Found problems: 6530
2002 China Girls Math Olympiad, 3
Find all positive integers $ k$ such that for any positive numbers $ a, b$ and $ c$ satisfying the inequality
\[ k(ab \plus{} bc \plus{} ca) > 5(a^2 \plus{} b^2 \plus{} c^2),\]
there must exist a triangle with $ a, b$ and $ c$ as the length of its three sides respectively.
2016 Hanoi Open Mathematics Competitions, 10
Let $h_a, h_b, h_c$ and $r$ be the lengths of altitudes and radius of the inscribed circle of $\vartriangle ABC$, respectively.
Prove that $h_a + 4h_b + 9h_c > 36r$.
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
\[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]
2011 Kyrgyzstan National Olympiad, 4
Given equation ${a^5} - {a^3} + a = 2$, with real $a$ . Prove that $3 < {a^6} < 4$.
2008 Federal Competition For Advanced Students, P1, 3
Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$.
Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.
2000 JBMO ShortLists, 20
Let $ABC$ be a triangle and let $a,b,c$ be the lengths of the sides $BC, CA, AB$ respectively. Consider a triangle $DEF$ with the side lengths $EF=\sqrt{au}$, $FD=\sqrt{bu}$, $DE=\sqrt{cu}$. Prove that $\angle A >\angle B >\angle C$ implies $\angle A >\angle D >\angle E >\angle F >\angle C$.
2024 PErA, P3
Let $x_1,x_2,\dots, x_n$ be positive real numbers such that $x_1+x_2+\cdots + x_n=1$. Prove that $$\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,$$ where we denote $x_0=x_n$ and $x_{n+1}=x_1$.
2008 Bulgaria Team Selection Test, 2
The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?
2005 All-Russian Olympiad, 1
Do there exist a bounded function $f: \mathbb{R}\to\mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies an inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$?
1987 Czech and Slovak Olympiad III A, 4
Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$
2022 Czech-Polish-Slovak Junior Match, 4
Let $a$ and $b$ be positive integers with the property that $\frac{a}{b} > \sqrt2$. Prove that
$$\frac{a}{b} - \frac{1}{2ab} > \sqrt2$$
2019 International Zhautykov OIympiad, 2
Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality :
$\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$
2023 Middle European Mathematical Olympiad, 2
If $a, b, c, d>0$ and $abcd=1$, show that $$\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. $$ When does equality hold?
2021 Canada National Olympiad, 2
Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$.
Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$
PEN I Problems, 6
Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]
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 $a < b$.
PEN P Problems, 6
Show that every integer greater than $1$ can be written as a sum of two square-free integers.
2008 Singapore Junior Math Olympiad, 2
Let $a.b,c,d$ be positive real numbers such that $cd = 1$. Prove that there is an integer $n$ such that $ab\le n^2\le (a + c)(b + d)$.
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
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 $a^2+ b^2 \le 2$.
1978 Czech and Slovak Olympiad III A, 1
Let $a_1,\ldots,a_n,b_1,\ldots,b_n$ be positive numbers. Show that
\[\sqrt{\left(a_1+\cdots+a_n\right)\left(b_1+\cdots+b_n\right)}\ge\sqrt{a_1b_1}+\cdots+\sqrt{a_nb_n}\]
and prove that equality holds if and only if
\[\frac{a_1}{b_1}=\cdots=\frac{a_n}{b_n}.\]
2008 Swedish Mathematical Competition, 6
A [i]sum decomposition[/i] of the number 100 is given by a positive integer $n$ and $n$ positive integers $x_1<x_2<\cdots <x_n$ such that $x_1 + x_2 + \cdots + x_n = 100$. Determine the largest possible value of the product $x_1x_2\cdots x_n$, and $n$ , as $x_1, x_2,\dots, x_n$ vary among all sum decompositions of the number $100$.
2008 Romania National Olympiad, 2
Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then
\[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]
2021 Kazakhstan National Olympiad, 1
Given $a,b,c>0$ such that
$$a+b+c+\frac{1}{abc}=\frac{19}{2}$$
What is the greatest value for $a$?
2002 China Team Selection Test, 2
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,
such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.
Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.
Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$