International Competitions 2016

Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA = FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen such that $DA = DC$ and $AC$ is the bisector of $\angle DAB$. Point $E$ is chosen such that $EA = ED$ and $AD$ is the bisector of $\angle EAC$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram (where $AM \parallel EX$ and $AE \parallel MX$). Prove that lines $BD$, $FX$, and $ME$ are concurrent.

Find all positive integers $n$ for which each cell of an $n \times n$ table can be filled with one of the letters $I$, $M$ and $O$ in such a way that:

• in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and
• in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.

Note: The rows and columns of an $n \times n$ table are each labelled 1 to $n$ in a natural order. Thus each cell corresponds to a pair of positive integers $(i,j)$ with $1 \leq i,j \leq n$. For $n > 1$, the table has $4n - 2$ diagonals of two types. A diagonal of the first type consists of all cells $(i,j)$ for which $i + j$ is a constant, and a diagonal of the second type consists of all cells $(i,j)$ for which $i - j$ is a constant.

Let $P = A_1A_2\ldots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n) = n^2 + n + 1$. What is the least possible value of the positive integer $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a + 1), P(a + 2), \ldots, P(a + b)\}$$ is fragrant?

The equation $$(x - 1)(x - 2) \cdots (x - 2016) = (x - 1)(x - 2) \cdots (x - 2016)$$ is written on the board, with 2016 linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these 4032 linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

There are $n \geq 2$ line segments in the plane such that every two segments cross, and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it, facing the other endpoint. Then he will clap his hands $n - 1$ times. Every time he claps, each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.

(a) Prove that Geoff can always fulfil his wish if $n$ is odd.

(b) Prove that Geoff can never fulfil his wish if $n$ is even.

Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be real numbers satisfying

(a) $x_j > -1$ for $j = 1, 2, \ldots, n$ and

(b) $x_1 + x_2 + \cdots + x_n = n$.

Prove the inequality $$\sum_{j=1}^n \frac{1}{1 + x_j} \geq \sum_{j=1}^n \frac{x_j}{1 + x_j^2}$$

and determine when equality holds.

There are $n \geq 3$ positive integers written on a blackboard. A move consists of choosing three numbers $a, b, c$ on the blackboard such that they are the sides of a non-degenerate non-equilateral triangle and replacing them by $a + b - c$, $b + c - a$ and $c + a - b$.

Show that an infinite sequence of moves cannot exist.

Let $ABC$ be an acute-angled triangle with $\measuredangle BAC > 45°$ and with circumcentre $O$. The point $P$ lies in its interior such that the points $A, P, O, B$ lie on a circle and $BP$ is perpendicular to $CP$. The point $Q$ lies on the segment $BP$ such that $AQ$ is parallel to $PO$.

Prove that $\measuredangle QCB = \measuredangle PCO$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$.

Remark: $\mathbb{N} = \{1, 2, 3, \ldots\}$ denotes the set of positive integers.

Determine all triples $(a, b, c)$ of real numbers satisfying the system of equations $$a^2 + ab + c = 0,$$ $$b^2 + bc + a = 0,$$ $$c^2 + ca + b = 0.$$

Let $\mathbb{R}$ denote the set of real numbers. Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that $$f(x)f(y) = x f(f(y - x)) + x f(2x) + f(x^2)$$ holds for all real numbers $x$ and $y$.

A tract of land in the shape of an $8 \times 8$ square, whose sides are oriented north-south and east-west, consists of $64$ smaller $1 \times 1$ square plots. There can be at most one house on each of the individual plots. A house can only occupy a single $1 \times 1$ square plot.

A house is said to be blocked from sunlight if there are three houses on the plots immediately to its east, west and south.

What is the maximum number of houses that can simultaneously exist, such that none of them is blocked from sunlight?

Remark: By definition, houses on the east, west and south borders are never blocked from sunlight.

A class of high school students wrote a test. Every question was graded as either $1$ point for a correct answer or $0$ points otherwise. It is known that each question was answered correctly by at least one student and the students did not all achieve the same total score.

Prove that there was a question on the test with the following property: The students who answered the question correctly got a higher average test score than those who did not.

Let $ABC$ be an acute-angled triangle with $AB \neq AC$, and let $O$ be its circumcentre. The line $AO$ intersects the circumcircle $\omega$ of $ABC$ a second time in point $D$, and the line $BC$ in point $E$. The circumcircle of $CDE$ intersects the line $CA$ a second time in point $P$. The line $PE$ intersects the line $AB$ in point $Q$. The line through $O$ parallel to $PE$ intersects the altitude of the triangle $ABC$ that passes through $A$ in point $F$.

Prove that $FP = FQ$.

Let $ABC$ be a triangle with $AB \neq AC$. The points $K, L, M$ are the midpoints of the sides $BC, CA, AB$, respectively. The inscribed circle of $ABC$ with centre $I$ touches the side $BC$ at point $D$. The line $g$, which passes through the midpoint of segment $ID$ and is perpendicular to $IK$, intersects the line $LM$ at point $P$.

Prove that $\measuredangle PIA = 90^{\circ}$.

A positive integer $n$ is called a Mozartian number if the numbers $1, 2, \ldots, n$ together contain an even number of each digit (in base $10$).

Prove:

(a) All Mozartian numbers are even.

(b) There are infinitely many Mozartian numbers.

We consider the equation $a^2 + b^2 + c^2 + n = abc$, where $a, b, c$ are positive integers.

Prove:

(a) There are no solutions $(a,b,c)$ for $n = 2017$.

(b) For $n = 2016$, $a$ must be divisible by $3$ for every solution $(a, b, c)$.

(c) The equation has infinitely many solutions $(a, b, c)$ for $n = 2016$.