Day \(1\)

 Friday, July 10, 2015

Problem 1. We say that a finite set \(\mathcal S\) of points in the plane is balanced if, for any two different points \(A\) and \(B\) in \(\mathcal{S}\), there is a point \(C\) in \(\mathcal{S}\) such that \(AC=BC\). We say that \(\mathcal{S}\) is centre-free if for any three different points \(A\), \(B\) and \(C\) in \(\mathcal{S}\), there is no points \(P\) in \(\mathcal{S}\) such that \(PA=PB=PC\).

(a) Show that for all integers \(n\ge 3\), there exists a balanced set consisting of \(n\) points.

(b) Determine all integers \(n\ge 3\) for which there exists a balanced centre-free set consisting of \(n\) points.

Problem 2.  Determine all triples \((a, b, c)\) of positive integers such that each of the numbers

\[ab-c,\;bc-a, \;ca-b\]

is a power of \(2\).

(A power of \(2\) is an integer of the form \(2^n\), where \(n\) is a non-negative integer. )

Problem 3. Let \(ABC\) be an acute triangle with \(AB \gt AC\). Let \(\Gamma\) be its cirumcircle, \(H\) its orthocenter, and \(F\) the foot of the altitude from \(A\). Let \(M\) be the midpoint of \(BC\). Let \(Q\) be the point on \(\Gamma\) such that \(\angle HQA = 90^{\circ}\) and let \(K\) be the point on \(\Gamma\) such that \(\angle HKQ = 90^{\circ}\). Assume that the points \(A\), \(B\), \(C\), \(K\) and \(Q\) are all different, and lie on \(\Gamma\) in this order.

Prove that the circumcircles of triangles \(KQH\) and \(FKM\) are tangent to each other.

                                      Day \(2\)

 Saturday, July 11, 2015

Problem 4. Triangle \(ABC\) has circumcircle \(\Omega\) and circumcenter \(O\). A circle \(\Gamma\) with center \(A\) intersects the segment \(BC\) at points \(D\) and \(E\), such that \(B\), \(D\), \(E\), and \(C\) are all different and lie on line \(BC\) in this order. Let \(F\) and \(G\) be the points of intersection of \(\Gamma\) and \(\Omega\), such that \(A\), \(F\), \(B\), \(C\), and \(G\) lie on \(\Omega\) in this order. Let \(K\) be the second point of intersection of the circumcircle of triangle \(BDF\) and the segment \(AB\). Let \(L\) be the second point of intersection of the circumcircle of triangle \(CGE\) and the segment \(CA\).

Suppose that the lines \(FK\) and \(GL\) are different and intersect at the point \(X\). Prove that \(X\) lies on the line \(AO\).

Problem 5. Let \(\Bbb R\) be the set of real numbers. Determine all functions \(f\colon\Bbb R\to\Bbb R\) satisfying the equation

\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]

for all real numbers \(x\) and \(y\).

Problem 6. The sequence \(a_1,a_2,\dotsc\) of integers satisfies the following conditions:

(i) \(1\leqslant a_j\leqslant2015\) for all \(j\geqslant1\);

(ii) \(k+a_k\neq \ell+a_\ell\) for all \(1\leqslant k\lt \ell\).

Prove that there exist two positive integers \(b\) and \(N\) such that

\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\leqslant1007^2\]

for all integers \(m\) and \(n\) satisfying \(n\gt m\geqslant N\).