Found problems: 6530
1953 Moscow Mathematical Olympiad, 253
Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.
2016 Junior Regional Olympiad - FBH, 1
If $a>b>c$ are real numbers prove that $$\frac{1}{a-b}+\frac{1}{b-c}>\frac{2}{a-c}$$
2012 Bogdan Stan, 4
Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following:
$$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$
1991 Federal Competition For Advanced Students, P2, 5
For all positive integers $ n$ prove the inequality:
$ \left( \frac{1\plus{}(n\plus{}1)^{n\plus{}1}}{n\plus{}2} \right)^{n\minus{}1}>\left( \frac{1\plus{}n^n}{n\plus{}1} \right)^n.$
2005 Germany Team Selection Test, 1
Find the smallest positive integer $n$ with the following property:
For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that
\[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]
2005 Rioplatense Mathematical Olympiad, Level 3, 1
Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that
\[ \frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.\]
2005 Iran Team Selection Test, 1
Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let
\[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1.
\]
Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that:
\[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2
\]
1976 Spain Mathematical Olympiad, 7
The price of a diamond is proportional to the square of its weight. Show that, breaking it into two parts, there is a depreciation of its value. When is it the maximum depreciation?
2020 OMMock - Mexico National Olympiad Mock Exam, 1
Let $a$, $b$, $c$ and $d$ positive real numbers with $a > c$ and $b < d$. Assume that
\[a + \sqrt{b} \ge c + \sqrt{d} \qquad \text{and} \qquad \sqrt{a} + b \le \sqrt{c} + d\]
Prove that $a + b + c + d > 1$.
[i]Proposed by Victor Domínguez[/i]
2002 China Team Selection Test, 2
For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always:
\[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]
2023 Junior Macedonian Mathematical Olympiad, 3
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality
$$ \left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.$$
When does equality hold?
[i]Authored by Anastasija Trajanova[/i]
2015 Federal Competition For Advanced Students, 1
Let $a$, $b$, $c$, $d$ be positive numbers. Prove that
$$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$
When does equality hold?
(Georg Anegg)
1994 Hungary-Israel Binational, 1
Let $ m$ and $ n$ be two distinct positive integers. Prove that there exists a real number $ x$ such that $ \frac {1}{3}\le\{xn\}\le\frac {2}{3}$ and $ \frac {1}{3}\le\{xm\}\le\frac {2}{3}$. Here, for any real number $ y$, $ \{y\}$ denotes the fractional part of $ y$. For example $ \{3.1415\} \equal{} 0.1415$.
VMEO IV 2015, 12.1
Find the largest constant $k$ such that the inequality
$$a^2+b^2+c^2-ab-bc-ca \ge k \left|\frac{a^3-b^3}{a+b}+\frac{b^3-c^3}{b+c}+\frac{c^3-a^3}{c+a}\right|$$
holds for any for non negative real numbers $a,b,c$ with $(a+b)(b+c)(c+a)>0$.
1996 Estonia Team Selection Test, 2
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
2024 Macedonian TST, Problem 2
Let $u,v,w$ be positive real numbers. Prove that there exists a cyclic permutation $(x,y,z)$ of $(u,v,w)$ such that for all positive real numbers $a,b,c$ the following holds:
\[
\frac{a}{x\,a + y\,b + z\,c}
\;+\;
\frac{b}{x\,b + y\,c + z\,a}
\;+\;
\frac{c}{x\,c + y\,a + z\,b}
\;\ge\;
\frac{3}{x + y + z}.
\]
1954 Polish MO Finals, 4
Find the values of $ x $ that satisfy the inequality
$$ \sqrt{x} - \sqrt{x- a} > 2,$$
where $ a $ is a gicen poistive number.
2014 Contests, 2
Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$.
Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).
2019 Stars of Mathematics, 4
For positive real numbers $a_1, a_2, ..., a_n$ with product 1 prove:
$$\left(\frac{a_1}{a_2}\right)^{n-1}+\left(\frac{a_2}{a_3}\right)^{n-1}+...+\left(\frac{a_{n-1}}{a_n}\right)^{n-1}+\left(\frac{a_n}{a_1}\right)^{n-1} \geq a_1^{2}+a_2^{2}+...+a_n^{2}$$
Proposed by Andrei Eckstein
1996 Taiwan National Olympiad, 2
Let $0<a\leq 1$ be a real number and let $a\leq a_{i}\leq\frac{1}{a_{i}}\forall i=\overline{1,1996}$ are real numbers. Prove that for any nonnegative real numbers $k_{i}(i=1,2,...,1996)$ such that $\sum_{i=1}^{1996}k_{i}=1$ we have $(\sum_{i=1}^{1996}k_{i}a_{i})(\sum_{i=1}^{1996}\frac{k_{i}}{a_{i}})\leq (a+\frac{1}{a})^{2}$.
2009 Indonesia TST, 1
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.
1996 Israel National Olympiad, 6
Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?
2018 Junior Balkan Team Selection Tests - Romania, 2
Let $x, y,z$ be positive real numbers satisfying $2x^2+3y^2+6z^2+12(x+y+z) =108$. Find the maximum value of $x^3y^2z$.
Alexandru Gırban
2008 Romania National Olympiad, 3
Let $ a,b \in [0,1]$. Prove that \[ \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.\]
2014 Iran Team Selection Test, 5
if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$
prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]