International Competitions 2015

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 point $P$ in $\mathcal{S}$ such that $PA = PB = PC$.

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

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

Determine all triples $(a, b, c)$ of positive integers such that each of the numbers $$ab - c, \quad bc - a, \quad ca - b$$ is a power of 2.

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

Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocentre, 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 $\angle HQA = 90°$, and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90°$. 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.

Triangle $ABC$ has circumcircle $\Omega$ and circumcentre $O$. A circle $\Gamma$ with centre $A$ 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 $AO$.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying the equation $$f \big (x + f (x + y) \big) + f (xy) = x + f (x + y) + yf (x)$$ for all real numbers $x$ and $y$.

The sequence $a_1, a_2, \ldots$ of integers satisfies the following conditions:

(i) $1 \leqslant a_{j} \leqslant 2015$ for all $j \geqslant 1$;

(ii) $k + a_{k} \neq \ell + a_{\ell}$ for all $1 \leqslant k < \ell$.

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

Find all surjective functions $f: \mathbb{N} \to \mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true: $$f(a) = f(b),$$ $$f(a + b) = \min\{f(a), f(b)\}.$$

Remarks: $\mathbb{N}$ denotes the set of all positive integers. A function $f: X \to Y$ is said to be surjective if for every $y \in Y$ there exists $x \in X$ such that $f(x) = y$.

Let $n \geq 3$ be an integer. An inner diagonal of a simple $n$-gon is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P) = D(n)$.

Remark: A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.

Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$, respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$, respectively. Prove that points $C$, $D$, $F$, and $G$ lie on a circle.

Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $$\frac{a^m + b^m}{a^n + b^n}$$ is an integer.

Prove that for all positive real numbers $a$, $b$, $c$ such that $abc = 1$ the following inequality holds: $$\frac{a}{2b + c^2} + \frac{b}{2c + a^2} + \frac{c}{2a + b^2} \leq \frac{a^2 + b^2 + c^2}{3}.$$

Determine all functions $f: \mathbb{R}\backslash\{0\} \to \mathbb{R}\backslash\{0\}$ such that $$f(x^2 y f(x)) + f(1) = x^2 f(x) + f(y)$$ holds for all nonzero real numbers $x$ and $y$.

There are $n$ students standing in line in positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i - j|$ steps. Determine the maximal sum of steps of all students that they can achieve.

Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N \times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N \times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$.

Example with $N = 3$, $K = 7$

Let $ABC$ be an acute triangle with $AB > AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and $$\angle AXB - \angle ACB = \angle CYA - \angle CBA$$ holds, the line $XY$ passes through $D$.

Let $I$ be the incentre of triangle $ABC$ with $AB > AC$ and let the line $AI$ intersect the side $BC$ at $D$. Suppose that point $P$ lies on the segment $BC$ and satisfies $PI = PD$. Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $BC$, and let $Q$ be the other intersection of the circumcircles of the triangles $ABC$ and $APD$. Prove that $\angle BAQ = \angle CAJ$.

Find all pairs of positive integers $(a,b)$ such that $$a! + b! = a^b + b^a.$$

Let $n \geq 2$ be an integer. Determine the number of positive integers $m$ such that $m \leq n$ and $m^2 + 1$ is divisible by $n$.