Let $A,B$ be sets of positive integers with $|A|=|B|=n$, where $n$ is a positive integer, and with equal sums of their elements. Consider an $n\times n$ square grid. Prove that nonnegative integers can be placed in its cells so that (i) the row sums are the elements of $A$; (ii) the column sums are the elements of $B$; and (iii) at least $(n-1)^2+k$ entries are zero, where $k=|A\cap B|$.
Let $ABC$ be an acute triangle with incircle $I$. Let $(k_a)$ be the circle whose center lies on the altitude from $A$, which passes through $A$ and is internally tangent to the incircle $(I)$ at $A_1$. Define $B_1,C_1$ analogously.
1/ Prove that $AA_1,BB_1,CC_1$ are concurrent at a point $P$.
2/ Let $(J_a),(J_b),(J_c)$ be the images of the $A$-, $B$-, and $C$-excircles of triangle $ABC$ under reflection in the midpoints of $BC,CA,AB$, respectively.
Prove that $P$ is the radical center of these three circles.
Given a triangle $ABC$, find the minimum of
$$S=\frac{\cos^2(A/2)\cos^2(B/2)}{\cos^2(C/2)}+\frac{\cos^2(B/2)\cos^2(C/2)}{\cos^2(A/2)}+\frac{\cos^2(C/2)\cos^2(A/2)}{\cos^2(B/2)}.$$
Let $A$ be a subset of $\{1,2,\ldots,4013,4014\}$ containing $2007$ elements such that for every $a,b\in A$, $a$ is not divisible by $b$. Let $m_A$ be the least element of $A$. Find the minimum value of $m_A$ over all such sets $A$.
Let $(H)$ be a regular nonagon. Consider three triangles whose vertices are vertices of $(H)$, with no two triangles sharing a vertex. Prove that one side can be chosen from each triangle so that the three chosen sides have equal lengths.