Found problems: 1269
1998 Kurschak Competition, 2
Prove that for every positive integer $n$, there exists a polynomial with integer coefficients whose values at points $1,2,\dots,n$ are pairwise different powers of $2$.
2001 Vietnam Team Selection Test, 1
Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by
\[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\]
for all $n \in \mathbb{N}^{*}$.
Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$.
(Here $[x]$ denotes the integral part of real number $x$).
2008 Germany Team Selection Test, 3
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that
\[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]
2014 Iran Team Selection Test, 3
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.
and for all $x\in \mathbb{R}$
$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .
prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$
$f(q(p(x))+h(x))=f(x)^{2}+1$
2011 China Girls Math Olympiad, 5
A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$, such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$, if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$, then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$.
2006 South africa National Olympiad, 6
Consider the function $f$ defined by
\[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\]
for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that
(a) $f(n+1)>f(n)$ for infinitely many $n$.
(b) $f(n+1)<f(n)$ for infinitely many $n$.
2003 Italy TST, 3
Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.
2014 Saudi Arabia IMO TST, 4
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(n+1)>\frac{f(n)+f(f(n))}{2}\] for all $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of strictly positive integers.
2013 Baltic Way, 5
Numbers $0$ and $2013$ are written at two opposite vertices of a cube. Some real numbers are to be written at the remaining $6$ vertices of the cube. On each edge of the cube the difference between the numbers at its endpoints is written. When is the sum of squares of the numbers written on the edges minimal?
2000 All-Russian Olympiad, 4
Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers. For $k=1,2,\cdots,n$ denote \[ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. \] Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$
2000 India National Olympiad, 2
Solve for integers $x,y,z$: \[ \{ \begin{array}{ccc} x + y &=& 1 - z \\ x^3 + y^3 &=& 1 - z^2 . \end{array} \]
2010 Contests, 2
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.
2012 Philippine MO, 2
Let $f$ be a polynomial function with integer coefficients and $p$ be a prime number. Suppose there are at least four distinct integers satisfying $f(x) = p$. Show that $f$ does not have integer zeros.
1990 China National Olympiad, 3
A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions:
i. for $x,y\ge 0$, $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$;
ii. there exists a constant $M$($M>0$), such that $|f(x)|\le M$ when $0\le x\le 1$.
Prove that $f(x)\le x^2$.
2011 Postal Coaching, 2
Let $x$ be a positive real number and let $k$ be a positive integer. Assume that $x^k+\frac{1}{x^k}$ and $x^{k+1}+\frac{1}{x^{k+1}}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is also a rational number.
2009 Germany Team Selection Test, 1
Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number).
(a) What is the smallest and biggest representable number?
(b) Construct 45 representable numbers.
1997 Irish Math Olympiad, 3
Find all polynomials $ p(x)$ satisfying the equation: $ (x\minus{}16)p(2x)\equal{}16(x\minus{}1)p(x)$ for all $ x$.
2008 Germany Team Selection Test, 3
Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that
\[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0.
\]
2014 Iran MO (3rd Round), 1
In each of (a) to (d) you have to find a strictly increasing surjective function from A to B or prove that there doesn't exist any.
(a) $A=\{x|x\in \mathbb{Q},x\leq \sqrt{2}\}$ and $B=\{x|x\in \mathbb{Q},x\leq \sqrt{3}\}$
(b) $A=\mathbb{Q}$ and $B=\mathbb{Q}\cup \{\pi \} $
In (c) and (d) we say $(x,y)>(z,t)$ where $x,y,z,t \in \mathbb{R}$ , whenever $x>z$ or $x=z$ and $y>t$.
(c) $A=\mathbb{R}$ and $B=\mathbb{R}^2$
(d) $X=\{2^{-x}| x\in \mathbb{N}\}$ , then $A=X \times (X\cup \{0\})$ and $B=(X \cup \{ 0 \}) \times X$
(e) If $A,B \subset \mathbb{R}$ , such that there exists a surjective non-decreasing function from $A$ to $B$ and a surjective non-decreasing function from $B$ to $A$ , does there exist a surjective strictly increasing function from $B$ to $A$?
Time allowed for this problem was 2 hours.
2012 Indonesia TST, 1
Given a positive integer $n$.
(a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$,
\[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\]
(b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.
2002 Indonesia MO, 5
Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table:
$\begin{array} {|c|c|c|} \cline{1-3}
10 & & \\ \cline{1-3}
& & 9 \\ \cline{1-3}
& 3 & \\ \cline{1-3}
11 & & 17 \\ \cline{1-3}
& 20 & \\ \cline{1-3}
\end{array}$
such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.
1987 Vietnam National Olympiad, 2
Let $ f : [0, \plus{}\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to\plus{}\infty}f(x)$.
1989 IMO Longlists, 96
Let $ f : \mathbb{N} \mapsto \mathbb{N}$ be such that
[b](i)[/b] $ f$ is strictly increasing;
[b](ii)[/b] $ f(mn) \equal{} f(m)f(n) \quad \forall m, n \in \mathbb{N};$ and
[b](iii)[/b] if $ m \neq n$ and $ m^n \equal{} n^m,$ then $ f(m) \equal{} n$ or $ f(n) \equal{} m.$
Determine $ f(30).$
1994 All-Russian Olympiad Regional Round, 10.2
The equation $ x^2 \plus{} ax \plus{} b \equal{} 0$ has two distinct real roots. Prove that the equation $ x^4 \plus{} ax^3 \plus{} (b \minus{} 2)x^2 \minus{} ax \plus{} 1 \equal{} 0$ has four distinct real roots.
2003 Indonesia MO, 3
Find all real numbers $x$ such that $\left\lfloor x^2 \right\rfloor + \left\lceil x^2 \right\rceil = 2003$.