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.
Let $x,y$ be integers with $x\ne-1$, $y\ne-1$, such that
$$\frac{x^4-1}{y+1}+\frac{y^4-1}{x+1}$$
is an integer. Prove that $x^4y^{44}-1$ is divisible by $x+1$.
Triangle $ABC$ has fixed vertices $B,C$ and a variable vertex $A$. Let $H,G$ be its orthocenter and centroid, respectively. Determine the locus of $A$, given that the midpoint $K$ of $HG$ lies on the line $BC$.
A regular polygon has $2007$ vertices. Find the smallest positive integer $k$ with the following property: in every choice of $k$ vertices, four can be chosen to form a convex quadrilateral of which three of its four sides are sides of the given polygon.
Let $b$ be a positive real number. Determine all functions $f:\mathbb R\to\mathbb R$ satisfying, for every $x,y\in\mathbb R$,
$$f(x+y)=f(x)3^{b^y+f(y)-1}+b^x\left(3^{b^y+f(y)-1}-b^y\right).$$
Let trapezoid $ABCD$, with larger base $BC$, be inscribed in circle $(O)$ with center $O$. A point $P$ varies on the line $BC$, outside segment $BC$, such that $PA$ is not tangent to $(O)$. The circle with diameter $PD$ meets $(O)$ again at $E$ ($E\ne D$). Let $M=BC\cap DE$, and let $N$ be the intersection of $PA$ with $(O)$ other than $A$. Prove that $MN$ passes through a fixed point.
Let $a\gt 2$ be a real number. Put
$$f_n(x)=a^{10}x^{n+10}+x^n+\cdots+x+1\qquad(n=1,2,\ldots).$$
Prove that for every $n$, the equation $f_n(x)=a$ has exactly one solution $x_n\in(0,+\infty)$. Prove that the sequence $(x_n)$ has a finite limit as $n\to+\infty$.