Found problems: 1311
2013 India IMO Training Camp, 2
Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.
2004 Vietnam National Olympiad, 1
Solve the system of equations $ \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}$.
1985 IMO Longlists, 40
Each of the numbers $x_1, x_2, \dots, x_n$ equals $1$ or $-1$ and
\[\sum_{i=1}^n x_i x_{i+1} x_{i+2} x_{i+3} =0.\]
where $x_{n+i}=x_i $ for all $i$. Prove that $4\mid n$.
1977 Canada National Olympiad, 4
Let
\[p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0\]
and
\[q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0\]
be two polynomials with integer coefficients. Suppose that all the coefficients of the product $p(x) \cdot q(x)$ are even but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.
2007 Tournament Of Towns, 2
The polynomial $x^3 + px^2 + qx + r$ has three roots in the interval $(0,2)$. Prove that $-2 <p + q + r < 0$.
2012 IberoAmerican, 1
Let $a,b,c,d$ be integers such that the number $a-b+c-d$ is odd and it divides the number $a^2-b^2+c^2-d^2$. Show that, for every positive integer $n$, $a-b+c-d$ divides $a^n-b^n+c^n-d^n$.
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 Czech-Polish-Slovak Match, 1
Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations
\[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]
2007 ITAMO, 2
We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q.
a) if P,Q are similar, then $P(2007)-Q(2007)$ is even
b) does there exist an integer $k > 2$ such that $k \mid P(2007)-Q(2007)$ for all similar polynomials P,Q?
1999 Turkey MO (2nd round), 4
Find all sequences ${{a}_{1}},{{a}_{2}},...,{{a}_{2000}}$ of real numbers such that $\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}$ and such that $\frac{1}{2}<{{a}_{n}}<1$ and ${{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})$ for all $n\ge 1$.
2013 India IMO Training Camp, 1
Find all functions $f$ from the set of real numbers to itself satisfying
\[ f(x(1+y)) = f(x)(1 + f(y)) \]
for all real numbers $x, y$.
2006 All-Russian Olympiad, 7
Assume that the polynomial $\left(x+1\right)^n-1$ is divisible by some polynomial $P\left(x\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0$, whose degree $k$ is even and whose coefficients $c_{k-1}$, $c_{k-2}$, ..., $c_1$, $c_0$ all are odd integers. Show that $k+1\mid n$.
1990 IMO Longlists, 48
Prove that $\sqrt 2 +\sqrt 3 +\sqrt{1990}$ is irrational.
2013 JBMO TST - Macedonia, 1
Let $ x $ be a real number such that $ x^3 $ and $ x^2+x $ are rational numbers. Prove that $ x $ is rational.
1997 Korea - Final Round, 3
Find all pairs of functions $ f, g: \mathbb R \to \mathbb R$ such that
[list]
(i) if $ x < y$, then $ f(x) < f(y)$;
(ii) $ f(xy) \equal{} g(y)f(x) \plus{} f(y)$ for all $ x, y \in \mathbb R$.
[/list]
2009 China Team Selection Test, 2
Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.
1986 Balkan MO, 3
Let $a,b,c$ be real numbers such that $ab\not= 0$ and $c>0$. Let $(a_{n})_{n\geq 1}$ be the sequence of real numbers defined by: $a_{1}=a, a_{2}=b$ and
\[a_{n+1}=\frac{a_{n}^{2}+c}{a_{n-1}}\]
for all $n\geq 2$.
Show that all the terms of the sequence are integer numbers if and only if the numbers $a,b$ and $\frac{a^{2}+b^{2}+c}{ab}$ are integers.
1985 IMO Longlists, 4
Let $x, y$, and $z$ be real numbers satisfying $x + y + z = xyz.$ Prove that
\[x(1 - y^2)(1 - z^2) + y(1 -z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) = 4xyz.\]
2017 Pakistan TST, Problem 3
Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$
$f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$
2024 Austrian MO National Competition, 5
Let $n$ be a positive integer and let $z_1,z_2,\dots,z_n$ be positive integers such that for $j=1,2,\dots,n$ the inequalites $z_j \le j$ hold and $z_1+z_2+\dots+z_n$ is even.
Prove that the number $0$ occurs among the values
\[z_1 \pm z_2 \pm \dots \pm z_n,\]
where $+$ or $-$ can be chosen independently for each operation.
[i](Walther Janous)[/i]
2008 Hong Kong TST, 1
Let $ f: Z \to Z$ be such that $ f(1) \equal{} 1, f(2) \equal{} 20, f(\minus{}4) \equal{} \minus{}4$ and $ f(x\plus{}y) \equal{} f(x) \plus{}f(y)\plus{}axy(x\plus{}y)\plus{}bxy\plus{}c(x\plus{}y)\plus{}4 \forall x,y \in Z$, where $ a,b,c$ are constants.
(a) Find a formula for $ f(x)$, where $ x$ is any integer.
(b) If $ f(x) \geq mx^2\plus{}(5m\plus{}1)x\plus{}4m$ for all non-negative integers $ x$, find the greatest possible value of $ m$.
2015 Romania Masters in Mathematics, 3
A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[
a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}.
\] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.
2007 Iran Team Selection Test, 1
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]
2005 District Olympiad, 4
Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$.
a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$;
b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.
2010 Contests, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]