\qquad \textbf{(C)}\ 35\qquad \textbf{(D)}\ 100\qquad \textbf{(E)}\ \text{none of these}$

$$Problem 2 :$$If ABCD is as convex quadrilateral with $\angle ADC = 30$ and $BD = AB+BC+CA$, prove that $BD$ bisects $\angle ABC$.

Each of $4$ stones weights the integer number of grams. A balance with arrow indicates the difference of weights on the left and the right sides of it. Is it possible to determine the weights of all stones in $4$ weighings, if the balance can make a mistake in $1$ gram in at most one weighing?

Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that \[ f(x)+f(y)+f(z)=1 \] holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$.

For any complex number $w = a + bi$, $|w|$ is defined to be the real number $\sqrt{a^2 + b^2}$. If $w = \cos{40^\circ} + i\sin{40^\circ}$, then \[ |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \] equals $\textbf{(A)}\ \frac{1}{9}\sin{40^\circ} \qquad \textbf{(B)}\ \frac{2}{9}\sin{20^\circ} \qquad \textbf{(C)}\ \frac{1}{9}\cos{40^\circ} \qquad \textbf{(D)}\ \frac{1}{18}\cos{20^\circ} \qquad \textbf{(E)}\text{ none of these}$

Let $ABCDE$ be a convex pentagon that satisfies all of the following:[list] [*]There is a circle $\Gamma$ tangent to each of the sides. [*]The lengths of the sides are all positive integers. [*]At least one of the sides of the pentagon has length $1$. [*]The side $AB$ has length $2$.[/list] Let $P$ be the point of tangency of $\Gamma$ with $AB$.[list] (a) Determine the lengths of the segments $AP$ and $BP$. (b) Give an example of a pentagon satisfying the given conditions.[/list]

Any circle in the $xy$-plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle. Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid. By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.

A sequence with first term $a_0$ is defined such that $a_{n+1}=2a_n^2-1$ for $n\geq0.$ Let $N$ denote the number of possible values of $a_0$ such that $a_0=a_{2020}.$ Find the number of factors of $N.$ [i]Proposed by Alex Li[/i]

Given a triangle with sides $a, b, c$ and medians $s_a, s_b, s_c$ respectively. Prove the following inequality: $$a + b + c> s_a + s_b + s_c> \frac34 (a + b + c) $$

Let $n \geq 2$ be an integer. Find the smallest positive integer $m$ for which the following holds: given $n$ points in the plane, no three on a line, there are $m$ lines such that no line passes through any of the given points, and for all points $X \neq Y$ there is a line with respect to which $X$ and $Y$ lie on opposite sides

Evaluate \[2\times (2\times (2\times (2\times (2\times (2\times 2-2)-2)-2)-2)-2)-2.\] [i]Proposed by Nathan Xiong[/i]

How many subsets of $\{1, 2, \cdots, 10\}$ are there that don't contain $2$ consecutive integers?