Found problems: 1269
2009 Italy TST, 1
Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that
\[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]
1989 IMO Longlists, 81
A real-valued function $ f$ on $ \mathbb{Q}$ satisfies the following conditions for arbitrary $ \alpha, \beta \in \mathbb{Q}:$
[b](i)[/b] $ f(0) \equal{} 0,$
[b](ii)[/b] $ f(\alpha) > 0 \text{ if } \alpha \neq 0,$
[b](iii)[/b] $ f(\alpha \cdot \beta) \equal{} f(\alpha)f(\beta),$
[b](iv)[/b] $ f(\alpha \plus{} \beta) \leq f(\alpha) \plus{} f(\beta),$
[b](v)[/b] $ f(m) \leq 1989$ $ \forall m \in \mathbb{Z}.$
Prove that \[ f(\alpha \plus{} \beta) \equal{} \max\{f(\alpha), f(\beta)\} \text{ if } f(\alpha) \neq f(\beta).\]
1985 IMO Longlists, 96
Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions:
(a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and
(b) $\lim_{x\to \infty} f(x) = 0$.
2003 IberoAmerican, 1
$(a)$There are two sequences of numbers, with $2003$ consecutive integers each, and a table of $2$ rows and $2003$ columns
$\begin{array}{|c|c|c|c|c|c|} \hline\ \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \end{array}$
Is it always possible to arrange the numbers in the first sequence in the first row and the second sequence in the second row, such that the sequence obtained of the $2003$ column-wise sums form a new sequence of $2003$ consecutive integers?
$(b)$ What if $2003$ is replaced with $2004$?
1984 Iran MO (2nd round), 2
Consider the function
\[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\]
Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.
1998 Vietnam National Olympiad, 3
Find all positive integer $n$ such that there exists a $P\in\mathbb{R}[x]$ satisfying $P(x^{1998}-x^{-1998})=x^{n}-x^{-n}\forall x\in\mathbb{R}-\{0\}$.
1990 China Team Selection Test, 2
Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$
2018 Latvia Baltic Way TST, P2
Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations:
$$\begin{cases}
y(x+y)^2=2\\
8y(x^3-y^3) = 13.
\end{cases}$$
1988 Vietnam National Olympiad, 2
Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0$, where $ q\neq 1$. Show that $ p \le \minus{} 3\sqrt 3$ and $ q > 1$.
2001 South africa National Olympiad, 2
Find all triples $(x,y,z)$ of real numbers that satisfy \[ \begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned} \]
2004 Greece National Olympiad, 2
If $m\geq 2$ show that there does not exist positive integers $x_1, x_2, ..., x_m,$ such that \[x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.\]
2006 Italy TST, 3
Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.
2005 Postal Coaching, 25
Find all pairs of cubic equations $x^3 +ax^2 +bx +c =0$ and $x^3 +bx^2 + ax +c = 0$ where $a,b,c$ are integers, such that each equation has three integer roots and both the equations have exactly one common root.
2002 Czech-Polish-Slovak Match, 3
Let $S = \{1, 2, \cdots , n\}, n \in N$. Find the number of functions $f : S \to S$ with the property that $x + f(f(f(f(x)))) = n + 1$ for all $x \in S$?
2006 Bulgaria Team Selection Test, 2
a) Let $\{a_n\}_{n=1}^\infty$ is sequence of integers bigger than 1. Proove that if $x>0$ is irrational, then $\ds x_n>\frac{1}{a_{n+1}}$ for infinitely many $n$, where $x_n$ is fractional part of $a_na_{n-1}\dots a_1x$.
b)Find all sequences $\{a_n\}_{n=1}^\infty$ of positive integers, for which exist infinitely many $x\in(0,1)$ such that $\ds x_n>\frac{1}{a_{n+1}}$ for all $n$.
[i]Nikolai Nikolov, Emil Kolev[/i]
2013 South East Mathematical Olympiad, 8
$n\geq 3$ is a integer. $\alpha,\beta,\gamma \in (0,1)$. For every $a_k,b_k,c_k\geq0(k=1,2,\dotsc,n)$ with $\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma$, we always have $\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda$.
Find the minimum of $\lambda$
2014 Romania Team Selection Test, 3
Let $n \in \mathbb{N}$ and $S_{n}$ the set of all permutations of $\{1,2,3,...,n\}$. For every permutation $\sigma \in S_{n}$ denote $I(\sigma) := \{ i: \sigma (i) \le i \}$.
Compute the sum $\sum_ {\sigma \in S_{n}} \frac{1}{|I(\sigma )|} \sum_ {i \in I(\sigma)} (i+ \sigma(i))$.
2007 Germany Team Selection Test, 1
For a multiple of $ kb$ of $ b$ let $ a \% kb$ be the greatest number such that $ a \% kb \equal{} a \bmod b$ which is smaller than $ kb$ and not greater than $ a$ itself. Let $ n \in \mathbb{Z}^ \plus{} .$ Determine all integer pairs $ (a,b)$ with:
\[ a\%b \plus{} a\%2b \plus{} a\%3b \plus{} \ldots \plus{} a\%nb \equal{} a \plus{} b
\]
2010 Germany Team Selection Test, 3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$
2004 India IMO Training Camp, 3
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
2002 Vietnam Team Selection Test, 2
Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.
1990 China Team Selection Test, 3
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And
(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.
(ii) $a \circ b \neq b \circ a$ when $a \neq b$.
Prove that:
a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.
b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.
2012 Albania National Olympiad, 4
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f(x^3)+f(y^3)=(x+y)f(x^2)+f(y^2)- f(xy)\]
for all $x\in\mathbb{R}$.
1990 IMO Longlists, 56
For positive integers $n, p$ with $n \geq p$, define real number $K_{n, p}$ as follows:
$K_{n, 0} = \frac{1}{n+1}$ and $K_{n, p} = K_{n-1, p-1} -K_{n, p-1}$ for $1 \leq p \leq n.$
(i) Define $S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots$ . Find $\lim_{n \to \infty} S_n.$
(ii) Find $T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots$.
2005 China Second Round Olympiad, 3
For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$.