Find all pairs \((m, n)\) of positive integers such that \(\frac{m^2}{2mn^2-n^3+1}\) is a positive integer.

S is the set \(\left\{1,\;2,\;3,\;\dots,\;1000000\right\}\). Show that for any subset \(A\) of \(S\) with \(101\) elements we can find \(100\) distinct elements \(x_i\) of \(S\), such that the sets \(\left\{a+x_i\vert a\in A\right\}\) are all pairwise disjoint.

\(\qquad\)We call a positive integer \(\it{alternating}\) if every two consecutive digits in its decimal representation are of different parity. \(\\\qquad\)Find all positive integers \(n\) such that \(n\) has a multiple which is alternating.

\(\qquad\)In a convex quadrilateral \(ABCD\) the diagonal \(BD\) does not bisect the angles \(ABC\) and \(CDA\). The point \(P\) lies inside \(ABCD\) and satisfies $$ \angle PBC = \angle DBA \;\; and \;\; \angle PDC = \angle BDA. $$ \(\qquad\)Prove that \(ABCD\) is a cyclic quadrilateral if and only if \(AP = CP\).

\(\qquad\)Let \(n \ge 3\) be an integer. Let \(t_1, t_2, \ldots, t_n\) be positive real numbers such that $$ n^2 + 1 > (t_1 + t_2 + \ldots + t_n) \left(\frac1{t_1}+\frac1{t_2}+\dots+\frac1{t_n}\right). $$ \(\qquad\)Show that \(t_i, t_j , t_k\) are side lengths of a triangle for all \(i, j, k\) with \(1 \le i \lt j \lt k \le n\).

\(\qquad\)Define a ”\(\it{hook}\)” to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. \(\\\it{attantion!\ ness.\ figure}\) \(\\\qquad\)Determine all \(m \times n\) rectangles that can be covered without gaps and without overlaps with hooks such that \(\\\qquad\bullet\) the rectangle is covered without gaps and without overlaps \(\\\qquad\bullet\) no part of a hook covers area outside the rectagle.

Find all polynomials \(f\) with real coefficients such that for all reals \(a, b, c\) such that \(ab + bc + ca = 0\) we have the following relations $$ f(a − b) + f(b − c) + f(c − a) = 2f(a + b + c). $$

\(\qquad\)Let \(ABC\) be an acute-angled triangle with \(AB \ne AC\). The circle with diameter \(BC\) intersects the sides \(AB\) and \(AC\) at \(M\) and \(N\) respectively. Denote by \(O\) the midpoint of the side \(BC\). The bisectors of the angles \(\angle BAC\) and \(\angle MON\) intersect at \(R\). \(\\\qquad\)Prove that the circumcircles of the triangles \(BMR\) and \(CNR\) have a common point lying on the side \(BC\).

\(\qquad\)In a mathematical competition, in which \(6\) problems were posed to the participants, every two of these problems were solved by more than \(\frac25\) of the contestants. Moreover, no contestant solved all the \(6\) problems. \(\\\qquad\)Show that there are at least \(2\) contestants who solved exactly \(5\) problems each.

\(\qquad\)Let \(ABCD\) be a fixed convex quadrilateral with \(BC = DA\) and \(BC\) not parallel with \(DA\). Let two variable points \(E\) and \(F\) lie of the sides \(BC\) and \(DA\), respectively and satisfy \(BE = DF\). The lines \(AC\) and \(BD\) meet at \(P\), the lines \(BD\) and \(EF\) meet at \(Q\), the lines \(EF\) and \(AC\) meet at \(R\). \(\\\qquad\)Prove that the circumcircles of the triangles \(PQR\), as \(E\) and \(F\) vary, have a common point other than \(P\).