Consider two integers $n \ge m \ge 4$ and $A = \{a_1, a_2, ..., a_m\}$ a subset of the set $\{1, 2, ..., n\}$ such that: [i]for all $a, b \in A, a \ne b$, if $a + b \le n$, then $a + b \in A$.[/i] Prove that $\frac{a_1 + a_2 + ... + a_m}{m} \ge \frac{n + 1}{2}$ .

Prove that for every real number $x$ and positive integer $n$ $$|\cos x|+|\cos2x|+|\cos2^2x|+\ldots+|\cos2^nx|\ge\frac n{2\sqrt2}.$$

Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

If $a,b,c,x$ are positive reals, show that $$\frac{a^{x+2}+1}{a^xbc+1}+\frac{b^{x+2}+1}{b^xac+1}+\frac{c^{x+2}+1}{c^xab+1}\geq 3$$

Let $f$ be a polynomial with real coefficients of degree $n$. Suppose that $\displaystyle \frac{f(x)-f(y)}{x-y}$ is an integer for all $0 \leq x<y \leq n$. Prove that $a-b | f(a)-f(b)$ for all distinct integers $a,b$.

[b]p1.[/b] Find a real number $x$ for which $x\lfloor x \rfloor = 1234.$ Note: $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. [b]p2.[/b] Let $C_1$ be a circle of radius $1$, and $C_2$ be a circle that lies completely inside or on the boundary of $C_1$. Suppose$ P$ is a point that lies inside or on $C_2$. Suppose $O_1$, and $O_2$ are the centers of $C_1$, and $C_2$, respectively. What is the maximum possible area of $\vartriangle O_1O_2P$? Prove your answer. [b]p3.[/b] The numbers $1, 2, . . . , 99$ are written on a blackboard. We are allowed to erase any two distinct (but perhaps equal) numbers and replace them by their nonnegative difference. This operation is performed until a single number $k$ remains on the blackboard. What are all the possible values of $k$? Prove your answer. Note: As an example if we start from $1, 2, 3, 4$ on the board, we can proceed by erasing $1$ and $2$ and replacing them by $1$. At that point we are left with $1, 3, 4$. We may then erase $3$ and $4$ and replacethem by $1$. The last step would be to erase $1$, $1$ and end up with a single $0$ on the board. [b]p4.[/b] Let $a, b$ be two real numbers so that $a^3 - 6a^2 + 13a = 1$ and $b^3 - 6b^2 + 13b = 19$. Find $a + b$. Prove your answer. [b]p5.[/b] Let $m, n, k$ be three positive integers with $n \ge k$. Suppose $A =\prod_{1\le i\le j\le m} gcd(n + i, k + j) $ is the product of $gcd(n + i, k + j)$, where $i, j$ range over all integers satisfying $1\le i\le j\le m$. Prove that the following fraction is an integer $$\frac{A}{(k + 1) \dots(k + m)}{n \choose k}.$$ Note: $gcd(a, b)$ is the greatest common divisor of $a$ and $b$, and ${n \choose k}= \frac{n!}{k!(n - k)!}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $n$ such that we can divide the set $\{1,2,3,\ldots,n\}$ into three sets with the same sum of members.

For arbitrary real numbers \( x_1,x_2,\ldots,x_n \), prove that \[ \left(\max_{1\leq i \leq n}x_i \right)^2 + 4\sum_{i=1}^{n-1}\left(\max_{1\leq j \leq i}x_j\right)\left(x_{i+1}-x_i\right) \leq 4x_n^2. \]

Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression \[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]

Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.

Prove that the system of equations $ \begin{cases} \ a^2 - b = c^2 \\ \ b^2 - a = d^2 \\ \end{cases} $ has no integer solutions $a, b, c, d$.

The sequence $ (x_n)_{n \geq 1}$ is defined by: $ x_1\equal{}1$ $ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$ Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.

Find all polynomials with real coefficients $P$ such that \[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\] for every $x\in\mathbb{R}$.

Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$

Let $a,b,c&gt;0$ and $abc=1$. Prove that $$\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.$$ [i](Zhuge Liang)[/i]

Find all functions $f : R^2 \to R$ that for all real numbers $x, y, z$ satisfies to the equation $f(f(x,z), f(z, y))= f(x, y) + z$

Suppose sequence $\{a_i\} = a_1, a_2, a_3, ....$ satisfies $a_{n+1} = \frac{1}{a_n+1}$ for all positive integers $n$. Define $b_k$ for positive integers $k \ge 2$ to be the minimum real number such that the product $a_1 \cdot a_2 \cdot ...\cdot a_k$ does not exceed $b_k$ for any positive integer choice of $a_1$. Find $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+...+\frac{1}{b_{10}}.$ .

Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

An infinite, strictly increasing sequence $\{a_n\}$ of positive integers satisfies the condition $a_{a_n}\le a_n + a_{n + 3}$ for all $n\ge 1$. Prove that there are infinitely many triples $(k, l, m)$ of positive integers such that $k <l <m$ and $a_k + a_m = 2a_l$.

Numbers $a, b, c$ are such that $3a + 4b = 3c$ and $4a - 3b = 4c$. Show that $a^2 + b^2 = c^2$.

Initially on a blackboard, the equation $a_1x^2+b_1x+c=0$ is written where $a_1, b_1, c_1$ are integers and $(a_1+c_1)b_1 > 0$. At each move, if the equation $ax^2+bx+c=0$ is written on the board and there is a $x \in \mathbb{R}$ satisfying the equation, Alice turns this equation into $(b+c)x^2+(c+a)x+(a+b)=0$. Prove that Alice will stop after a finite number of moves.

Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have \[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\] Determine the value of $f(2023)/f(2022)$.

$a_1,a_2,a_3,...$ is a monotone increasing sequence of natural numbers. It is known that for any $k, a_{a_k} = 3k$. a) Find $a_{100}$. b) Find $a_{1983}$. (A Andjans, Riga) PS. (a) for Juniors, (b) for Seniors

The sequence of integers $a_1,a_2,a_3 ,.. $such that $a_1 = 1$, $a_2 = 2$ and for every natural $n \ge 1$ $$a_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\ a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}$$ Is there such a natural $m$ that $a_m= 2000$?

Find all integer $b$ such that $[x^2]-2012x+b=0$ has odd number of roots.