International Competitions 2013

Prove that for any pair of positive integers $k$ and $n$, there exist $k$ positive integers $m_1, m_2, \ldots, m_k$ (not necessarily different) such that

$$1 + \frac{2^k - 1}{n} = \left(1 + \frac{1}{m_1}\right)\left(1 + \frac{1}{m_2}\right) \cdots \left(1 + \frac{1}{m_k}\right).$$

A configuration of 4027 points in the plane is called Colombian if it consists of 2013 red points and 2014 blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied:

• no line passes through any point of the configuration;
• no region contains points of both colours.

Find the least value of $k$ such that for any Colombian configuration of 4027 points, there is a good arrangement of $k$ lines.

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled.

The excircle of triangle $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$. The excircles opposite $B$ and $C$ are similarly defined.

Let $ABC$ be an acute-angled triangle with orthocentre $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of $CWM$, and let $Y$ be the point on $\omega_2$ such that $WY$ is a diameter of $\omega_2$. Prove that $X$, $Y$ and $H$ are collinear.

Let $\mathbb{Q}_{>0}$ be the set of positive rational numbers. Let $f: \mathbb{Q}_{>0} \to \mathbb{R}$ be a function satisfying the following three conditions:

(i) for all $x, y \in \mathbb{Q}_{>0}$, we have $f(x)f(y) \geq f(xy)$;

(ii) for all $x, y \in \mathbb{Q}_{>0}$, we have $f(x + y) \geq f(x) + f(y)$;

(iii) there exists a rational number $a > 1$ such that $f(a) = a$.

Prove that $f(x) = x$ for all $x \in \mathbb{Q}_{>0}$.

Let $n \geq 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, \ldots, n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels $a < b < c < d$ with $a + d = b + c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$.

Let $M$ be the number of beautiful labellings, and let $N$ be the number of ordered pairs $(x,y)$ of positive integers such that $x + y \leq n$ and $\gcd(x,y) = 1$. Prove that

$$M = N + 1.$$

Let $a, b, c$ be positive real numbers such that $$a + b + c = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$$

Prove that $$2(a + b + c) \geq \sqrt[3]{7a^2b + 1} + \sqrt[3]{7b^2c + 1} + \sqrt[3]{7c^2a + 1}.$$

Find all triples $(a,b,c)$ for which equality holds.

Let $n$ be a positive integer. On a board consisting of $4n \times 4n$ squares, exactly $4n$ tokens are placed so that each row and each column contains one token. In a step, a token is moved horizontally or vertically to a neighbouring square. Several tokens may occupy the same square at the same time. The tokens are to be moved to occupy all the squares of one of the two diagonals.

Determine the smallest number $k(n)$ such that for any initial situation, we can do it in at most $k(n)$ steps.

Let $ABC$ be an isosceles triangle with $AC = BC$. Let $N$ be a point inside the triangle such that $2\angle ANB = 180° + \angle ACB$. Let $D$ be the intersection of the line $BN$ and the line parallel to $AN$ that passes through $C$. Let $P$ be the intersection of the angle bisectors of the angles $CAN$ and $ABN$.

Show that the lines $DP$ and $AN$ are perpendicular.

Let $a$ and $b$ be positive integers. Prove that there exist positive integers $x$ and $y$ such that $$\binom{x + y}{2} = ax + by.$$

Find all functions $f\colon\mathbb{R}\to\mathbb{R}$ such that $$f(xf(x)+2y)=f(x^{2})+f(y)+x+y-1$$ for all $x$, $y\in\mathbb{R}$.

Let $x$, $y$, $z$, $w\in\mathbb{R}\setminus\{0\}$ such that $x+y\neq 0$, $z+w\neq 0$, and $xy+zw\geq 0$. Prove the inequality $$\left(\frac{x+y}{z+w}+\frac{z+w}{x+y}\right)^{-1}+\frac{1}{2}\geq\left(\frac{x}{z}+\frac{z}{x}\right)^{-1}+\left(\frac{y}{w}+\frac{w}{y}\right)^{-1}.$$

There are $n\geq 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from $1$ to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the current number plates are swapped exactly once during the day.

How many different sequences of number plates are possible at the end of the day?

Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.

What is the maximal possible number of points with this property?

Let $ABC$ be an acute triangle. Construct a triangle $PQR$ such that $AB = 2PQ$, $BC = 2QR$, $CA = 2RP$, and the lines $PQ$, $QR$, and $RP$ pass through the points $A$, $B$, and $C$, respectively. (All six points $A$, $B$, $C$, $P$, $Q$, and $R$ are distinct.)

Let $K$ be a point inside an acute triangle $ABC$, such that $BC$ is a common tangent of the circumcircles of $AKB$ and $AKC$. Let $D$ be the intersection of the lines $CK$ and $AB$, and let $E$ be the intersection of the lines $BK$ and $AC$. Let $F$ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $F$ and radius $FD$ intersect at points $P$ and $Q$.

Prove that the segment $PQ$ is a diameter of $k$.

The numbers from $1$ to $2013^{2}$ are written row by row into a table consisting of $2013 \times 2013$ cells. Afterwards, all columns and all rows containing at least one of the perfect squares $1,4,9,\ldots,2013^{2}$ are simultaneously deleted.

How many cells remain?

The expression $$\pm \square \pm \square \pm \square \pm \square \pm \square \pm \square$$ is written on the blackboard. Two players, $A$ and $B$, play a game, taking turns. Player $A$ takes the first turn. In each turn, the player on turn replaces a symbol $\square$ by a positive integer. After all the symbols $\square$ are replaced, player $A$ replaces each of the signs $\pm$ by either $+$ or $-$, independently of each other. Player $A$ wins if the value of the expression on the blackboard is not divisible by any of the numbers $11, 12, \ldots, 18$. Otherwise, player $B$ wins.

Determine which player has a winning strategy.