I need to put these somewhere
Find all monic polynomials \(Q\) with integer coefficients such that for all integers \(a\), \(b\) there exists an integer \(c\) such that \(Q(a)Q(b) = Q(c)\).
Find all functions \(f : \mathbf{Z}_{> 0} \to \mathbf{Z}_{> 0}\) such that for all positive integers \(m\), \(n\): \[mf(n+1) + f(n)f(m+1) \mid f(mn+1) + mf(n)\] and \(\gcd(f(n), f(n+1)) = 1\).
Solution moved to israel_fe.html.
Let \(b_1 < b_2 < \dots\) be the sequence of natural numbers that are the sum of two squares. Prove that there exist infinitely many \(m\) such that \(b_{m+1} - b_m = 2015\).
Turns out this is 2015 Iran TST 1 Day 1 P3.
Let \(ABC\) be a triangle, and let \(AD\), \(BE\), \(CF\) be its altitudes. Let \(H\) be its orthocenter, and let \(O_B\) and \(O_C\) be the circumcenters of triangles \(AHC\) and \(AHB\). Let \(G\) be the second intersection of the circumcircles of triangles \(FDO_B\) and \(EDO_C\). Prove that the lines \(DG\), \(EF\), and \(A\)-median of \(\triangle ABC\) are concurrent.
Find all surjective functions \(f:\mathbf{R}\to \mathbf{R}\) such that \[f(2xf(y)+2yf(x))=f(2x)f(2y)\] for all \(x\), \(y\in\mathbf{R}\).
Let \( \mathbf{Z}^2 \) denote the set of all integer lattice points in the plane with Cartesian coordinates. A graph \( G = (\mathbf{Z}^2, E) \) is constructed by connecting two lattice points with an edge if their Euclidean distance is \(1\). For a positive integer \( n \), define \( f(n) \) as the number of connected subgraphs \( H = (V(H), E(H)) \) of \( G \) satisfying: \[ \{(0, 0)\} \subseteq V(H) \subseteq \mathbf{Z}^2, \quad |V(H)| = n, \quad E(H) \subseteq E. \] Prove that there exists a positive constant \( C \) such that for any positive integer \( n \), \( f(n) \leq C \cdot 7^n \).
Canmoo is trying to do constructions, but doesn't have a ruler or compass. Instead, Canmoo has a device that, given four distinct points \(A\), \(B\), \(C\), \(P\) in the plane, will mark the isogonal conjugate of \(P\) with respect to triangle \(ABC\), if it exists. Show that if two points are marked on the plane, then Canmoo can construct their midpoint using this device, a pencil for marking additional points, and no other tools.
(Additional points marked by the pencil can be assumed to be in general position, meaning they don't lie on any line through two existing points or any circle through three existing points.)
Let \(H\) be the orthocenter of an acute \(\triangle ABC\). The line through \(H\) parallel to \(BC\) intersects sides \(AB\) and \(AC\) at \(D\) and \(E\) respectively. The line through \(H\) parallel to \(CA\) intersects sides \(BC\) and \(BA\) at \(P\) and \(Q\) respectively. The line through \(H\) parallel to \(AB\) intersects sides \(CA\) and \(CB\) at \(X\) and \(Y\) respectively. Prove that the line through \(A\) perpendicular to \(QX\), the line through \(B\) perpendicular to \(DY\), and the line through \(C\) perpendicular to \(EP\) are concurrent.