MathDB

2008 IMO Shortlist

Part of IMO Shortlist

Subcontests

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IMO ShortList 2008, Combinatorics problem 1

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest n n for which there exist n n boxes B1 B_1, … \ldots, Bn B_n such that Bi B_i and Bj B_j intersect if and only if i≢j±1(modn) i\not\equiv j\pm 1\pmod n. Proposed by Gerhard Woeginger, Netherlands

IMO ShortList 2008, Number Theory problem 1

Let nn be a positive integer and let pp be a prime number. Prove that if aa, bb, cc are integers (not necessarily positive) satisfying the equations an+pb=bn+pc=cn+pa a^n + pb = b^n + pc = c^n + pa then a=b=ca = b = c.
Proposed by Angelo Di Pasquale, Australia
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IMO ShortList 2008, Algebra problem 6

Let f:R→N f: \mathbb{R}\to\mathbb{N} be a function which satisfies f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right) for all x x, y∈R y\in\mathbb{R}. Prove that there is a positive integer which is not a value of f f.
Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania

IMO Shortlist 2008, Geometry problem 6

There is given a convex quadrilateral ABCD ABCD. Prove that there exists a point P P inside the quadrilateral such that \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} if and only if the diagonals AC AC and BD BD are perpendicular.
Proposed by Dusan Djukic, Serbia

IMO ShortList 2008, Combinatorics problem 6

For n≥2 n\ge 2, let S1 S_1, S2 S_2, … \ldots, S2n S_{2^n} be 2n 2^n subsets of A \equal{} \{1, 2, 3, \ldots, 2^{n \plus{} 1}\} that satisfy the following property: There do not exist indices a a and b b with a<b a < b and elements x x, y y, z∈A z\in A with x<y<z x < y < z and y y, z∈Sa z\in S_a, and x x, z∈Sb z\in S_b. Prove that at least one of the sets S1 S_1, S2 S_2, … \ldots, S2n S_{2^n} contains no more than 4n 4n elements. Proposed by Gerhard Woeginger, Netherlands
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IMO ShortList 2008, Algebra problem 4

For an integer m m, denote by t(m) t(m) the unique number in {1,2,3} \{1, 2, 3\} such that m \plus{} t(m) is a multiple of 3 3. A function f:Z→Z f: \mathbb{Z}\to\mathbb{Z} satisfies f( \minus{} 1) \equal{} 0, f(0) \equal{} 1, f(1) \equal{} \minus{} 1 and f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m) for all integers m m, n≥0 n\ge 0 with 2n>m 2^n > m. Prove that f(3p)≥0 f(3p)\ge 0 holds for all integers p≥0 p\ge 0. Proposed by Gerhard Woeginger, Austria

IMO ShortList 2008, Number Theory problem 4

Let n n be a positive integer. Show that the numbers \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1} are congruent modulo 2n 2^n to 1 1, 3 3, 5 5, … \ldots, 2^n \minus{} 1 in some order. Proposed by Duskan Dukic, Serbia

IMO Shortlist 2008, Geometry problem 4

In an acute triangle ABC ABC segments BE BE and CF CF are altitudes. Two circles passing through the point A A and F F and tangent to the line BC BC at the points P P and Q Q so that B B lies between C C and Q Q. Prove that lines PE PE and QF QF intersect on the circumcircle of triangle AEF AEF.
Proposed by Davood Vakili, Iran
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Permutations and divisibility

Let n∈Nn \in \mathbb N and AnA_n set of all permutations (a1,…,an)(a_1, \ldots, a_n) of the set {1,2,…,n}\{1, 2, \ldots , n\} for which k∣2(a1+⋯+ak), for all 1≤k≤n.k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n. Find the number of elements of the set AnA_n.
Proposed by Vidan Govedarica, Serbia

IMO Shortlist 2008, Geometry problem 2

Given trapezoid ABCD ABCD with parallel sides AB AB and CD CD, assume that there exist points E E on line BC BC outside segment BC BC, and F F inside segment AD AD such that \angle DAE \equal{} \angle CBF. Denote by I I the point of intersection of CD CD and EF EF, and by J J the point of intersection of AB AB and EF EF. Let K K be the midpoint of segment EF EF, assume it does not lie on line AB AB. Prove that I I belongs to the circumcircle of ABK ABK if and only if K K belongs to the circumcircle of CDJ CDJ. Proposed by Charles Leytem, Luxembourg

IMO ShortList 2008, Number Theory problem 2

Let a1 a_1, a2 a_2, … \ldots, an a_n be distinct positive integers, n≥3 n\ge 3. Prove that there exist distinct indices i i and j j such that a_i \plus{} a_j does not divide any of the numbers 3a1 3a_1, 3a2 3a_2, … \ldots, 3an 3a_n. Proposed by Mohsen Jamaali, Iran