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Shortlisted Problems with Solutions

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Shortlisted Problems with Solutions

54th International Mathematical Olympiad
Santa Marta, Colombia 2013

Note of Confidentiality

The Shortlisted Problems should be kept
strictly confidential until IMO 2014.

Contributing Countries
The Organizing Committee and the Problem Selection Committee of IMO 2013 thank the following
50 countries for contributing 149 problem proposals.

Argentina, Armenia, Australia, Austria, Belgium, Belarus, Brazil, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, El Salvador, Estonia, Finland,
France, Georgia, Germany, Greece, Hungary, India, Indonesia, Iran, Ireland,
Israel, Italy, Japan, Latvia, Lithuania, Luxembourg, Malaysia, Mexico,
Netherlands, Nicaragua, Pakistan, Panama, Poland, Romania, Russia,
Saudi Arabia, Serbia, Slovenia, Sweden, Switzerland, Tajikistan, Thailand,
Turkey, U.S.A., Ukraine, United Kingdom

Problem Selection Committee
Federico Ardila (chairman)
Ilya I. Bogdanov
G´eza K´os
Carlos Gustavo Tamm de Ara´
ujo Moreira (Gugu)
Christian Reiher

Shortlisted problems

3

Problems
Algebra
A1.

Let n be a positive integer and let a1 , . . . , an´1 be arbitrary real numbers. Define the
sequences u0 , . . . , un and v0 , . . . , vn inductively by u0 “ u1 “ v0 “ v1 “ 1, and
uk`1 “ uk ` ak uk´1 ,

vk`1 “ vk ` an´k vk´1

for k “ 1, . . . , n ´ 1.

Prove that un “ vn .

(France)
A2. Prove that in any set of 2000 distinct real numbers there exist two pairs a ą b and c ą d
with a ‰ c or b ‰ d, such that
ˇ
ˇ
ˇa ´ b
ˇ
1
ˇ
ˇ
ˇ c ´ d ´ 1ˇ ă 100000 .

A3. Let Qą0 be the set of positive rational numbers. Let f : Qą0
the conditions

(Lithuania)
Ñ R be a function satisfying

f pxqf pyq ě f pxyq and f px ` yq ě f pxq ` f pyq

for all x, y P Qą0 . Given that f paq “ a for some rational a ą 1, prove that f pxq “ x for all
x P Qą0 .

(Bulgaria)
A4. Let n be a positive integer, and consider a sequence a1 , a2 , . . . , an of positive integers.
Extend it periodically to an infinite sequence a1 , a2 , . . . by defining an`i “ ai for all i ě 1. If
a1 ď a2 ď ¨ ¨ ¨ ď an ď a1 ` n
and
aai ď n ` i ´ 1

prove that

for i “ 1, 2, . . . , n,

a1 ` ¨ ¨ ¨ ` an ď n2 .
(Germany)
A5. Let Zě0 be the set of all nonnegative integers. Find all the functions f : Zě0 Ñ Zě0
satisfying the relation
f pf pf pnqqq “ f pn ` 1q ` 1

for all n P Zě0 .

(Serbia)
A6. Let m ‰ 0 be an integer. Find all polynomials P pxq with real coefficients such that
px3 ´ mx2 ` 1qP px ` 1q ` px3 ` mx2 ` 1qP px ´ 1q “ 2px3 ´ mx ` 1qP pxq
for all real numbers x.
(Serbia)

4

IMO...
Shortlisted Problems with Solutions
54
th
Internation al Mathematical Olympiad
Santa Marta, Colombia 2013
Shortlisted Problems with Solutions - Trang 2
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