Found problems: 15925
2011 Greece Team Selection Test, 3
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
2017 Iran MO (3rd round), 1
Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that
$$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$
for all nonnegative real numbers $x,y$ and $z$.
2011 Princeton University Math Competition, A1 / B5
A polynomial $p$ can be written as
\begin{align*}
p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.
\end{align*}
Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$.
2000 Mediterranean Mathematics Olympiad, 3
Let $c_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n$ $(n\geq 2)$ be positive real numbers. Prove that the equation
\[ \sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i\]
has a unique solution $(x_1,\ldots ,x_n)$ if and only if $\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i$.
1988 IMO Shortlist, 19
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$
2009 Middle European Mathematical Olympiad, 6
Let $ a$, $ b$, $ c$ be real numbers such that for every two of the equations
\[ x^2\plus{}ax\plus{}b\equal{}0, \quad x^2\plus{}bx\plus{}c\equal{}0, \quad x^2\plus{}cx\plus{}a\equal{}0\]
there is exactly one real number satisfying both of them. Determine all possible values of $ a^2\plus{}b^2\plus{}c^2$.
2015 Miklos Schweitzer, 5
Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$
2014 District Olympiad, 1
Prove that:
[list=a][*]$\displaystyle\left( \frac{1}{2}\right) ^{3}+\left( \frac{2}{3}\right)^{3}+\left( \frac{5}{6}\right) ^{3}=1$
[*]$3^{33}+4^{33}+5^{33}<6^{33}$[/list]
2002 Austria Beginners' Competition, 2
Prove that there are no $x\in\mathbb{R}^+$ such that $$x^{\lfloor x \rfloor }=\frac92.$$
2014 Peru Iberoamerican Team Selection Test, P4
Determine the minimum value of
$$x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015$$
where $x$ is a real number.
1996 VJIMC, Problem 2
Let $\{x_n\}^\infty_{n=0}$ be the sequence such that $x_0=2$, $x_1=1$ and $x_{n+2}$ is the remainder of the number $x_{n+1}+x_n$ divided by $7$. Prove that $x_n$ is the remainder of the number
$$4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k$$
2014 Bosnia And Herzegovina - Regional Olympiad, 1
Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$
2018 Hanoi Open Mathematics Competitions, 4
Find the number of distinct real roots of the following equation $x^2 +\frac{9x^2}{(x + 3)^2} = 40$.
A. $0$ B. $1$ C. $2$ D. $3$ E. $4$
2024 Junior Balkan Team Selection Tests - Moldova, 7
Find all the real numbers $x,y,z$ which satisfy the following conditions:
$$
\begin{cases}
3(x^2+y^2+z^2)=1\\
x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\\
\end{cases}
$$
2019 CHMMC (Fall), 6
Compute $$\prod^{2019}_{i=1} (2^{2^i}- 2^{2^{i-1}} + 1).$$
1990 IMO Longlists, 72
Let $n \geq 5$ be a positive integer. $a_1, b_1, a_2, b_2, \ldots, a_n, b_n$ are integers. $( a_i, b_i)$ are pairwisely distinct for $i = 1, 2, \ldots, n$, and $|a_1b_2 - a_2b_1| = |a_2b_3 -a_3b_2| = \cdots = |a_{n-1}b_n -a_nb_{n-1}| = 1$. Prove that there exists a pair of indexes $i, j$ satisfying $2 \leq |i - j| \leq n - 2$ and $|a_ib_j -a_jb_i| = 1.$
2005 Georgia Team Selection Test, 4
Find all polynomials with real coefficients, for which the equality
\[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\]
holds for any real number $ x$.
2019 Purple Comet Problems, 14
For real numbers $a$ and $b$, let $f(x) = ax + b$ and $g(x) = x^2 - x$. Suppose that $g(f(2)) = 2, g(f(3)) = 0$, and $g(f(4)) = 6$. Find $g(f(5))$.
2011 Dutch IMO TST, 4
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.
2013 District Olympiad, 2
Let the matrices of order 2 with the real elements $A$ and $B$ so that $AB={{A}^{2}}{{B}^{2}}-{{\left( AB \right)}^{2}}$ and $\det \left( B \right)=2$.
a) Prove that the matrix $A$ is not invertible.
b) Calculate $\det \left( A+2B \right)-\det \left( B+2A \right)$.
2010 Saudi Arabia Pre-TST, 4.3
Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that
$$\frac{a-2}{a+1}+\frac{b-2}{b+1}+\frac{c-2}{c+1} \le 0$$
2013 Online Math Open Problems, 45
Let $N$ denote the number of ordered 2011-tuples of positive integers $(a_1,a_2,\ldots,a_{2011})$ with $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$ such that there exists a polynomial $f$ of degree $4019$ satisfying the following three properties:
[list] [*] $f(n)$ is an integer for every integer $n$; [*] $2011^2 \mid f(i) - a_i$ for $i=1,2,\ldots,2011$; [*] $2011^2 \mid f(n+2011) - f(n)$ for every integer $n$. [/list]
Find the remainder when $N$ is divided by $1000$.
[i]Victor Wang[/i]
2006 Serbia Team Selection Test, 1
$$Problem 1 $$The set S = {1,2,3,...,2006} is partitioned into two disjoint subsets A and B
such that:
(i) 13 ∈ A;
(ii) if a ∈ A, b ∈ B, a+b ∈ S, then a+b ∈ B;
(iii) if a ∈ A, b ∈ B, ab ∈ S, then ab ∈ A.
Determine the number of elements of A
2012 Ukraine Team Selection Test, 6
For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.
2003 Abels Math Contest (Norwegian MO), 1a
Let $x$ and $y$ are real numbers such that $$\begin{cases} x + y = 2 \\ x^3 + y^3 = 3\end{cases} $$ What is $x^2+y^2$?