Found problems: 6530
2007 IberoAmerican Olympiad For University Students, 3
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
2002 Moldova National Olympiad, 2
The coefficients of the equation $ ax^2\plus{}bx\plus{}c\equal{}0$, where $ a\ne 0$, satisfy the inequality $ (a\plus{}b\plus{}c)(4a\minus{}2b\plus{}c)<0$. Prove that this equation has $ 2$ real distinct solutions.
JOM 2015 Shortlist, A3
Let $ a, b, c $ be positive real numbers less than or equal to $ \sqrt{2} $ such that $ abc = 2 $, prove that $$ \sqrt{2}\displaystyle\sum_{cyc}\frac{ab + 3c}{3ab + c} \ge a + b + c $$
2007 Postal Coaching, 3
Let $a$ and $b$ be two positive real numbers such that $a^{2007} = a + 1$ and $b^{4014} = b + 3a$. Determine whether $a > b$ or $b > a$.
1999 Akdeniz University MO, 3
For all $x> \sqrt 2$, $y> \sqrt 2$ numbers, prove that
$$x^4-x^3y+x^2y^2-xy^3+y^4>x^2+y^2$$
2003 Costa Rica - Final Round, 3
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.
2012 Mexico National Olympiad, 4
The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied three times the result is $1$, and applying it four times the result is $0$. When the process is applied one or more times to an integer $n$, the result is eventually $0$. The number obtained before obtaining $0$ is called the [i]house[/i] of $n$.
How many integers less than $26000$ share the same [i]house[/i] as $2012$?
2013 Hanoi Open Mathematics Competitions, 6
Let be given $a\in\{0,1,2, 3,..., 100\}.$
Find all $n \in\{1,2, 3,..., 2013\}$ such that $C_n^{2013} > C_a^{2013}$ , where $C_k^m=\frac{m!}{k!(m -k)!}$.
2018 Switzerland - Final Round, 9
Let $n$ be a positive integer and let $G$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called [i]parallelogram-free[/i] if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can have?
1999 Greece Junior Math Olympiad, 1
Show that if $a,b$ are positive real numbers such that $a^{2000}+b^{2000}=a^{1998}+b^{1998}$ then $a^2+ b^2 \le 2$.
2009 Korea Junior Math Olympiad, 6
If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$
2012 VJIMC, Problem 3
Determine the smallest real number $C$ such that the inequality
$$\frac x{\sqrt{yz}}\cdot\frac1{x+1}+\frac y{\sqrt{zx}}\cdot\frac1{y+1}+\frac z{\sqrt{xy}}\cdot\frac1{x+1}\le C$$holds for all positive real numbers $x,y$ and $z$ with $\frac1{x+1}+\frac1{y+1}+\frac1{z+1}=1$.
2016 Romania National Olympiad, 2
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions:
$$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$
for all real numbers $ x,y,t $ with $ t\in [0,1] . $
Prove that:
[b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $
[b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds.
$$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$
1987 Greece National Olympiad, 3
Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$
2007 Mexico National Olympiad, 3
Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that
\[\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2\]
2009 Jozsef Wildt International Math Competition, W. 29
Prove that for all triangle $\triangle ABC$ holds the following inequality $$\sum \limits_{cyc} \left (1-\sqrt{\sqrt{3}\tan \frac{A}{2}}+\sqrt{3}\tan \frac{A}{2}\right )\left (1-\sqrt{\sqrt{3}\tan \frac{B}{2}}+\sqrt{3}\tan \frac{B}{2}\right )\geq 3$$
2005 Taiwan National Olympiad, 3
$a_1, a_2, ..., a_{95}$ are positive reals. Show that
$\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$
2005 Greece JBMO TST, 2
Prove that for each $x,y,z \in R$ it holds that
$$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$
1991 IMTS, 4
Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that
\[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]
2016 Bosnia And Herzegovina - Regional Olympiad, 1
If $\mid ax^2+bx+c \mid \leq 1$ for all $x \in [-1,1]$ prove that:
$a)$ $\mid c \mid \leq 1$
$b)$ $\mid a+c \mid \leq 1$
$c)$ $a^2+b^2+c^2 \leq 5$
2007 Bulgarian Autumn Math Competition, Problem 11.2
Find all values of the parameter $a$ for which the inequality
\[\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}\]
has a unique solution.
1995 Cono Sur Olympiad, 3
Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function).
1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$.
2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$
1977 Poland - Second Round, 1
Let $ a $ and $ b $ be different real numbers. Prove that for any real numbers $ c_1, c_2, \ldots,c_n $ there exists a sequence of $ n $-elements $ (x_i) $, each term of which is equal to one of the numbers $ a $ or $ b $ such that $$
|x_1c_1 + x_2c_2 + \ldots + x_nc_n| \geq \frac{|b-a|}{2}(|c_1|+|c_2|+\ldots+|c_n|).$$
2011 ELMO Shortlist, 4
In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
\[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\]
[i]Calvin Deng.[/i]
1979 Poland - Second Round, 2
Prove that if $ a, b, c $ are non-negative numbers, then $$
a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b).$$