MathDB

2

Part of 2009 Ukraine National Mathematical Olympiad

Problems(6)

Nice equation 2p^2 + p + 9 = m^2 - [UKRMO 2009 Grade 9]

Source:

1/23/2011
Find all prime numbers pp and positive integers mm such that 2p2+p+9=m2.2p^2 + p + 9 = m^2.
number theory proposednumber theory
Convex 2009−gon - [UKRMO 2009 Grade 8]

Source:

1/20/2011
There is convex 20092009-gon on the plane.
a) Find the greatest number of vertices of 20092009-gon such that no two forms the side of the polygon.
b) Find the greatest number of vertices of 20092009-gon such that among any three of them there is one that is not connected with other two by side.
Prove that if AK = KF then AB = CF - [UKRMO 2009 Grade 8]

Source:

1/21/2011
In acute-angled triangle ABC,ABC, let MM be the midpoint of BCBC and let KK be a point on side AB.AB. We know that AMAM meet CKCK at F.F. Prove that if AK=KFAK = KF then AB=CF.AB = CF .
The knight in 2009 x 2009 board - [UKRMO 2009 Grade 10]

Source:

1/23/2011
There is a knight in the left down corner of 2009×20092009 \times 2009 chessboard. The row and the column containing this corner are painted. The knight cannot move into painted cell and after its move new row and column that contains a square with knight become painted. Is it possible to paint all rows and columns of the chessboard?
combinatorics proposedcombinatorics
Find all Z → Z functions - [UKRMO 2009 Grade 11]

Source:

1/23/2011
Find all functions f:Z→Zf : \mathbb Z \to \mathbb Z such that f(n∣m∣)+f(n(∣m∣+2))=2f(n(∣m∣+1))∀m,n∈Z.f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.
Note. ∣x∣|x| denotes the absolute value of the integer x.x.
functionabsolute valuealgebra proposedalgebra
Four Elements Subsets - [UKRMO 2009 Grade 11]

Source:

1/23/2011
Let M={1,2,3,4,6,8,12,16,24,48}.M = \{1, 2, 3, 4, 6, 8,12,16, 24, 48\} . Find out which of four-element subsets of MM are more: those with product of all elements greater than 20092009 or those with product of all elements less than 2009.2009.
combinatorics unsolvedcombinatorics