The 54th International Mathematical Olympiad: Problems and Solutions

Day 1 (July 23th, 2013)

Problem 1
 

Assume that \( k \) and \( n \) are two positive integers. Prove that there exist positive integers \( m_1 \), \( \dots \), \( m_k \) such that \[ 1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\]

Problem 2
 

Given \( 2013 \) red and \( 2014 \) blue points in the plane, assume that no three of the given points lie on a line. A partition of the plane is called perfect if no region contains points of different colors. Determine the smallest number \( k \) for which a perfect partition can always be achieved by drawing \( k \) straight lines.

Problem 3
 

Let \( A_1 \), \( B_1 \), and \( C_1 \) be the points at which the excircles touch the sides \( BC \), \( CA \), and \( AB \) of the triangle \( ABC \). Prove that if the circumcenter of \( \triangle A_1B_1C_1 \) belongs to the circumcircle of \( \triangle ABC \), then one of the angles of \( \triangle ABC \) is \( 90^{\circ} \).

Day 2 (July 24th, 2013)

Problem 4
 

Let \( ABC \) be an acute triangle with orthocenter \( H \), and let \( W \) be a point on the side \( BC \) between \( B \) and \( C \). The points \( M \) and \( N \) are the feet of perpendiculars from \( B \) and \( C \), respectively. Let \( \omega_1 \) be the circumcircle of \( \triangle BWN \), and let \( X \) be the point such that \( WX \) is a diameter of \( \omega_1 \). Let \( \omega_2 \) be the circumcircle of \( \triangle CWM \), and let \( Y \) be the point such that \( WY \) is a diameter of \( \omega_2 \). Prove that the points \( X \), \( Y \), and \( H \) are collinear.

Problem 5
 

Let \( \mathbb Q_+ \) be the set of all positive rational numbers. Let \( f:\mathbb Q_+\to\mathbb R \) be a function that satisfies the following three conditions:

  • (i) \( f(x)f(y)\geq f(xy) \) for all \( x,y\in\mathbb Q_+ \),

  • (ii) \( f(x+y)\geq f(x)+f(y) \) for all \( x,y\in\mathbb Q_+ \),

  • (iii) There exists a rational number \( a> 1 \) such that \( f(a)=a \).

Prove that \( f(x)=x \) for all \( x\in\mathbb Q_+ \).

Problem 6
 

Given an integer \( n\geq 3 \), assume that \( n+1 \) equally spaced points are marked on a circle. Consider all labelings of these points with the numbers \( 0 \), \( 1 \), \( \dots \), \( n \) such that each label is used exactly once. Two labelings are considered the same if one can be obtained from the other by a rotation of the circle. A labeling is beautiful if, for any four labels \( a< b< c< d \) with \( a+d=b+c \), the chord joining points labeled \( a \) and \( d \) does not intersect the chord joining points labeled \( b \) and \( c \).

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


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