Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.

The company employs 50,000 people. For each of them, the sum of the number of his immediate superiors and his immediate subordinates is equal to 7. On Monday, each employee of the enterprise issues an order and gives a copy of this order to each of his direct subordinates (if there are any). Further, every day an employee takes all the basics he received on the previous day and either distributes them copies to all your direct subordinates, or, if any, he is not there, he carries out orders himself. It turned out that on Friday no papers were transferred to the institution. Prove that the enterprise has at least 97 bosses over whom there are no bosses.

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$. [i]Proposed by Iman Maghsoudi[/i]

In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment

Let $a, b, c$ be positive reals with $abc = 1$. Prove the inequality $$\frac{a^5}{a^3 + 1}+\frac{b^5}{b^3 + 1}+\frac{c^5}{c^3 + 1} \ge \frac32$$ and determine all values of a, b, c for which equality is attained

(a) Prove that for every positive integer $m$ there exists an integer $n\ge m$ such that $$\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\ (*)$$ (b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation $ (*)$ holds. Prove that $p(2018) = p(2019).$ Remark: For a real number $x,$ we denote by $\left \lfloor x \right \rfloor$ the largest integer not larger than $x.$

For positive integers $i = 2, 3, \ldots, 2020$, let \[ a_i = \frac{\sqrt{3i^2+2i-1}}{i^3-i}. \]Let $x_2$, $\ldots$, $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \cdots + x_{2020}^4 = 1-\frac{1}{1010\cdot 2020\cdot 2021}$. Let $S$ be the maximum possible value of \[ \sum_{i=2}^{2020} a_i x_i (\sqrt{a_i} - 2^{-2.25} x_i) \] and let $m$ be the smallest positive integer such that $S^m$ is rational. When $S^m$ is written as a fraction in lowest terms, let its denominator be $p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ for prime numbers $p_1 < \cdots < p_k$ and positive integers $\alpha_i$. Compute $p_1\alpha_1+p_2\alpha_2 + \cdots + p_k\alpha_k$. [i]Proposed by Edward Wan and Brandon Wang[/i]

Determine all commutative rings $R$ with at least four elements that are not fields, such that for any pairwise distinct and nonzero elements $a,b,c\in R$, $ab+bc+ca$ is invertible. [i]Vlad Matei[/i]

For two complex numbers $z_1,z_2$ satisfy that $|z_1|=|z_1+z_2|=3,|z_1-z_2|=3\sqrt3$, then $\log_3|(z_1\overline{z_2})^{2000}+(\overline{z_1}z_2)^{2000}|=$________.

Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$.

Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

Let $f$ be the function of the set of positive integers into itself, defined by $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the number of positive odd integers m such that $f(m) = n$ is equal to the number of positive integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$. [color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]

The diagram below shows equilateral $\triangle ABC$ with side length $2$. Point $D$ lies on ray $\overrightarrow{BC}$ so that $CD = 4$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $E$, $F$, and $D$ are collinear, and the area of $\triangle AEF$ is half of the area of $\triangle ABC$. Then $\tfrac{AE}{AF}=\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + 2n$. [asy] import math; size(7cm); pen dps = fontsize(10); defaultpen(dps); dotfactor=4; pair A,B,C,D,E,F; B=origin; C=(2,0); D=(6,0); A=(1,sqrt(3)); E=(1/3,sqrt(3)/3); F=extension(A,C,E,D); draw(C--A--B--D,linewidth(1.1)); draw(E--D,linewidth(.7)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("$E$",E,NW); label("$F$",F,NE); [/asy]

A finite collection of positive real numbers (not necessarily distinct) is [i]balanced [/i]if each number is less than the sum of the others. Find all $m \ge 3$ such that every balanced finite collection of $m$ numbers can be split into three parts with the property that the sum of the numbers in each part is less than the sum of the numbers in the two other parts.

Does exist polynoms of one variable that are irreducible over the field of integers, have degree $ 60 $ and have multiples of the form $ X^n-1? $ If so, how many of them?

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

The following sentece is written on a board: [center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center] Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?

Suppose $f$ is a function that assigns to each real number $x$ a value $f(x)$, and suppose the equation $$f(x_1 + x_2 + x_3 + x_4 + x_5) = f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5) - 8$$ holds for all real numbers $x_1, x_2,x_3, x_4, x_5$. What is $f(0)$?

On the coordinate plane, draw a set of points $M(x;y)$, whose coordinates satisfy the equation $$\sqrt{(x - 1)^2+ y^2} +\sqrt{x^2 + (y -1)^2} = \sqrt2.$$

a) Given a prime number $p$ and $2$ polynomials$$P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.$$ We know that the product $P(x)Q(x)$ is a polynomial whose coefficents are all divisible by $p.$ Prove that: at least $1$ in $2$ polynomials $P(x),Q(x)$ has all coefficents are all divisible by $p.$ b) Prove that the product of $2$ original polynomials is a original polynomial.

Find the maximal possible $n$, where $A_1, \dots, A_n \subseteq \{1, 2, \dots, 2016\}$ satisfy the following properties. - For each $1 \le i \le n$, $\lvert A_i \rvert = 4$. - For each $1 \le i < j \le n$, $\lvert A_i \cap A_j \rvert$ is even.

Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying: \[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\] Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.

Let $n$ red and $n$ blue subarcs of a circle be given such that each red subarc intersects each blue subarc. Prove that there is a point which is covered by at least $n$ of the given (red or blue) subarcs.

Before Ashley started a three-hour drive, her car’s odometer reading was $ 29792$, a palindrome. At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of $ 75$ miles per hour, which of the following was her greatest possible average speed? $ \textbf{(A)}\ 33\frac 13 \qquad \textbf{(B)}\ 53\frac 13\qquad \textbf{(C)}\ 60\frac 23\qquad \textbf{(D)}\ 70\frac 13\qquad \textbf{(E)}\ 74\frac 13$