An unordered triple of numbers $(a,b,c)$ in one move you can change to either $(a,b,2a+2b-c)$, $(a,2a+2c-b,c)$ or $(2b+2c-a,b,c)$. Can you from the triple $(3,5,14)$ get the tripel $(3,13,6)$ in finite amount of moves?

The driver starts driving every morning at the same time from office to the house of his boss, picks up the boss and then drives back to the office. He always drives with the same speed on the same road. Because the time of arrival of the car to the boss's house is predetermined, the boss always leaves the house on time, and thus the driver does not spend any time waiting for his boss. Once the driver started driving from the office $42$ minutes later, than usual. The boss saw that the car didn't come and started walking in the direction of office. When he met the car on the road, the driver picked him up and started driving back to the office. The speed of the boss is 20 times lower than the speed of the car, and the time usually spent on the route from office to the house is at least an hour. Determine did the car come earlier or later to the office and by how many minutes.

In the triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. $I$ is the incenter of the triangle. It is known that the angle $MIC$ is a right angle. Find the angle $NIB$.

Paca-Vaca decided to note every day a single quadratic polynomial of the form $x^2+ax+b$, where $a$ and $b$ are positive integers, less or equal than $100$. He follows the rule that the polynomial he writes must not have any common roots with the polynomials previously noted. What is the maximum amount of days Paca-Vaca can follow this plan?

In every cell of the table $3 \times 3$ a monomial with a positive coefficient is written (cells (1,1); (2,3); (3,2) have the degree of two, cells (1,2);(2,1);(3,3) have a degree of one, cells (3,1);(2,2);(1,3) have a constant). Vuga added up monomials in every row and got three quadratic polynomials. It turned out that exactly $N$ of them have real roots. Leka added up monomials in every column and got three quadratic polynomials. It turned out that exactly $M$ of them have real roots. Find the maximum possible value of $N-M$.

On the side $BC$ of a triangle $ABC$ the midpoint $M$ and arbitrary point $K$ is marked. Lines that pass through $K$ parallel to the sides of the triangle intersect the line $AM$ at $L$ and $N$. Prove that $ML=MN$.

A sequence $(a_n)$ positive integers is determined by equalities $a_1=20,a_2=22$ and $a_{n+1}=4a_n^2+5a_{n-1}^3$ for all $n \geq 2$. Find the maximum power of two which divides $a_{2023}$.

The fence consists of $25$ vertical bars. The heights of the bars are pairwise distinct positive integers from $1$ to $25$. The width of every bar is $1$. Find the maximum $S$ for which regardless of the order of the bars one can find a rectangle of area $S$ formed by the fence.

Real numbers $a,b,c,d$ satisfy the equality $$\frac{1-ab}{a+b}=\frac{bc-1}{b+c}=\frac{1-cd}{c+d}=\sqrt{3}$$Find all possible values of $ad$.

An unordered triple $(a,b,c)$ in one move can be changed to either of the triples: $(a,b,2a+2b-c)$,$(a,2a+2c-b,c)$ or $(2b+2c-a,b,c)$. Can one get from triple $(3,5,14)$ the triple $(9,8,11)$ in finite amount of moves?

The triangle $ABC$ has perimeter $36$, and the length of $BC$ is $9$. Point $M$ is the midpoint of $AC$, and $I$ is the incenter. Find the angle $MIC$.

A circle is divided into $2n$ equal sectors, $n \in \mathbb{N}$. Vitya and Masha are playing the following game. At first, Vitya writes one number in every sector from the set $\{1,2,\ldots,n\}$ and every number is used exatly twice. After that Masha chooses $n$ consecutive sectors and writes $1$ in the first sector, $2$ in the second, $n$ in the last. Vitya wins if at least in one sector two equal number will be written, otherwise Masha wins. Find all $n$ for which Vitya can guarantee his win.

The polynomial $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_1x+a_0$ ($a_{2n} \neq 0$) doesn't have any real roots. Prove that the polynomial $Q(x)=a_{2n}x^{2n}+a_{2n-2}x^{2n-2}+\ldots+a_2x^2+a_0$ also doesn't have any real roots.

Find the biggest positive integer $n$ for which the number $(n!)^6-6^n$ is divisible by $2022$.

