Found problems: 15925
1981 Canada National Olympiad, 4
$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.
2019 CMIMC, 7
For all positive integers $n$, let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.
2022 Macedonian Team Selection Test, Problem 3
We consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)+n)=n$ and $f(a+b-1) \leq f(a)+f(b)$ for all positive integers $a, b, n$. Prove that there are at most two values for $f(2022)$.
$\textit {Proposed by Ilija Jovcheski}$
2021 Belarusian National Olympiad, 11.1
Find all functions $f: \mathbb{R} \to \mathbb{R}$, such that for all real $x,y$ the following equation holds:$$f(x-0.25)+f(y-0.25)=f(x+\lfloor y+0.25 \rfloor - 0.25)$$
2013 Korea Junior Math Olympiad, 3
$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$).
For positive integer $n$, de fine as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$
Prove that $b_n$ is positive integer.
2016 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.
2010 IMO, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
2003 All-Russian Olympiad, 1
Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b \in M$, the number $a^2 +b\sqrt 2$ is rational. Prove that $a\sqrt 2$ is rational for all $a \in M.$
2006 MOP Homework, 1
Let a,b, and c be positive reals. Prove:
$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\ge (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$
2005 USAMTS Problems, 4
Find, with proof, all triples of real numbers $(a, b, c)$ such that all four roots of the polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+b$ are positive integers. (The four roots need not be distinct.)
2001 Taiwan National Olympiad, 1
Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$. it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$. And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$. Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$.
2014 Postal Coaching, 5
Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.
2018-IMOC, N6
If $f$ is a polynomial sends $\mathbb Z$ to $\mathbb Z$ and for $n\in\mathbb N_{\ge2}$, there exists $x\in\mathbb Z$ so that $n\nmid f(x)$, show that for every $k\in\mathbb Z$, there is a non-negative integer $t$ and $a_1,\ldots,a_t\in\{-1,1\}$ such that
$$a_1f(1)+\ldots+a_tf(t)=k.$$
1997 Czech and Slovak Match, 3
Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$
.
2014 Contests, 1
Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\]
in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.
(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)
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 $n$.
2010 Romania Team Selection Test, 1
A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$.
[i]AMM Magazine[/i]
1969 Swedish Mathematical Competition, 2
Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.
1979 Chisinau City MO, 171
Are there numbers $a, b$ such that $| a -b |\le 1979$ and the equation $ax^2 + (a + b) x + b = x$ has no roots?
2014 Cezar Ivănescu, 1
[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $
[b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $
2009 Singapore Junior Math Olympiad, 5
Let $a, b$ be positive real numbers satisfying $a + b = 1$. Show that if $x_1,x_2,...,x_5$ are positive real numbers such that $x_1x_2...x_5 = 1$, then $(ax_1+b)(ax_2+b)...(ax_5+b)>1$
2012 USAMTS Problems, 3
Let $f(x) = x-\tfrac1{x}$, and defi ne $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.
2021 Balkan MO Shortlist, A4
Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $x_0 = (0, 0)$, and for every $n \ge 1$, his jump is
$x_n - x_{n - 1} = (\pm f(n), \pm g(n))$ or $(\pm g(n), \pm f(n)),$
with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0, 0)$ infinitely many times when
(a) $f, g$ are polynomials with integer coefficients?
(b) $f, g$ are any pair of functions from the positive integers to the integers?
2014 District Olympiad, 2
Solve in real numbers the equation
\[ x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right) \]
2012 District Olympiad, 1
Let $ f:[0,\infty )\longrightarrow\mathbb{R} $ a bounded and periodic function with the property that
$$ |f(x)-f(y)|\le |\sin x-\sin y|,\quad\forall x,y\in[0,\infty ) . $$
Show that the function $ [0,\infty ) \ni x\mapsto x+f(x) $ is monotone.