The function $f$, with domain on the set of non-negative integers, is defined by the following : $\bullet$ $f (0) = 2$ $\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value. Determine $f (n)$.

Given $a_0, a_1, ... , a_n$. It is known that $$a_0=a_n=0, a_{k-1}-2a_k+a_{k+1}\ge 0$$ for all $k = 1, 2, ... , k-1$.Prove that all the numbers are nonnegative.

Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation $$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$ for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$. [i]Proposed by Maximiliano Sánchez[/i]

The function $F (x)$ is defined on $R$ and has a second derivative for each value of the variable. Prove that there is a point $x_0$ such that the product $ F(x_0) F''(x_0)$ is non-negative. PS. In my [url=http://www.1543.su/olympiads/soros/20002001/1/1soros00.htm]source[/url], it is not clear if it means $ F(x_0) F''(x_0)$ or $ F(x_0) F'(x_0)$.

Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions: (1) $a_0+a_n=0$; (2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$; (3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.

Prove that there are no integers $a,b,c,d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$, and equals $2$ at $x=62$.

Without using a calculator, find out what is greater: $\sin 28^o$ or $tg21^o$?

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.

Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that \[ \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p}, \]where $ r \equiv p\minus{}d \pmod 3$, $ r\in\{\minus{}1,0,1\}$.

Distinct real numbers \( a, b, c \) satisfy the following condition: \[ \frac{a - b}{a^3b^3} + \frac{b - c}{b^3c^3} + \frac{c - a}{c^3a^3} = 0. \] What are the possible values of the expression \[ \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2}? \] [i]Proposed by Vadym Solomka[/i]

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2,...,x_n \in (-1, 1)$ such that $|x_1| + |x_2| +...+ |x_n| = m + |x_1 + x_2 + ... + x_n|$.

Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number.

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

There are $8$ different quadratic trinomials written on the board, among them there are no two that add up to a zero polynomial. It turns out that if we choose any two trinomials $g_1(x), g_2(X)$ from the board, then the remaining $6$ trinomials can be denoted as $g_3(x),g_4(x),\dots,g_8(x)$ so that all four polynomials $g_1(x)+g_2(x),g_3(x)+g_4(x),g_5(x)+g_6(x)$ and $g_7(x)+g_8(x)$ have a common root. Do all trinomials on the board necessarily have a common root? [i]Proposed by S. Berlov[/i]

(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$. (b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple of $\pi$. Using (a) or otherwise show that if $G_1=G_2=0$, then $G_n=0$ for all positive integers $n$.

For any natural number $ n, $ denote $ x_n $ as being the number of natural numbers of $ n $ digits that are divisible by $ 4 $ and formed only with the digits $ 0,1,2 $ or $ 6. $ [b]a)[/b] Calculate $ x_1,x_2,x_3,x_4. $ [b]b)[/b] Find the natural number $ m $ such that $$ 1+\left\lfloor \frac{x_2}{x_1}\right\rfloor +\left\lfloor \frac{x_3}{x_2}\right\rfloor +\left\lfloor \frac{x_4}{x_3}\right\rfloor +\cdots +\left\lfloor \frac{x_{m+1}}{x_m}\right\rfloor =2016 , $$ where $ \lfloor\rfloor $ is the usual integer part.

For a given positive integer $n,$ find $$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.

find all $f : \mathbb{C} \to \mathbb{C}$ st: $$f(f(x)+yf(y))=x+|y|^2$$ for all $x,y \in \mathbb{C}$

Let $p$ be a prime number, and define a sequence by: $x_i=i$ for $i=,0,1,2...,p-1$ and $x_n=x_{n-1}+x_{n-p}$ for $n \geq p$ Find the remainder when $x_{p^3}$ is divided by $p$.

[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$ [b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests? [b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side. [b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of? [b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes? [b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes? [b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$? [img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img] [b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $. [b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$? [b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes? [b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for some positive integer \(N\), \(N\) points can be chosen on a circle such that there are at least \(1000N^2\) unordered quadruples \((A,B,C,D)\) of distinct selected points for which \(\displaystyle \frac{AC}{BC} = \frac{AD}{BD}\).

Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.

Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $