Found problems: 1269
2008 China Second Round Olympiad, 2
Let $f(x)$ be a periodic function with periods $T$ and $1$($0<T<1$).Prove that:
(1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$;
(2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence {$a_n$},with $1>a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$.
2006 Vietnam National Olympiad, 5
Find all polynomyals $P(x)$ with real coefficients which satisfy the following equality for all real numbers $x$: \[ P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 . \]
2014 India IMO Training Camp, 2
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
2000 China Team Selection Test, 2
[b]a.)[/b] Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that
\[x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots \]
Prove that
\[x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}\]
where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$
[b]b.)[/b] Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
2002 China Team Selection Test, 1
Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \]
Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.
2005 Regional Competition For Advanced Students, 4
Prove: if an infinte arithmetic sequence ($ a_n\equal{}a_0\plus{}nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n\equal{}b_0q^n$) of real numbers.
2001 Baltic Way, 14
There are $2n$ cards. On each card some real number $x$, $(1\le x\le 2n)$, is written (there can be different numbers on different cards). Prove that the cards can be divided into two heaps with sums $s_1$ and $s_2$ so that $\frac{n}{n+1}\le\frac{s_1}{s_2}\le 1$.
1990 Polish MO Finals, 1
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy
\[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]
1979 IMO Longlists, 56
Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.
2010 India IMO Training Camp, 9
Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$.
Show that there exists $j<k$ and $l<m$ such that
\[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]
2006 Estonia Math Open Senior Contests, 10
Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i \minus{} x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.
2014 IberoAmerican, 2
Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$:
$xP(x-c) = (x - 2014)P(x)$
1989 IMO Longlists, 12
Let $ P(x)$ be a polynomial such that the following inequalities are satisfied:
\[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\]
and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\]
Prove that for every positive natural number $ n,$ $ P(n)$ is positive.
2010 Contests, 2
A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:
[list]$f(7) = 77$
$f(b) = 85$, where $b$ is Beth's age,
$f(c) = 0$, where $c$ is Charles' age.[/list]
How old is each child?
2004 China Team Selection Test, 3
Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$, such that $ \displaystyle n \equal{} \sum_{i\equal{}1}^{m\minus{}1} a_i(m\minus{}a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$ may not distinct and $ 1 \leq a_i \leq m\minus{}1$.
1991 China Team Selection Test, 1
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have
\[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]
1972 IMO Longlists, 19
Let $S$ be a subset of the real numbers with the following
properties:
$(i)$ If $x \in S$ and $y \in S$, then $x - y \in S$;
$(ii)$ If $x \in S$ and $y \in S$, then $xy \in S$;
$(iii)$ $S$ contains an exceptional number $x'$ such that there is no number $y$ in $S$ satisfying $x'y + x' + y = 0$;
$(iv)$ If $x \in S$ and $x \neq x'$ , there is a number $y$ in $S$ such that $xy+x+y = 0$.
Show that
$(a)$ $S$ has more than one number in it;
$(b)$ $x' \neq -1$ leads to a contradiction;
$(c)$ $x \in S$ and $x \neq 0$ implies $1/x \in S$.
1999 Vietnam Team Selection Test, 2
Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that $f(x) = q \cdot g(x)$ for all $x \in R$.
[b]I.[/b] Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar.
[b]II.[/b] How many polynomials of degree 1999 are there which have above mentioned property.
1986 Canada National Olympiad, 5
Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$. Suppose $u_1 = 39$ and $u_2 = 45$. Prove that 1986 divides infinitely many terms of the sequence.
1984 Iran MO (2nd round), 1
Let $f$ and $g$ be two functions such that
\[f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor x \rfloor |}.\]
Find the domains of $f$ and $g$ and then prove that
\[\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).\]
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 Iran MO (2nd round), 2
Let $n$ be a positive integer. Prove that the polynomial
\[P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1 \]
Does not have any rational root.
1997 Taiwan National Olympiad, 1
Let $a$ be rational and $b,c,d$ are real numbers, and let $f: \mathbb{R}\to [-1.1]$ be a function satisfying $f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-[b]]+d$ for all $x$. Show that $f$ is periodic.
2000 Moldova National Olympiad, Problem 5
Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ that satisfy $f(x+y)-f(x-y)=2y(3x^2+y^2)$ for all $x,y{\in}R$
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Azerbaijan Land of the Fire :lol:
2001 Greece National Olympiad, 3
A function $f : \Bbb{N}_0 \to \Bbb{R}$ satisfies $f(1) = 3$ and \[f(m + n) + f(m - n) - m + n - 1 =\frac{f(2m) + f(2n)}{2},\]
for any non-negative integers $m$ and $n$ with $m \geq n.$ Find all such functions $f$.