,Shortlisted Problems with Solutions
56th International Mathematical Olympiad
Chiang Mai, Thailand, 4–16
,Note of Confidentiality
The shortlisted problems should be kept
strictly confidential until IMO 2016.
Contributing Countries
The Organizing Committee and the Problem Selection Committee of IMO 2015 thank the
following 53 countries for contributing 155 problem proposals:
Albania, Algeria, Armenia, Australia, Austria, Brazil, Bulgaria,
Canada, Costa Rica, Croatia, Cyprus, Denmark, El Salvador,
Estonia, Finland, France, Georgia, Germany, Greece, Hong Kong,
Hungary, India, Iran, Ireland, Israel, Italy, Japan, Kazakhstan,
Lithuania, Luxembourg, Montenegro, Morocco, Netherlands,
Pakistan, Poland, Romania, Russia, Saudi Arabia, Serbia,
Singapore, Slovakia, Slovenia, South Africa, South Korea, Sweden,
Turkey, Turkmenistan, Taiwan, Tanzania, Ukraine, United Kingdom,
U.S.A., Uzbekistan
Problem Selection Committee
Dungjade Shiowattana, Ilya I. Bogdanov, Tirasan Khandhawit,
Wittawat Kositwattanarerk, Géza Kós, Weerachai Neeranartvong,
Nipun Pitimanaaree, Christian Reiher, Nat Sothanaphan,
Warut Suksompong, Wuttisak Trongsiriwat, Wijit Yangjit
Assistants: Jirawat Anunrojwong, Pakawut Jiradilok
, Shortlisted problems 3
Problems
Algebra
A1. Suppose that a sequence a1 , a2 , . . . of positive real numbers satisfies
kak
ak`1 ě
a2k ` pk ´ 1q
for every positive integer k. Prove that a1 ` a2 ` ¨ ¨ ¨ ` an ě n for every n ě 2.
(Serbia)
A2. Determine all functions f : Z Ñ Z with the property that
` ˘ ` ˘
f x ´ f pyq “ f f pxq ´ f pyq ´ 1
holds for all x, y P Z.
(Croatia)
A3. Let n be a fixed positive integer. Find the maximum possible value of
ÿ
ps ´ r ´ nqxr xs ,
1ďrăsď2n
where ´1 ď xi ď 1 for all i “ 1, 2, . . . , 2n.
(Austria)
A4. Find all functions f : R Ñ R satisfying the equation
` ˘
f x ` f px ` yq ` f pxyq “ x ` f px ` yq ` yf pxq
for all real numbers x and y.
(Albania)
A5. Let 2Z ` 1 denote the set of odd integers. Find all functions f : Z Ñ 2Z ` 1 satisfying
` ˘ ` ˘
f x ` f pxq ` y ` f x ´ f pxq ´ y “ f px ` yq ` f px ´ yq
for every x, y P Z.
(U.S.A.)
A6. Let n be a fixed integer with n ě 2. We say that two polynomials P and Q with real
coefficients are block-similar if for each i P t1, 2, . . . , nu the sequences
P p2015iq, P p2015i ´ 1q, . . . , P p2015i ´ 2014q and
Qp2015iq, Qp2015i ´ 1q, . . . , Qp2015i ´ 2014q
are permutations of each other.
paq Prove that there exist distinct block-similar polynomials of degree n ` 1.
pbq Prove that there do not exist distinct block-similar polynomials of degree n.
(Canada)
56th International Mathematical Olympiad
Chiang Mai, Thailand, 4–16
,Note of Confidentiality
The shortlisted problems should be kept
strictly confidential until IMO 2016.
Contributing Countries
The Organizing Committee and the Problem Selection Committee of IMO 2015 thank the
following 53 countries for contributing 155 problem proposals:
Albania, Algeria, Armenia, Australia, Austria, Brazil, Bulgaria,
Canada, Costa Rica, Croatia, Cyprus, Denmark, El Salvador,
Estonia, Finland, France, Georgia, Germany, Greece, Hong Kong,
Hungary, India, Iran, Ireland, Israel, Italy, Japan, Kazakhstan,
Lithuania, Luxembourg, Montenegro, Morocco, Netherlands,
Pakistan, Poland, Romania, Russia, Saudi Arabia, Serbia,
Singapore, Slovakia, Slovenia, South Africa, South Korea, Sweden,
Turkey, Turkmenistan, Taiwan, Tanzania, Ukraine, United Kingdom,
U.S.A., Uzbekistan
Problem Selection Committee
Dungjade Shiowattana, Ilya I. Bogdanov, Tirasan Khandhawit,
Wittawat Kositwattanarerk, Géza Kós, Weerachai Neeranartvong,
Nipun Pitimanaaree, Christian Reiher, Nat Sothanaphan,
Warut Suksompong, Wuttisak Trongsiriwat, Wijit Yangjit
Assistants: Jirawat Anunrojwong, Pakawut Jiradilok
, Shortlisted problems 3
Problems
Algebra
A1. Suppose that a sequence a1 , a2 , . . . of positive real numbers satisfies
kak
ak`1 ě
a2k ` pk ´ 1q
for every positive integer k. Prove that a1 ` a2 ` ¨ ¨ ¨ ` an ě n for every n ě 2.
(Serbia)
A2. Determine all functions f : Z Ñ Z with the property that
` ˘ ` ˘
f x ´ f pyq “ f f pxq ´ f pyq ´ 1
holds for all x, y P Z.
(Croatia)
A3. Let n be a fixed positive integer. Find the maximum possible value of
ÿ
ps ´ r ´ nqxr xs ,
1ďrăsď2n
where ´1 ď xi ď 1 for all i “ 1, 2, . . . , 2n.
(Austria)
A4. Find all functions f : R Ñ R satisfying the equation
` ˘
f x ` f px ` yq ` f pxyq “ x ` f px ` yq ` yf pxq
for all real numbers x and y.
(Albania)
A5. Let 2Z ` 1 denote the set of odd integers. Find all functions f : Z Ñ 2Z ` 1 satisfying
` ˘ ` ˘
f x ` f pxq ` y ` f x ´ f pxq ´ y “ f px ` yq ` f px ´ yq
for every x, y P Z.
(U.S.A.)
A6. Let n be a fixed integer with n ě 2. We say that two polynomials P and Q with real
coefficients are block-similar if for each i P t1, 2, . . . , nu the sequences
P p2015iq, P p2015i ´ 1q, . . . , P p2015i ´ 2014q and
Qp2015iq, Qp2015i ´ 1q, . . . , Qp2015i ´ 2014q
are permutations of each other.
paq Prove that there exist distinct block-similar polynomials of degree n ` 1.
pbq Prove that there do not exist distinct block-similar polynomials of degree n.
(Canada)