On one of the sides of the $60$ degree angle with vertex $O$ a fixed point $F$ is marked. On the other side of the angle a point $A$ is chosen, and on the ray $OF$, but not the segment $OF$, a point $B$ such that $OA=FB$. On the segment $AB$ equilateral triangle $ABC$ and $ABD$ are built such that points $O$ and $C$ lie in the same half-plane with respect to $AB$, and $D$ in the other. a) Prove that the point $C$ does not depend on $A$. b) Prove that all points $D$ lie on a line.

On the faces of a cube several positive integer numbers are written. On every edge the sum of the numbers of it's two faces is written, and in every vertex the sum of numbers on the three faces that have this vertex. It turned out that all the written numbers are different. Find the smallest possible amount of the sum of all written numbers.

A circle $\omega$ with center $I$ is located inside the circle $\Omega$ with center $O$. Ray $IO$ intersects $\omega$ and $\Omega$ at $P_1$ and $P_2$ respectively. On $\Omega$ an arbitrary point $A \neq P_2$ is chosen. The circumcircle of the triangle $P_1P_2A$ intersects $\omega$ for the second time at $X$. Line $AX$ intersects $\Omega$ for the second time at $Y$. Prove that lines $XP_1$ and $YP_2$ are perpendicular to each other

A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one. Find the maximum possible number of distinct prime divisors of $n$

Let $a,b,c$ be positive real numbers, that satisfy $abc=1$. Prove the inequality: $$\frac{ab}{1+c}+\frac{bc}{1+a}+\frac{ca}{1+b} \geq \frac{27}{(a+b+c)(3+a+b+c)}$$

Find the maximal possible numbers one can choose from $1,\ldots,100$ such that none of the products of non-empty subset of this numbers was a perfect square.

On hyperbola $y=\frac{1}{x}$ points $A_1,\ldots,A_{10}$ are chosen such that $(A_i)_x=2^{i-1}a$, where $a$ is some positive constant. Find the area of $A_1A_2 \ldots A_{10}$

Prove that for any positive integer $n$ there exists a positive integer $k$ such that $3^k+4^k-1 \vdots 12^n$

Point $D$ is the midpoint of $BC$, where $ABC$ is an isosceles triangle ($AB=AC$). On circle $(ABD)$ a point $P \neq A$ is chosen. $O$ is the circumcenter of $ACP$, $Q$ is the foot of the perpendicular from $C$ onto $AO$. Prove that the circumcenter of triangle $ABQ$ lies on the line $AP$

On the Alphamegacentavra planet there are $2023$ cities, some of which are connected by non-directed flights. It turned out that among any $4$ cities one can find two with no flight between them. Find the maximum number of triples of cities such that between any two of them there is a flight.

On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$. Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$

On a blackboard triangle $ABC$ is drawn. Vlad draws a random point $D$ inside it and then reflects $A,C,B$ across the midpoints of $CD, BD, AD$, gets $C_1, A_1, B_1$. When Vlad wasn't looking at the board, Dima deleted from it everything, except for $A_1,B_1,C_1$. Can Vlad now using only chalk, ruler and compass draw the original point $D$?

Prove that for any fixed integer $a$ equation $$(m!+a)^2=n!+a^2$$has finitely many solutions in positive integers $m,n$

Denote by $R_{>0}$ the set of all positive real numbers. Find all functions $f: R_{>0} \to R_{>0}$ such that for all $x,y \in R_{>0}$ the following equation holds $$f(y)f(x+f(y))=f(1+xy)$$

A sequence of positive integers is given such that the sum of any $6$ consecutive terms does not exceed $11$. Prove that for any positive integer $a$ in the sequence one can find consecutive terms with sum $a$

Let $a$ be some integer. Prove that the polynomial $x^4(x-a)^4+1$ can not be a product of two non-constant polynomials with integer coefficients

Let $\omega$ be the incircle of triangle $ABC$. Line $l_b$ is parallel to side $AC$ and tangent to $\omega$. Line $l_c$ is parallel to side $BC$ and tangent to $\omega$. It turned out that the intersection point of $l_b$ and $l_c$ lies on circumcircle of $ABC$ Find all possible values of $\frac{AB+AC}{BC}$

Positive integer $n$ is called good if there exist $n$ points on plane($X_1, \ldots, X_n$), such that for all $1 \leq i \leq n$ vectors $X_iX_1, \ldots, X_iX_n$ can be partitioned into two groups with equal sums. Find all good numbers