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.

We are given $n>1$ piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin. Find the minimal number of weightings on digital scale that we need in order to conclude: which piles consists of which type of coins and also the mass of fake coin. (We assume that every pile consists from infinite number of coins.)

Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?

Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?

(a) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $2f(f(x)) = f(x) \geq 0$ for all $x \in R$. (b) Find out, whether there exists a differentiable function $f: R \longrightarrow R$, not equaling $0$ for all $x \in R$, satisfying the conditions $-1 \leq 2f(f(x)) = f(x) \leq 1$ for all $x \in R$.

Let $ S(t)$ be the area of the traingle $ OAB$ with $ O(0,\ 0,\ 0),\ A(2,\ 2,\ 1),\ B(t,\ 1,\ 1 \plus{} t)$. Evaluate $ \int_1^ e S(t)^2\ln t\ dt$.

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$ [asy] size(6cm); pair A = (0,10); label("$A$", A, N); pair B = (0,0); label("$B$", B, S); pair C = (10,0); label("$C$", C, S); pair D = (10,10); label("$D$", D, SW); pair EE = (15,11.8); label("$E$", EE, N); pair F = (3,10); label("$F$", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label("$110^\circ$", (15,9), SW); [/asy] $\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$

The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?

A natural number is written in each box of the $13 \times 13$ grid area. Prove that you can choose $2$ rows and $4$ columns such that the sum of the numbers written at their $8$ intersections is divisible by $8$.

Compute $$\prod^{2019}_{i=1} (2^{2^i}- 2^{2^{i-1}} + 1).$$

Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle. Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$. [i]Australia[/i]

Let $ n $ be a natural number. Find all integer numbers that can be written as $$ \frac{1}{a_1} +\frac{2}{a_2} +\cdots +\frac{n}{a_n} , $$ where $ a_1,a_2,...,a_n $ are natural numbers.

Let $p$ be a prime number such that $p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1$. Show that $d$ is a prime.

How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?

We consider all $7$-digit numbers that are obtained by swapping in all ways Possible digits of $1234567$. How many of them are divisible by $7$?