Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality $$ f(f(x)+f(y))=(x+y)f(x+y) $$ for all real $x$ and $y$. [i](B. Serankou)[/i]

Let $a$ and $b$ satisfy the conditions $\begin{cases} a^3 - 6a^2 + 15a = 9 \\ b^3 - 3b^2 + 6b = -1 \end{cases}$ . The value of $(a - b)^{2014}$ is: (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$  

The ship [i]Meridiano do Bacalhau[/i] does its fishing business during $64$ days. Each day the capitain chooses a direction which may be either north or south and the ship sails that direction in that day. On the first day of business the ship sails $1$ mile, on the second day sails $2$ miles; generally, on the $n$-th day it sails $n$ miles. After of the $64$-th day, the ship was $2014$ miles north from its initial position. What is the greatest number of days that the ship could have sailed south?

Given a finite set $X$, two positive integers $n,k$, and a map $f:X\to X$. Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$,$i=1,2,3,\ldots$. It is known that for any $x\in X$,$f^{(n)}(x)=x$. Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$. Prove that: (1)$\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$ (2)$\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.

[b]p1.[/b] Prove that for every point inside a regular polygon, the average of the distances to the sides equals the radius of the inscribed circle. The distance to a side means the shortest distance from the point to the line obtained by extending the side. [b]p2.[/b] Suppose we are given positive (not necessarily distinct) integers $a_1, a_2,..., a_{1997}$ . Show that it is possible to choose some numbers from this list such that their sum is a multiple of $1997$. [b]p3.[/b] You have Blue blocks, Green blocks and Red blocks. Blue blocks and green blocks are $2$ inches thick. Red blocks are $1$ inch thick. In how many ways can you stack the blocks into a vertical column that is exactly $12$ inches high? (For example, for height $3$ there are $5$ ways: RRR, RG, GR, RB, BR.) [b]p4.[/b] There are $1997$ nonzero real numbers written on the blackboard. An operation consists of choosing any two of these numbers, $a$ and $b$, erasing them, and writing $a+b/2$ and $b-a/2$ instead of them. Prove that if a sequence of such operations is performed, one can never end up with the initial collection of numbers. [b]p5.[/b] An $m\times n$ checkerboard (m and n are positive integers) is covered by nonoverlapping tiles of sizes $2\times 2$ and $1\times 4$. One $2\times 2$ tile is removed and replaced by a $1\times 4$ tile. Is it possible to rearrange the tiles so that they cover the checkerboard? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each $x$ with $|x|<1$, compute the sum of the series $$1+4x+9x^2+\ldots+n^2x^{n-1}+\ldots.$$

$f(x) = [x] + [2x] + [3x] + [4x] + [5x] + [6x]$. What values does $f$ take?

Consider the set of sequences $\{S_i\}$ that start with $S_0 = 12$, $S_1 = 21$, $S_2 = 28$, and for $n > 2$, $S_n$ is the sum of two (not necessarily distinct) $S_{k_n}$ and $S_{j_n}$ with $k_n, j_n < n$. Find the largest integer that cannot be found in any sequence $S_i$.

$32$ stones, with pairwise different weights, and lever scales without weights are given. How to determine by $35$ scaling, which stone is the heaviest and which is the second by weight?

Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

Show that the inequality \[\prod_{k=2}^n\left(1-\frac{1}{k^3}\right)>\frac12\] holds for every positive integer $n>1.$

Solve the following equation in $\mathbb{R}$: $$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$

Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$

How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$?

$x,y,z\in\mathbb{R^+}$. If $xyz=1$, then prove the following: $$\sum\frac{x^6+2}{x^3}\geq3(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})$$

Find all functions $f: \mathbb Q \to \mathbb Q$ such that $ f(x^{2}\plus{}y\plus{}f(xy)) \equal{} 3\plus{}(x\plus{}f(y)\minus{}2)f(x)$ for all $x,y \in \mathbb Q$.

It is known that for $0\le x \le 1$ the square trinomial $f (x)$ satisfies the condition $|f(x) | \le 1$. Show that $| f '(0) | \le 8.$

Determine all real-coefficient polynomials $P(x)$ such that \[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.

What is the minimum distance between $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \in Z$?

Solve the system equations \[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]

Find all nonnegative integers $a, b, c$ such that $$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$

Is there a positive integer $ n$ such that among 200th digits after decimal point in the decimal representations of $ \sqrt{n}$, $ \sqrt{n\plus{}1}$, $ \sqrt{n\plus{}2}$, $ \ldots,$ $ \sqrt{n\plus{}999}$ every digit occurs 100 times? [i]Proposed by A. Golovanov[/i]

There exists a unique pair of polynomials $(P(x),Q(x))$ such that \begin{align*} P(Q(x))&= P(x)(x^2-6x+7) \\ Q(P(x))&= Q(x)(x^2-3x-2) \end{align*} Compute $P(10)+Q(-10)$. [i]Proposed by Connor Gordon[/i]

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]