Find all ordered 4-tuples of integers $(a,b,c,d)$ (not necessarily distinct) satisfying the following system of equations: \begin{align*}a^2-b^2-c^2-d^2&=c-b-2\\2ab&=a-d-32\\2ac&=28-a-d\\2ad&=b+c+31.\end{align*}

Alex calculated the value of function $f(n) = n^2 + n + 1$ for each integer from $1$ to $100$. Marina calculated the value of function $g(n) = n^2-n+1$ for the same numbers. Who of them has greater product of values and what is their ratio?

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f(x)(x+f(f(y))) = f(x^2)+xf(y)$$

$a,b,c$ are nonnegative integers that satisfy $a^2+b^2+c^2=3$. Find the minimum and maximum value the sum $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a}$$ may achieve and find all $a,b,c$ for which equality occurs.\\ \\ [i](Andrei Bâra)[/i]

Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?

Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge 1$ for all real numbers $x$, where $0 < x < 1$

Let $p$ and $q$ be prime numbers. The sequence $(x_n)$ is defined by $x_1 = 1$, $x_2 = p$ and $x_{n+1} = px_n - qx_{n-1}$ for all $n \geq 2$. Given that there is some $k$ such that $x_{3k} = -3$, find $p$ and $q$.

Evaluate $\sum^{15}_{k=0}\left(2^{560}(-1)^k \cos^{560}\left( \frac{k\pi}{16}\right)\right) \pmod{17}.$

The sequence $x_0,x_1,x_2,.....$ of real numbers is called with period $p$,with $p$ being a natural number, when for each $p\ge2$, $x_n=x_{n+p}$. Prove that,for each $p\ge2$ there exists a sequence such that $p$ is its least period and $x_{n+1}=x_n-\frac{1}{x_n}$ $(n=0,1,....)$

A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds: For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials. $a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$. $b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials. [i]Proposed by Alireza Shavali[/i]

Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought?

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

The positive real numbers $a, b, c$ with $abc = 1$ Show that: $\sqrt{a + \frac{1}{a}} + \sqrt{b + \frac{1}{b}} + \sqrt{c + \frac{1}{c}}\geq 2(\sqrt{a} + \sqrt{b} + \sqrt{c})$

A sequence of real numbers $a_1, a_2, \ldots $ satisfies the following conditions. $a_1 = 2$, $a_2 = 11$. for all positive integer $n$, $2a_{n+2} =3a_n + \sqrt{5 (a_n^2+a_{n+1}^2)}$ Prove that $a_n$ is a rational number for each of positive integer $n$.

If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$

Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$ i) $f(1)=2016$ ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$

Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by $$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$

Find, with proof, all integers $n$ such that there is a solution in nonnegative real numbers $(x,y,z)$ to the system of equations \[2x^2+3y^2+6z^2=n\text{ and }3x+4y+5z=23.\]

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

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 $\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+...+\frac{1}{b_{10}}.$ .

The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of $ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $

A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.

Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\] [i]Proposed by Azerbaijan[/i] [hide=Second Suggested Version]Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\][/hide]

Find the function $f(x)$ such that \[f(x)^2f\left(\frac{1-x}{x+1}\right) =64x \] for $x\not=0,x\not=1,x\not=-1$.

Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$ [i]Proposed by Nikola Velov[/i]