positive reals $a_1, a_2, . . . $ satisfying (i) $a_{n+1}=a_1^2\cdot a_2^2 \cdot . . . \cdot a_n^2-3$(all positive integers $n$) (ii) $\frac{1}{2}(a_1+\sqrt{a_2-1})$ is positive integer. prove that $\frac{1}{2}(a_1 \cdot a_2 \cdot . . . \cdot a_n + \sqrt{a_{n+1}-1})$ is positive integer

A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$. Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$. Prove that $IE \cdot IK= IC \cdot IL$.

$n$ players participated in a competition. Any two players have played exactly one game, and there was no tie game. For a set of $k(\le n)$ players, if it is able to line the players up so that each player won every player at the back, we call the set ranked. For each player who participated in the competition, the set of players who lost to the player is ranked. Prove that the whole set of players can be split into three or less ranked sets.

find all positive integer $n$, satisfying $$\frac{n(n+2016)(n+2\cdot 2016)(n+3\cdot 2016) . . . (n+2015\cdot 2016)}{1\cdot 2 \cdot 3 \cdot . . . . . \cdot 2016}$$is positive integer.

$n \in \mathbb {N^+}$ Prove that the following equation can be expressed as a polynomial about $n$. $$\left[2\sqrt {1}\right]+\left[2\sqrt {2}\right]+\left[2\sqrt {3}\right]+ . . . +\left[2\sqrt {n^2}\right]$$

circle $O_1$ is tangent to $AC$, $BC$(side of triangle $ABC$) at point $D, E$. circle $O_2$ include $O_1$, is tangent to $BC$, $AB$(side of triangle $ABC$) at point $E, F$ The tangent of $O_2$ at $P(DE \cap O_2, P \neq E)$ meets $AB$ at $Q$. A line passing through $O_1$(center of $O_1$) and parallel to $BO_2$($O_2$ is also center of $O_2$) meets $BC$ at $G$, $EQ \cap AC=K, KG \cap EF=L$, $EO_2$ meets circle $O_2$ at $N(\neq E)$, $LO_2 \cap FN=M$. IF $N$ is a middle point of $FM$, prove that $BG=2EG$

positive integers $a_1, a_2, . . . , a_9$ satisfying $a_1+a_2+ . . . +a_9 =90$ find maximum of $$\frac{1^{a_1} \cdot 2^{a_2} \cdot . . . \cdot 9^{a_9}}{a_1! \cdot a_2! \cdot . . . \cdot a_9!}$$ mentionI was really shocked because there are no inequality problems at KJMO and the test difficulty even more lower...

One moving point in the coordinate plane can move right or up one position. $N$ is a number of all paths : paths that moving point starts from $(0, 0)$, without passing $(1, 0), (2, 1), . . . , (n, n-1)$ and moves $2n$ times to $(n, n)$. $a_k$ is a number of special paths : paths include in $N$, but $k$th moves to the right, $k+1$th moves to the up. find $$\frac{1}{N} (a_1+a_2+ . . . + a_{2n-1})$$