2015 IMO Problems

Problem 1

We say that a finite set $\mathcal{S}$ in the plane is balanced if, for any two different points $A$ , $B$ 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 points $A$ , $B$ , $C$ in $\mathcal{S}$ , there is no point $P$ in $\mathcal{S}$ such that $PA=PB=PC$ .

Show that for all integers $n\geq 3$ , there exists a balanced set consisting of $n$ points.
Determine all integers $n\geq 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Solution

Problem 2

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

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

Solution

Problem 3

Let $ABC$ be an acute triangle with $AB>AC$ . 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 $\angle HKQ=90^\circ$ . 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.

Solution

Problem 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 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 $AO$ .

Solution

Problem 5

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f$ : $\mathbb{R}\rightarrow\mathbb{R}$ satisfying the equation

$f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)$

for all real numbers $x$ and $y$ .

Solution

Problem 6

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

(i) $1\le a_j\le2015$ for all $j\ge1$ ,
(ii) $k+a_k\neq \ell+a_\ell$ 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 $n>m\ge N$ .

Solution

Problem 1

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$ and $b$ , $f(2a) + 2f(b) = f(f(a + b)).$

Solution

Problem 2

In triangle $ABC$ , point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$ . Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$ , respectively, such that $PQ$ is parallel to $AB$ . Let $P_1$ be a point on line $PB_1$ , such that $B_1$ lies strictly between $P$ and $P_1$ , and $\angle PP_1C=\angle BAC$ . Similarly, let $Q_1$ be the point on line $QA_1$ , such that $A_1$ lies strictly between $Q$ and $Q_1$ , and $\angle CQ_1Q=\angle CBA$ .

Prove that points $P,Q,P_1$ , and $Q_1$ are concyclic.

Solution

Problem 3

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$ , user $B$ is also friends with user $A$ . Events of the following kind may happen repeatedly, one at a time: Three users $A$ , $B$ , and $C$ such that $A$ is friends with both $B$ and $C$ , but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$ , and no longer friends with $C$ . All other friendship statuses are unchanged. Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

Solution

Problem 4

Find all pairs $(k,n)$ of positive integers such that

$k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).$

Solution

Problem 5

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation:

If there are exactly $k > 0$ coins showing $H$ , then he turns over the $k^{th}$ coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n = 3$ the process starting with the configuration $THT$ would be $THT \rightarrow HHT \rightarrow HTT \rightarrow TTT$ , which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$ , let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$ . Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$ .

Solution

Problem 6

Let $I$ be the incenter of acute triangle $ABC$ with $AB \neq AC$ . The incircle $\omega$ of $ABC$ is tangent to sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. The line through $D$ perpendicular to $EF$ meets ω again at $R$ . Line $AR$ meets ω again at $P$ . The circumcircles of triangles $PCE$ and $PBF$ meet again at $Q$ . Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$ .

Solution

2015 IMO (Problems) • Resources
Preceded by 2014 IMO Problems	1 • 2 • 3 • 4 • 5 • 6	Followed by 2016 IMO Problems
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2015 IMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6