Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$: \[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]

You are given some $12$ non-zero, not necessarily distinct real numbers. Find all positive integers $k$ from $1$ to $12$, such that among these numbers you can always choose $k$ numbers whose sum has the same sign as their product, that is, either both the sum and the product are positive, or both are negative. [i]Proposed by Anton Trygub[/i]

Prove that for every integer $S\ge100$ there exists an integer $P$ for which the following story could hold true: The mathematician asks the shop owner: ``How much are the table, the cabinet and the bookshelf?'' The shop owner replies: ``Each item costs a positive integer amount of Euros. The table is more expensive than the cabinet, and the cabinet is more expensive than the bookshelf. The sum of the three prices is $S$ and their product is $P$.'' The mathematician thinks and complains: ``This is not enough information to determine the three prices!'' (Proposed by Gerhard Woeginger, Austria)

Prove that for all positive reals $a, b, c$ the following double inequality holds: $$\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}$$

[b]p1.[/b] Due to the crisis, the salaries of the company's employees decreased by $1/5$. By what percentage should it be increased in order for it to reach its previous value? [b]p2.[/b] Can the sum of six different positive numbers equal their product? [b]p3.[/b] Points$ A, B, C$ and $B$ are marked on the straight line. It is known that $AC = a$ and $BP = b$. What is the distance between the midpoints of segments $AB$ and $CB$? List all possibilities. [b]p4.[/b] Find the last three digits of $625^{19} + 376^{99}$. [b]p5.[/b] Citizens of five different countries sit at the round table (there may be several representatives from one country). It is known that for any two countries (out of the given five) there will be citizens of these countries sitting next to each other. What is the smallest number of people that can sit at the table? [b]p6.[/b] Can any rectangle be cut into $1999$ pieces, from which you can form a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Whether there are integers $a_1$, $a_2$, $\cdots$, that are different from each other, satisfying: (1) For $\forall k\in\mathbb N_+$, $a_{k^2}>0$ and $a_{k^2+k}<0$; (2) For $\forall n\in\mathbb N_+$, $\left| a_{n+1}-a_n\right|\leqslant 2023\sqrt n$?

There are $2500$ people in Lexington High School, who all start out healthy. After $1$ day, $1$ person becomes infected with coronavirus. Each subsequent day, there are twice as many newly infected people as on the previous day. How many days will it be until over half the school is infected?

If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left \lfloor n \plus{} \sqrt {3n} \plus{} \frac {1}{2} \right \rfloor$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} \left \lfloor \frac {n^2 \plus{} 2n}{3} \right \rfloor$.

Consider polynomial $f(x)={{x}^{2}}-\alpha x+1$ with $\alpha \in \mathbb{R}.$ a) For $\alpha =\frac{\sqrt{15}}{2}$, let write $f(x)$ as the quotient of two polynomials with nonnegative coefficients. b) Find all value of $\alpha $ such that $f(x)$ can be written as the quotient of two polynomials with nonnegative coefficients.

Prove that the Fibonacci sequence $\{F_n\}^\infty_{n=1}$ defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for all $n \geq 1$ is a divisibility sequence, that is, if $m\mid n$ then $F_m \mid F_n$ for all positive integers $m$ and $n$.

Suppose $f$ is a nonconstant polynomial with integer coefficients with the following property: [list] [*]$f(0)$ and $f(1)$ are both odd. [*]Define a sequence of integers with $a_k = f(1)f(2) \cdots f(k)+1$ [/list] Prove that there are infinitely many prime numbers dividing at least one element of the sequence. [i]Proposed by Sayandeep Shee[/i]

Find the value of \[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]

Given the following real numbers $a. b, c $ greater than one that $a + b + c = 6$. Prove the inequality $$\frac{a}{b^2-1}+\frac{b}{c^2-1}+\frac{c}{a^2-1}\ge 2$$

While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw $3$ squares $\square_1, \square_2, \square_3$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\{1,2,3,4,5\}$ such that no integer is used more than once. Ana's goal is to minimize the minimum value that the polynomial $a_1x^2 + a_2x + a_3$ attains over all real $x$, where $a_1, a_2, a_3$ are the integers written in $\square_1, \square_2, \square_3$ respectively. Banana aims to maximize $M$. Assuming both play optimally, compute the final value of $100a_1+10a_2+a_3$.

Let $x_1 , \dots , x_{2024}$ be non negative real numbers with $\displaystyle{\sum_{i=1}^{2024}}x_i = 1$. Find, with proof, the minimum and maximum possible values of the following expression \[\sum_{i=1}^{1012} x_i + \sum_{i=1013}^{2024} x_i^2 .\]

Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows: (i) $A(0, m, n) = m'$ for all $m, n \in N$ (ii) $A(k', 0, n) =\left\{ \begin{array}{ll} n & if \, \, k = 0 \\ 0 & if \, \,k = 1, \\ 1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$ (iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$. Compute $A(5, 3, 2)$. (H. Nestra)

Let $f(x)$ be a linear function and $k,l,m$ - pairwise different real numbers. It is known that $f(k)=l^3+m^3$, $f(l)=m^3+k^3$ and $f(m)=k^3+l^3$. Find the value of $k+l+m$.

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

Find all values ​​of $ r$ such that the inequality $$r (ab + bc + ca) + (3- r) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right) \ge 9$$ is true for $a,b,c$ arbitrary positive reals

Let $a$ and $b$ be natural numbers. Prove that the number of polynomials $P(x)$ with integer coefficients such that $|P(n)| \leq a^n$ for every natural number $n \geq b$ is finite.

Function $f(x)$. which is defined on the set of non-negative real numbers, acquires real values. It is known that $f(0)\le 0$ and the function $f(x)/x$ is increasing for $x>0$. Prove that for arbitrary $x\ge 0$ and $y\ge 0$, holds the inequality $f(x+y)\ge f(x)+ f(y)$ .

The fraction $\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}$ is (with suitable restrictions of the values of $a$, $b$, and $c$): $ \textbf{(A) }\text{irreducible}\qquad\textbf{(B) }\text{reducible to negative 1}\qquad$ $\textbf{(C) }\text{reducible to a polynomial of three terms} \qquad\textbf{(D) }\text{reducible to} \frac{a-b+c}{a+b-c} \qquad\textbf{(E) }\text{reducible to} \frac{a+b-c}{a-b+c} $