Found problems: 6530
1988 Tournament Of Towns, (199) 2
Prove that $a^2pq + b^2qr + c^2rp \le 0$, whenever $a, b$ and $c$ are the lengths of the sides of a triangle and $p + q + r = 0$ .
( J. Mustafaev , year 12 student, Baku)
2019 IFYM, Sozopol, 5
Let $a>0$ and $12a+5b+2c>0$. Prove that it is impossible for the equation
$ax^2+bx+c=0$ to have two real roots in the interval $(2,3)$.
2014 Saint Petersburg Mathematical Olympiad, 3
$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$
VI Soros Olympiad 1999 - 2000 (Russia), 9.4
For real numbers $x \ge 0$ and $y \ge 0$, prove the inequality $$x^4+y^3+x^2+y+1 >\frac92 xy.$$
2004 Turkey MO (2nd round), 5
The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.
2021 IMO, 2
Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$
2012 Irish Math Olympiad, 3
Suppose $a,b,c$ are positive numbers. Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2\ge (2a+b+c) \left(\frac{2}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ with equality if and only if $a=b=c$.
2022 Austrian MO National Competition, 1
Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality?
[i](Walther Janous)[/i]
1997 Estonia Team Selection Test, 2
Prove that for all positive real numbers $a_1,a_2,\cdots a_n$ \[\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}\] When does the inequality hold?
2018 Oral Moscow Geometry Olympiad, 5
Two ants sit on the surface of a tetrahedron. Prove that they can meet by breaking the sum of a distance not exceeding the diameter of a circle is circumscribed around the edge of a tetrahedron.
2010 Contests, 1
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]
2001 Flanders Math Olympiad, 4
A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.
2021 Austrian MO Regional Competition, 1
Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying
$$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds.
(Karl Czakler)
1983 All Soviet Union Mathematical Olympiad, 368
The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the triangle $ABC$ respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the triangles $DEF$, $DEA$, $DBF$, $CEF$, respectively. Prove that $$d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$$ When the equality takes place?
1988 Austrian-Polish Competition, 2
If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ($n \ge 2$), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$-tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$.
2023 Baltic Way, 5
Find the smallest positive real $\alpha$, such that $$\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$.
2006 China National Olympiad, 1
Let $a_1,a_2,\ldots,a_k$ be real numbers and $a_1+a_2+\ldots+a_k=0$. Prove that
\[ \max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right). \]
2020-21 IOQM India, 5
Find the number of integer solutions to $||x| - 2020| < 5$.
2019 Saint Petersburg Mathematical Olympiad, 3
Prove that the distance between the midpoint of side $BC$ of triangle $ABC$ and the midpoint of the arc $ABC$ of its circumscribed circle is not less than $AB / 2$
2009 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c$ be positive real number such that $a + b + c \ge \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$ .
Prove that $ \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\ge \frac{1}{ab}+ \frac{1}{bc}+ \frac{1}{ca}$ .
2007 Today's Calculation Of Integral, 225
2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.
2013 Baltic Way, 16
We call a positive integer $n$ [i]delightful[/i] if there exists an integer $k$, $1 < k < n$, such that
\[1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n\]
Does there exist a delightful number $N$ satisfying the inequalities
\[2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?\]
2003 Alexandru Myller, 2
For three positive real numbers $ a,b,c $ satisfying the condition $ \frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca} =1, $ prove that
$$ 3/2\le \frac{ab-1}{ab+1} +\frac{bc-1}{bc+1} +\frac{ca-1}{ca+1} <2. $$
[i]Mircea Becheanu[/i]
1987 Traian Lălescu, 2.1
Let $ \lambda \in (0,2) $ and $ a,b,c,d\in\mathbb{R} $ so that $ a\le b\le c. $ Prove the inequality:
$$ (a+b+c+d)^2\ge 4\lambda (ac+bd). $$
2013 District Olympiad, 4
For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$