Shortlisted Problems with Solutions
53rd International Mathematical Olympiad
Mar del Plata, Argentina 2012
,
,Note of Confidentiality
The shortlisted problems should be kept
strictly confidential until IMO 2013
Contributing Countries
The Organizing Committee and the Problem Selection Committee of IMO 2012 thank the
following 40 countries for contributing 136 problem proposals:
Australia, Austria, Belarus, Belgium, Bulgaria, Canada, Cyprus,
Czech Republic, Denmark, Estonia, Finland, France, Germany,
Greece, Hong Kong, India, Iran, Ireland, Israel, Japan,
Kazakhstan, Luxembourg, Malaysia, Montenegro, Netherlands,
Norway, Pakistan, Romania, Russia, Serbia, Slovakia, Slovenia,
South Africa, South Korea, Sweden, Thailand, Ukraine,
United Kingdom, United States of America, Uzbekistan
Problem Selection Committee
Martı́n Avendaño
Carlos di Fiore
Géza Kós
Svetoslav Savchev
, 4
Algebra
A1. Find all the functions f : Z → Z such that
f (a)2 + f (b)2 + f (c)2 = 2f (a)f (b) + 2f (b)f (c) + 2f (c)f (a)
for all integers a, b, c satisfying a + b + c = 0.
A2. Let Z and Q be the sets of integers and rationals respectively.
a) Does there exist a partition of Z into three non-empty subsets A, B, C such that the sets
A + B, B + C, C + A are disjoint?
b) Does there exist a partition of Q into three non-empty subsets A, B, C such that the sets
A + B, B + C, C + A are disjoint?
Here X + Y denotes the set {x + y | x ∈ X, y ∈ Y }, for X, Y ⊆ Z and X, Y ⊆ Q.
A3. Let a2 , . . . , an be n − 1 positive real numbers, where n ≥ 3, such that a2 a3 · · · an = 1.
Prove that
(1 + a2 )2 (1 + a3 )3 · · · (1 + an )n > nn .
A4. Let f and g be two nonzero polynomials with integer coefficients and deg f > deg g.
Suppose that for infinitely many primes p the polynomial pf + g has a rational root. Prove
that f has a rational root.
A5. Find all functions f : R → R that satisfy the conditions
f (1 + xy) − f (x + y) = f (x)f (y) for all x, y ∈ R
and f (−1) 6= 0.
A6. Let f : N → N be a function, and let f m be f applied m times. Suppose that for
every n ∈ N there exists a k ∈ N such that f 2k (n) = n + k, and let kn be the smallest such k.
Prove that the sequence k1 , k2 , . . . is unbounded.
A7. We say that a function f : Rk → R is a metapolynomial if, for some positive integers m
and n, it can be represented in the form
f (x1 , . . . , xk ) = max min Pi,j (x1 , . . . , xk )
i=1,...,m j=1,...,n
where Pi,j are multivariate polynomials. Prove that the product of two metapolynomials is also
a metapolynomial.
53rd International Mathematical Olympiad
Mar del Plata, Argentina 2012
,
,Note of Confidentiality
The shortlisted problems should be kept
strictly confidential until IMO 2013
Contributing Countries
The Organizing Committee and the Problem Selection Committee of IMO 2012 thank the
following 40 countries for contributing 136 problem proposals:
Australia, Austria, Belarus, Belgium, Bulgaria, Canada, Cyprus,
Czech Republic, Denmark, Estonia, Finland, France, Germany,
Greece, Hong Kong, India, Iran, Ireland, Israel, Japan,
Kazakhstan, Luxembourg, Malaysia, Montenegro, Netherlands,
Norway, Pakistan, Romania, Russia, Serbia, Slovakia, Slovenia,
South Africa, South Korea, Sweden, Thailand, Ukraine,
United Kingdom, United States of America, Uzbekistan
Problem Selection Committee
Martı́n Avendaño
Carlos di Fiore
Géza Kós
Svetoslav Savchev
, 4
Algebra
A1. Find all the functions f : Z → Z such that
f (a)2 + f (b)2 + f (c)2 = 2f (a)f (b) + 2f (b)f (c) + 2f (c)f (a)
for all integers a, b, c satisfying a + b + c = 0.
A2. Let Z and Q be the sets of integers and rationals respectively.
a) Does there exist a partition of Z into three non-empty subsets A, B, C such that the sets
A + B, B + C, C + A are disjoint?
b) Does there exist a partition of Q into three non-empty subsets A, B, C such that the sets
A + B, B + C, C + A are disjoint?
Here X + Y denotes the set {x + y | x ∈ X, y ∈ Y }, for X, Y ⊆ Z and X, Y ⊆ Q.
A3. Let a2 , . . . , an be n − 1 positive real numbers, where n ≥ 3, such that a2 a3 · · · an = 1.
Prove that
(1 + a2 )2 (1 + a3 )3 · · · (1 + an )n > nn .
A4. Let f and g be two nonzero polynomials with integer coefficients and deg f > deg g.
Suppose that for infinitely many primes p the polynomial pf + g has a rational root. Prove
that f has a rational root.
A5. Find all functions f : R → R that satisfy the conditions
f (1 + xy) − f (x + y) = f (x)f (y) for all x, y ∈ R
and f (−1) 6= 0.
A6. Let f : N → N be a function, and let f m be f applied m times. Suppose that for
every n ∈ N there exists a k ∈ N such that f 2k (n) = n + k, and let kn be the smallest such k.
Prove that the sequence k1 , k2 , . . . is unbounded.
A7. We say that a function f : Rk → R is a metapolynomial if, for some positive integers m
and n, it can be represented in the form
f (x1 , . . . , xk ) = max min Pi,j (x1 , . . . , xk )
i=1,...,m j=1,...,n
where Pi,j are multivariate polynomials. Prove that the product of two metapolynomials is also
a metapolynomial.
