Found problems: 1269
2013 ELMO Shortlist, 7
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
1984 Vietnam National Olympiad, 2
Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy
\[xP(x - a) = (x - b)P(x).\]
1977 Bundeswettbewerb Mathematik, 4
Find all functions $f : \mathbb R \to \mathbb R$ such that
\[f(x)+f\left(1-\frac{1}{x}\right)=x,\]
holds for all real $x$.
2007 South East Mathematical Olympiad, 1
Determine the number of real number $a$, such that for every $a$, equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:
(a) $x_0$ is an even integer;
(b) $|x_0|<1000$.
1985 Iran MO (2nd round), 5
Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying
\[\forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \quad \text{and} \quad \lim_{x \to 0} g(x)=1.\]
Find the derivative of $f$ in an arbitrary point $x.$
2006 Vietnam National Olympiad, 1
Solve the following system of equations in real numbers:
\[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]
2005 South East Mathematical Olympiad, 7
(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$, where $x \in R$ and $a$ is parameter.
(2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$, and satisfy the condition
\[ \sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507. \]
Find the maximum value of $n$.
2007 Germany Team Selection Test, 2
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]
2005 MOP Homework, 7
Let $x_{1,1}$, $x_{2,1}$, ..., $x_{n,1}$, $n \ge 2$, be a sequence of integers and assume that not all $x_{i,1}$ are equal. For $k \ge 2$, if sequence $\{x_{i,k}\}^n_{i=1}$ is defined, we define sequence $\{x_{i,k+1}\}^n_{i=1}$ as \[x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k}),\] for $i=1, 2, ..., n$, (where $x_{n+1,k}=x_{1,k}$). Show that if $n$ is odd then there exist indices $j$ and $k$ such that $x_{j,k}$ is not an integer.
2008 Silk Road, 4
Find all polynomials $ P\in\mathbb{R}[x]$ such that for all $ r\in\mathbb{Q}$,there exist
$ d\in\mathbb{Q}$ such that $ P(d)\equal{}r$
2004 Czech and Slovak Olympiad III A, 1
Find all triples $(x,y,z)$ of real numbers such that
\[x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).\]
1978 IMO Longlists, 14
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:
$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.
$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.
$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.
Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.
1998 Greece JBMO TST, 4
(a) A polynomial $P(x)$ with integer coefficients takes the value $-2$ for at least seven distinct integers $x$. Prove that it cannot take the value $1996$.
(b) Prove that there are irrational numbers $x,y$ such that $x^y$ is rational.
2011 Poland - Second Round, 3
There are two given different polynomials $P(x),Q(x)$ with real coefficients such that $P(Q(x))=Q(P(x))$. Prove that $\forall n\in \mathbb{Z_{+}}$ polynomial:
\[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\]
is divisible by $P(x)-Q(x)$.
2005 Lithuania Team Selection Test, 3
The sequence $a_1, a_2,..., a_{2000}$ of real numbers satisfies the condition
\[a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2\]
for all $n$, $1\leq n \leq 2000$. Prove that every element of the sequence is an integer.
2013 Romania Team Selection Test, 3
Given an integer $n\geq 2$, determine all non-constant polynomials $f$ with complex coefficients satisfying the condition
\[1+f(X^n+1)=f(X)^n.\]
2001 China Team Selection Test, 1
For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0
< x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of
\[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j +
1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k
x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l
x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.
2004 Nordic, 3
Given a finite sequence $x_{1,1}, x_{2,1}, \dots , x_{n,1}$ of integers $(n\ge 2)$, not all equal, define the sequences $x_{1,k}, \dots , x_{n,k}$ by
\[ x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k})\quad\text{where }x_{n+1,k}=x_{1,k}.\]
Show that if $n$ is odd, then not all $x_{j,k}$ are integers. Is this also true for even $n$?
1983 IMO Longlists, 20
Let $f$ and $g$ be functions from the set $A$ to the same set $A$. We define $f$ to be a functional $n$-th root of $g$ ($n$ is a positive integer) if $f^n(x) = g(x)$, where $f^n(x) = f^{n-1}(f(x)).$
(a) Prove that the function $g : \mathbb R \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$-th functional roots for each positive integer $n.$
(b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$
2004 Spain Mathematical Olympiad, Problem 1
We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.
2002 Switzerland Team Selection Test, 5
Find all $f: R\rightarrow R$ such that
(i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite
(ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$
2001 India IMO Training Camp, 3
Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that
\[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]
2013 Baltic Way, 1
Let $n$ be a positive integer. Assume that $n$ numbers are to be chosen from the table
$\begin{array}{cccc}0 & 1 & \cdots & n-1\\ n & n+1 & \cdots & 2n-1\\ \vdots & \vdots & \ddots & \vdots\\(n-1)n & (n-1)n+1 & \cdots & n^2-1\end{array} $
with no two of them from the same row or the same column. Find the maximal value of the product of these $n$ numbers.
2000 IberoAmerican, 3
Find all the solutions of the equation \[\left(x+1\right)^y-x^z=1\] For $x,y,z$ integers greater than 1.
2004 Vietnam Team Selection Test, 1
Let $ \left\{x_n\right\}$, with $ n \equal{} 1, 2, 3, \ldots$, be a sequence defined by $ x_1 \equal{} 603$, $ x_2 \equal{} 102$ and $ x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}$ $ \forall n \geq 1$. Show that:
[b](1)[/b] The number $ x_n$ is a positive integer for every $ n \geq 1$.
[b](2)[/b] There are infinitely many positive integers $ n$ for which the decimal representation of $ x_n$ ends with 2003.
[b](3)[/b] There exists no positive integer $ n$ for which the decimal representation of $ x_n$ ends with 2004.