Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.

Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$. [b](1)[/b] If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$. [b](2)[/b] If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$.

In a game, a participant chooses a nine-digit positive integer $\overline{ABCDEFGHI}$ with distinct non-zero digits. The score of the participant is $A^{B^{C^{D^{E^{F^{G^{H^{I}}}}}}}}$. Which nine-digit number should be chosen in order to maximise the score?

Let $n$ be a natural number, and suppose that the equation $$x_1x_2+x_2x_3+x_3x_4+x_4x_5+\dots +x_{n-1}x_n+x_nx_1=0$$ has a solution with all the $x_i$s equal to $\pm 1$. Prove that $n$ is divisible by $4$.

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

Let $u, v$ and$ w$ be any positive numbers smaller than $1$. Prove that among the numbers $u(1 -v)$, $v(1 -w)$, $w(1 - u)$ there is always at least one value not greater than $\frac14$ .

Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: \[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]

Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that \[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\] for all real numbers $a,b,c,d$. [i]Proposed by usjl[/i]

Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.

Given six positive numbers $a,b,c,d,e,f$ such that $a < b < c < d < e < f.$ Let $a+c+e=S$ and $b+d+f=T.$ Prove that \[2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.\] [i]Proposed by Sung Yun Kim, South Korea[/i]

Let $n$ be a positive integer. Find the number of polynomials $P(x)$ with coefficients in $\{0, 1, 2, 3\}$ for which $P(2) = n$.

Let $x_{1,1}$, $x_{2,1}$, ..., $x_{n,1}$, $n \ge 2$, be a sequence of integers and assume that not all $x_{i,1}$ are equal. For $k \ge 2$, if sequence $\{x_{i,k}\}^n_{i=1}$ is defined, we define sequence $\{x_{i,k+1}\}^n_{i=1}$ as \[x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k}),\] for $i=1, 2, ..., n$, (where $x_{n+1,k}=x_{1,k}$). Show that if $n$ is odd then there exist indices $j$ and $k$ such that $x_{j,k}$ is not an integer.

[b]p1.[/b] The following figure represents a rectangular piece of paper $ABCD$ whose dimensions are $4$ inches by $3$ inches. When the paper is folded along the line segment $EF$, the corners $A$ and $C$ coincide. (a) Find the length of segment $EF$. (b) Extend $AD$ and $EF$ so they meet at $G$. Find the area of the triangle $\vartriangle AEG$. [img]https://cdn.artofproblemsolving.com/attachments/d/4/e8844fd37b3b8163f62fcda1300c8d63221f51.png[/img] [b]p2.[/b] (a) Let $p$ be a prime number. If $a, b, c$, and $d$ are distinct integers such that the equation $(x -a)(x - b)(x - c)(x - d) - p^2 = 0$ has an integer solution $r$, show that $(r - a) + (r - b) + (r - c) + (r - d) = 0$. (b) Show that $r$ must be a double root of the equation $(x - a)(x - b)(x - c)(x - d) - p^2 = 0$. [b]p3.[/b] If $\sin x + \sin y + \sin z = 0$ and $\cos x + \cos y + \cos z = 0$, prove the following statements. (a) $\cos (x - y) = -\frac12$ (b) $\cos (\theta - x) + \cos(\theta - y) + \cos (\theta - z) = 0$, for any angle $\theta$. (c) $\sin^2 x + \sin^2 y + \sin^2 z =\frac32$ [b]p4.[/b] Let $|A|$ denote the number of elements in the set $A$. (a) Construct an infinite collection $\{A_i\}$ of infinite subsets of the set of natural numbers such that $|A_i \cap A_j | = 0$ for $i \ne j$. (b) Construct an infinite collection $\{B_i\}$ of infinite subsets of the set of natural numbers such that $|B_i \cap B_j |$ gives a distinct integer for every pair of $i$ and $j$, $i \ne j$. [b]p5.[/b] Consider the equation $x^4 + y^4 = z^5$. (a) Show that the equation has a solution where $x, y$, and $z$ are positive integers. (b) Show that the equation has infinitely many solutions where $x, y$, and $z$ are positive integers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold [list] [*] $f(x+y)\leq f(x)+f(y)$, [*]$f(xy)=f(x)f(y)$, [*]$f(2)=1/2$. [/list]

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

Solve in $R$ the system: $$\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\ \dfrac{xyz}{y + z - 1}= 1998000 \\ \\ \dfrac{xyz}{z+x}= 1998000 \end{cases}$$

Let $n \ge 2$ be a fixed integer. [list=a] [*]Determine the largest positive integer $m$ (in terms of $n$) such that there exist complex numbers $r_1$, $\dots$, $r_n$, not all zero, for which \[ \prod_{k=1}^n (r_k+1) = \prod_{k=1}^n (r_k^2+1) = \dots = \prod_{k=1}^n (r_k^m+1) = 1. \] [*]For this value of $m$, find all possible values of \[ \prod\limits_{k=1}^n (r_k^{m+1}+1). \] [/list] [i]Kaixin Wang[/i]

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number. Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive. Prove, that if exist at least one $leading$ element, then sum of all $leading$'s elements is positive. Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds \[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \] where $ A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.

Let $a_1, . . . , a_n$ be positive integers satisfying the inequality $\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$. Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.

Let $a, b, c$ be non-negative real numbers. Prove that $$a\sqrt{3a^2+6b^2}+b\sqrt{3b^2+6c^2}+c\sqrt{3c^2+6a^2}=>(a+b+c)^2$$

Solve the system of equations: $$\begin{cases} xy+zu=14 \\ xz+yu=11 \\ xu+yz=10 \\ x+y+z+u=10 \end{cases}$$

Let $ a,b,c$ be positive integers. If $ 30|a\plus{}b\plus{}c$, prove that $ 30|a^5\plus{}b^5\plus{}c^5$.