IMO 2015 Problems

Day 1

1. We say that 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 $n \geq 3$, there exists a balanced set consisting of $n$ points.

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

2. Find all positive integers $(a,b,c)$ such that $$ab - c, \ bc - a, \ ca - b$$ are all powers of $2$.

3. Let $ABC$ be an acute triangle with $AB > BC$. Let $\Gamma$ be its circumcircle, $H$ 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 $KHQ$ and $FKM$ are tangent to each other.

Day 2

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 cicumcircle 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 all different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

5. Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the equation $$f(x + f(x+y)) +f(xy) = x + f(x+y) + yf(x)$$
for all real numbers $x$ and $y$.

6. The sequnce $a_{1}, a_{2}, . . .$ of integers satisfies the conditions :
(i) $1 \leq a_{j} \leq 2015$ for all $j \geq 1$,
(ii) $k + a_{k} \neq l + a_{l}$ for all $1 \leq k < l$.

Prove that there exist two positive integers $b$ and $N$ for which $$\left | \sum_{j = m + 1}^{n}(a_{j} - b) \right | \leq 1007^{2}$$ for all integers $m$ and $n$ such that $n > m \geq N$.