Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real $x, y$ holds equality $$f(xf(y)) + f(xy) = 2f(x)y$$ [i]Proposed by Arseniy Nikolaev[/i]

[u]Set 4[/u] [b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$. [b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$. [b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$. [u]Set 5[/u] [b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$. [b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$. [b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 6[/u] [b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$. [b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$? [b]G18.[/b] Deja vu? Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$.

Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffery replaces them with \[\frac{x + y}{2} \quad \text{and} \quad 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}.\] After $2017$ minutes the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer.

If $a, b, c$ and $d$ are complex non-zero numbers such that $$2|a-b|\leq |b|, 2|b-c|\leq |c|, 2|c-d| \leq |d| , 2|d-a|\leq |a|.$$ Prove that $$\frac{7}{2} <\left| \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d} \right| .$$

Solve the inequality $$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$

Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and \[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\] Show that for all $k$, \[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\] where $[x]$ denotes the greatest integer not exceeding $x.$

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \] [i]Romania[/i]

$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds: $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$

Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.

For a given real number $a$, define the sequence $(a_n)$ by $a_1 = a$ and $$a_{n+1} =\begin{cases} \dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\ 0 \,\,\, if \,\,\, a_n = 0 \end{cases}$$ Prove that the sequence $(a_n)$ contains infinitely many nonpositive terms.

Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials $f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$ have integer coefficients. Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$. Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j)\right|=1$

Solve in the real numbers $x, y, z$ a system of the equations: \[ \begin{cases} x^2 - (y+z+yz)x + (y+z)yz = 0 \\ y^2 - (z + x + zx)y + (z+x)zx = 0 \\ z^2 - (x+y+xy)z + (x+y)xy = 0. \\ \end{cases} \]

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

[b]a)[/b] Given two positive reals $ x,y, $ prove that $ \min\left( x,1/x+y,1/y \right)\le\sqrt 2. $ and determine when equality holds. [b]b)[/b] Find all triplets of real numbers $ (a,b,c) $ having the property that for every triplet of real numbers $ (x,y,z) , $ the following equality holds: $$ |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z| $$

Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$

Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le 0$.

Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that \[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \] for all real number $x$.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$: $ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.

We call a number [i]pal[/i] if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example, $122$ and $34$ are pal but $304$ and $12$ are not pal. Prove that there exists a pal number with $n$ digits, $n > 1$.

Prove that $$\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}}=1$$ This means $1/(2+ (1/(3+ (1/(4+(...+1/1991)))))) +1/(1 + (1/(1 + (1/(3 + (1/(4 + (...+ 1/1991...)))))))) = 1.)$ (G. Galperin, Moscow-Tel Aviv)

Let $ p\ge 2 $ be a fixed natural number, and let the sequence of functions $ \left( f_n\right)_{n\ge 2}:[0,1]\longrightarrow\mathbb{R} $ defined as $ f_n (x)=f_{n-1}\left( f_1 (x)\right) , $ where $ f_1 (x)=\sqrt[p]{1-x^p} . $ Find $ a\in (0,1) $ such that: [b]a)[/b] exists $ b\ge a $ so that $ f_1:[a,b]\longrightarrow [a,b] $ is bijective. [b]b)[/b] $ \forall x\in [0,1]\quad\exists y\in [0,1]\quad m\in\mathbb{N}\implies \left| f_m(x)-f_m(y)\right| >a|x-y| $

Find all $(a, b, c, d)\in \mathbb{R}$ satisfying \[\begin{aligned} \begin{cases} a+b+c+d=0,\\ a^2+b^2+c^2+d^2=12,\\ abcd=-3.\\ \end{cases} \end{aligned}\]

Let $n$ be a positive integer. Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$. Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\ b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$ Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.