MathDB

5

Part of 2010 IMO Shortlist

Problems(3)

f(f(x)^2*y)=x^3f*(xy)

Source: IMO Shortlist 2010, Algebra 5

7/17/2011
Denote by Q+\mathbb{Q}^+ the set of all positive rational numbers. Determine all functions f:Q+↦Q+f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+ which satisfy the following equation for all x,y∈Q+:x, y \in \mathbb{Q}^+: f(f(x)2y)=x3f(xy).f\left( f(x)^2y \right) = x^3 f(xy).
Proposed by Thomas Huber, Switzerland
functionalgebraIMO Shortlistfunctional equation
Number of wins and losses of the ith player

Source: IMO Shortlist 2010, Combinatorics 5

7/17/2011
n≥4n \geq 4 players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players badbad if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let wiw_i and lil_i be respectively the number of wins and losses of the ii-th player. Prove that ∑i=1n(wi−li)3≥0.\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.
Proposed by Sung Yun Kim, South Korea
combinatoricsIMO Shortlistgraph theoryTournament graphsvertex degreeHi
IMO Shortlist 2010 - Problem G5

Source:

7/17/2011
Let ABCDEABCDE be a convex pentagon such that BC∥AE,BC \parallel AE, AB=BC+AE,AB = BC + AE, and ∠ABC=∠CDE.\angle ABC = \angle CDE. Let MM be the midpoint of CE,CE, and let OO be the circumcenter of triangle BCD.BCD. Given that ∠DMO=90∘,\angle DMO = 90^{\circ}, prove that 2∠BDA=∠CDE.2 \angle BDA = \angle CDE.
Proposed by Nazar Serdyuk, Ukraine
geometrycircumcirclereflectionparallelogramIMO Shortlist