Found problems: 1311
2007 Bulgaria Team Selection Test, 2
Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$
2006 Switzerland Team Selection Test, 3
Find all the functions $f : \mathbb{R} \to \mathbb{R}$ satisfying for all $x,y \in \mathbb{R}$ $f(f(x)-y^2) = f(x)^2 - 2f(x)y^2 + f(f(y))$.
2011 Cono Sur Olympiad, 4
A number $\overline{abcd}$ is called [i]balanced[/i] if $a+b=c+d$. Find all balanced numbers with 4 digits that are the sum of two palindrome numbers.
2012 China Team Selection Test, 1
Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.
2010 Slovenia National Olympiad, 1
Let $a,b$ be real numbers such that $|a| \neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$ Find the value of the expression $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$
2005 ISI B.Stat Entrance Exam, 3
Let $f$ be a function defined on $\{(i,j): i,j \in \mathbb{N}\}$ satisfying
(i) $f(i,i+1)=\frac{1}{3}$ for all $i$
(ii) $f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j)$ for all $k$ such that $i <k<j$.
Find the value of $f(1,100)$.
2006 Poland - Second Round, 1
Positive integers $a,b,c,x,y,z$ satisfy:
$a^2+b^2=c^2$, $x^2+y^2=z^2$
and
$|x-a| \leq 1$ , $|y-b| \leq 1$.
Prove that sets $\{a,b\}$ and $\{x,y\}$ are equal.
2011 All-Russian Olympiad Regional Round, 10.4
Non-zero real numbers $a$, $b$ and $c$ are such that any two of the three equations $ax^{11}+bx^4+c=0$, $bx^{11}+cx^4+a=0$, $cx^{11}+ax^4+b=0$ have a common root. Prove that all three equations have a common root. (Author: I. Bogdanov)
2001 Hungary-Israel Binational, 4
Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the fifth powers of the roots of $P$.
2024 Austrian MO National Competition, 1
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(\alpha f(x)+f(y))=\beta x+f(y)\]
holds for all real $x$ and $y$.
[i](Walther Janous)[/i]
2006 IMS, 5
Suppose that $a_{1},a_{2},\dots,a_{k}\in\mathbb C$ that for each $1\leq i\leq k$ we know that $|a_{k}|=1$. Suppose that \[\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.\] Prove that $c=k$ and $a_{i}=1$ for each $i$.
2011 Iran MO (3rd Round), 1
We define the recursive polynomial $T_n(x)$ as follows:
$T_0(x)=1$
$T_1(x)=x$
$T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$.
[b]a)[/b] find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$.
[b]b)[/b] find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$.
[i]Proposed by Morteza Saghafian[/i]
2007 ISI B.Math Entrance Exam, 5
Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$.
$(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$.
$(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.
1985 Iran MO (2nd round), 3
Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that
\[f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.\]
Prove that
\[f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.\]
[i]Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have $g(x)\leq{g(y)}$.[/i]
2007 Moldova National Olympiad, 12.3
For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]
2010 Benelux, 2
Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]
2011 ISI B.Math Entrance Exam, 4
Let $t_1 < t_2 < t_3 < \cdots < t_{99}$ be real numbers. Consider a function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|$ . Show that $f(x)$ will attain minimum value at $x=t_{50}$.
2014 Moldova Team Selection Test, 1
Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$.
1996 All-Russian Olympiad, 7
Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$?
[i]E. Malinnikova[/i]
2003 Romania Team Selection Test, 7
Find all integers $a,b,m,n$, with $m>n>1$, for which the polynomial $f(X)=X^n+aX+b$ divides the polynomial $g(X)=X^m+aX+b$.
[i]Laurentiu Panaitopol[/i]
1995 All-Russian Olympiad, 1
A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$.
[i]S. Tokarev[/i]
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
2008 Romania National Olympiad, 2
Let $ a,b,c$ be 3 complex numbers such that \[ a|bc| \plus{} b|ca| \plus{} c|ab| \equal{} 0.\] Prove that \[ |(a\minus{}b)(b\minus{}c)(c\minus{}a)| \geq 3\sqrt 3 |abc|.\]
2007 Indonesia MO, 6
Find all triples $ (x,y,z)$ of real numbers which satisfy the simultaneous equations
\[ x \equal{} y^3 \plus{} y \minus{} 8\]
\[y \equal{} z^3 \plus{} z \minus{} 8\]
\[ z \equal{} x^3 \plus{} x \minus{} 8.\]
1986 Iran MO (2nd round), 2
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\]
Is the function $g$ continuous in the point $x=0 \ ?$
[b](d)[/b] Sketch the diagram of $g.$