2010 Switzerland - Final Round

Part of Switzerland - Final Round

Subcontests

(8)

1

Triangulation of an n-gon

n\geqslant 3

P

n

-gon. Show that

P

can be, by n \minus{} 3 non-intersecting diagonals, partitioned in triangles such that the circumcircle of each triangle contains the whole area of

P

. Under which conditions is there exactly one such triangulation?

1

Equal numbers of members

In a village with at least one inhabitant, there are several associations. Each inhabitant is a member of at least

k

associations, and any two associations have at most one common member. Prove that at least

k

associations have the same number of members.

1

m+n+1 divides 2(m^2+n^2)-1

m

n

be natural numbers such that m\plus{}n\plus{}1 is prime and divides 2(m^2\plus{}n^2)\minus{}1. Prove that m\equal{}n.

1

Functional equation

Find all functions

f: \mathbb{R}\mapsto\mathbb{R}

such that for all

x

y

\in\mathbb{R}

, f(f(x))\plus{}f(f(y))\equal{}2y\plus{}f(x\minus{}y) holds.

1

Closed path and parallel edges

Some sides and diagonals of a regular

n

-gon form a connected path that visits each vertex exactly once. A parallel pair of edges is a pair of two different parallel edges of the path. Prove that (a) if

n

is even, there is at least one parallel pair. (b) if

n

is odd, there can't be one single parallel pair.

1

Cyclic inequality in x,y,z

x

y

z \in\mathbb{R}^+

xyz = 1

. Prove that \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}

1

Number of natural solutions (a,b)

n\in\mathbb{N}

, determine the number of natural solutions

(a,b)

such that (4a\minus{}b)(4b\minus{}a)\equal{}2010^n holds.

1

Coins on number line

Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one

1

unit to the right and the second one

1

unit to the left. Under which initial conditions is it possible to move all coins to one single point?