MathDB

IMO Shortlist 2011, Algebra 4

Source: IMO Shortlist 2011, Algebra 4

7/11/2012

Determine all pairs

(f,g)

of functions from the set of positive integers to itself that satisfy

f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1

for every positive integer

n

. Here,

f^k(n)

means

\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))

.
Proposed by Bojan Bašić, Serbia

functionalgebrafunctional equationIMO Shortlist

IMO Shortlist 2011, Combinatorics 4

Source: IMO Shortlist 2011, Combinatorics 4

7/12/2012

Determine the greatest positive integer

k

that satisfies the following property: The set of positive integers can be partitioned into

k

subsets

A_1, A_2, \ldots, A_k

such that for all integers

n \geq 15

and all

i \in \{1, 2, \ldots, k\}

there exist two distinct elements of

A_i

whose sum is

n.

Proposed by Igor Voronovich, Belarus

arithmetic sequencecombinatoricsIMO ShortlistAdditive combinatorics

IMO Shortlist 2011, G4

Source: IMO Shortlist 2011, G4

7/13/2012

Let

ABC

be an acute triangle with circumcircle

\Omega

. Let

B_0

be the midpoint of

AC

and let

C_0

be the midpoint of

AB

. Let

D

be the foot of the altitude from

A

and let

G

be the centroid of the triangle

ABC

. Let

\omega

be a circle through

B_0

and

C_0

that is tangent to the circle

\Omega

at a point

X\not= A

. Prove that the points

D,G

and

X

are collinear.
Proposed by Ismail Isaev and Mikhail Isaev, Russia

geometrycircumcirclesymmetryIMO Shortlisthomothetyradical axisgeometry solved

IMO Shortlist 2011, Number Theory 4

Source: IMO Shortlist 2011, Number Theory 4

7/11/2012

For each positive integer

k,

let

t(k)

be the largest odd divisor of

k.

Determine all positive integers

a

for which there exists a positive integer

n,

such that all the differences

t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)

are divisible by 4.
Proposed by Gerhard Wöginger, Austria

number theoryIMO Shortlistcombinatorics

4

Problems(4)