Found problems: 6530
2006 Irish Math Olympiad, 4
Find the greatest value and the least value of $x+y$ where $x,y$ are real numbers, with $x\ge -2$, $y\ge -3$ and $$x-2\sqrt{x+2}=2\sqrt{y+3}-y$$
2000 Stanford Mathematics Tournament, 25
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart?
2004 Mediterranean Mathematics Olympiad, 1
Find all natural numbers $m$ such that
\[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]
2012 239 Open Mathematical Olympiad, 4
For positive real numbers $a$, $b$, and $c$ with $a+b+c=1$, prove that:
$$ (a-b)^2 + (b-c)^2 + (c-a)^2 \geq \frac{1-27abc}{2}. $$
2022 OMpD, 2
We say that a sextuple of positive real numbers $(a_1, a_2, a_3, b_1, b_2, b_3)$ is $\textit{phika}$ if $a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = 1$.
(a) Prove that there exists a $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$ such that:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) > 1 - \frac{1}{2022^{2022}}$$
(b) Prove that for every $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$, we have:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) < 1$$