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 $$n\gt m\geqslant N$$.

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