Found problems: 6530
2003 CHKMO, 3
Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.
1994 Hong Kong TST, 1
Suppose, $x, y, z \in \mathbb{R}_+$ such that $xy+yz+zx=1$. Prove that, \[x(1-y^2)(1-z^2)+y(1-z^2)(1-x^2)+z(1-x^2)(1-y^2)\leq \frac{4\sqrt{3}}{9}\]
2010 Postal Coaching, 1
In a family there are four children of different ages, each age being a positive integer not less than $2$ and not greater than $16$. A year ago the square of the age of the eldest child was equal to the sum of the squares of the ages of the remaining children. One year from now the sum of the squares of the youngest and the oldest will be equal to the sum of the squares of the other two. How old is each child?
2003 Hong kong National Olympiad, 1
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
1988 Flanders Math Olympiad, 4
Be $R$ a positive real number. If $R, 1, R+\frac12$ are triangle sides, call $\theta$ the angle between $R$ and $R+\frac12$ (in rad).
Prove $2R\theta$ is between $1$ and $\pi$.
1992 Polish MO Finals, 3
Show that for real numbers $x_1, x_2, ... , x_n$ we have:
\[ \sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0 \]
When do we have equality?
2005 India Regional Mathematical Olympiad, 3
If $a,b,c$ are positive three real numbers such that $| a-b | \geq c , | b-c | \geq a, | c-a | \geq b$ . Prove that one of $a,b,c$ is equal to the sum of the other two.
2014 Saudi Arabia Pre-TST, 4.1
Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.
1999 Greece JBMO TST, 2
For $a,b,c>0$, prove that
(i) $\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0$
(ii) $a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}$
1992 Turkey Team Selection Test, 3
$x_1, x_2,\cdots,x_{n+1}$ are posive real numbers satisfying the equation
$\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1$
Prove that $x_1x_2 \cdots x_{n+1} \geq n^{n+1}$.
2010 ELMO Shortlist, 2
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]
[i]Calvin Deng.[/i]
2008 Bosnia and Herzegovina Junior BMO TST, 1
Let $ a,b,c$ be real positive numbers such that absolute difference between any two of them is less than $ 2$. Prove that: $ a \plus{} b \plus{} c < \sqrt {ab \plus{} 1} \plus{} \sqrt {ac \plus{} 1} \plus{} \sqrt {bc \plus{} 1}$
2019 Hanoi Open Mathematics Competitions, 14
Let $a, b, c$ be nonnegative real numbers satisfying $a + b + c =3$.
a) If $c > \frac32$, prove that $3(ab + bc + ca) - 2abc < 7$.
b) Find the greatest possible value of $M =3(ab + bc + ca) - 2abc $.
2013 Iran Team Selection Test, 11
Let $a,b,c$ be sides of a triangle such that $a\geq b \geq c$. prove that:
$\sqrt{a(a+b-\sqrt{ab})}+\sqrt{b(a+c-\sqrt{ac})}+\sqrt{c(b+c-\sqrt{bc})}\geq a+b+c$
2013 239 Open Mathematical Olympiad, 8
The product of the positive numbers $a, b, c, d$ and $e$ is equal to $1$. Prove that
$$ \frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{e^2}+\frac{e^2}{a^2} \geq a+b+c+d+e .$$
2001 Korea - Final Round, 3
Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that
\[(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|\]
and find all cases of equality.
1980 Miklós Schweitzer, 1
For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{}
m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\]
[i]V. T. Sos[/i]
1995 Baltic Way, 8
The real numbers $a,b$ and $c$ satisfy the inequalities $|a|\ge |b+c|,|b|\ge |c+a|$ and $|c|\ge |a+b|$. Prove that $a+b+c=0$.
2021 Israel National Olympiad, P3
Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle.
Prove that
\[AX+AY+BC>AB+AC\]
2004 Tournament Of Towns, 5
Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].
1963 Dutch Mathematical Olympiad, 3
Twenty numbers $a_1,a_2,..,a_{20}$ satisfy:
$$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$
$$a_1+a_2+...+a_{20}=1518$$
Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.
2002 IMO Shortlist, 2
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
1985 Balkan MO, 2
Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that
$\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$.
Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$
2004 Greece Junior Math Olympiad, 4
Determine the rational number $\frac{a}{b}$, where $a,b$ are positive integers, with minimal denominator, which is such that
$ \frac{52}{303} < \frac{a}{b}< \frac{16}{91}$
2005 Baltic Way, 5
Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that
\[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]