# [Shortlists] International Mathematical Olympiad 2015

### Algebra

1. Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies $a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}$for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
2. Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that $f(x-f(y))=f(f(x))-f(y)-1$holds for all $x,y\in\mathbb{Z}$.
3. Let $n$ be a fixed positive integer. Find the maximum possible value of $\sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s,$where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
4. Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation$f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$for all real numbers $x$ and $y$.
5. Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying $f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y)$for every $x, y \in \mathbb{Z}$.
6. Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are block-similar if for each $i \in \{1, 2, \ldots, n\}$ the sequences $$\begin{eqnarray*} P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \\ Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014) \end{eqnarray*}$$ are permutations of each other.
a) Prove that there exist distinct block-similar polynomials of degree $n + 1$.
b) Prove that there do not exist distinct block-similar polynomials of degree $n$.

### Combinatorics

1. In Lineland there are $n\geq1$ towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the $2n$ bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes. Let $A$ and $B$ be two towns, with $B$ to the right of $A$. We say that town $A$ can sweep town $B$ away if the right bulldozer of $A$ can move over to $B$ pushing off all bulldozers it meets. Similarly town $B$ can sweep town $A$ away if the left bulldozer of $B$ can move over to $A$ pushing off all bulldozers of all towns on its way. Prove that there is exactly one town that cannot be swept away by any other one.
2. 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.
3. For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is good if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
4. Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
• A player cannot choose a number that has been chosen by either player on any previous turn.
• A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
• The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
5. The sequence $a_1,a_2,\dots$ of integers satisfies the conditions
• $1\le a_j\le2015$ for all $j\ge1$,
• $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$.
Prove that there exist two positive integers $b$ and $N$ for which$\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2$for all integers $m$ and $n$ such that $n>m\ge N$.
6. Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is clean if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
7. In a company of people some pairs are enemies. A group of people is called unsociable if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part.

### Geometry

1. Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
2. 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 $AO$.
3. Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
4. Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
5. Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
6. Let $ABC$ be an acute triangle with $AB > AC$. 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.
7. Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
8. A triangulation of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a Thaiangulation if all triangles in it have the same area. Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)

### Number Theory

1. Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by $a_0 = M + \frac{1}{2}, \quad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots$contains at least one integer term.
2. Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
3. Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
4. Suppose that $a_0, a_1, \cdots$ and $b_0, b_1, \cdots$ are two sequences of positive integers such that $a_0, b_0 \ge 2$ and $a_{n+1} = \gcd{(a_n, b_n)} + 1, \quad b_{n+1} = \operatorname{lcm}{(a_n, b_n)} - 1.$ Show that the sequence $a_n$ is eventually periodic; in other words, there exist integers $N \ge 0$ and $t > 0$ such that $a_{n+t} = a_n$ for all $n \ge N$.
5. Find all postive integers $(a,b,c)$ such that $$ab-c,\quad bc-a,\quad ca-b$$are all powers of $2$.
6. Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties
• if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
• the set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1$, $f(2) - 2$, $f(3) - 3$, $\ldots$ is periodic.
7. Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
8. For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define $\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.$That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that $\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.$
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