MathDB

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Part of 2010 Tuymaada Olympiad

Problems(3)

Tuymaada 2010, Junior League, Problem 1

Source:

7/18/2010
Misha and Sahsa play a game on a 100×100100\times 100 chessboard. First, Sasha places 5050 kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins):
At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture.
Is there a winning strategy available for Sasha?
combinatorics unsolvedcombinatorics
Tuymaada 2010, Junior League, Problem 5

Source:

7/18/2010
We have a set MM of real numbers with ∣M∣>1|M|>1 such that for any x∈Mx\in M we have either 3x−2∈M3x-2\in M or −4x+5∈M-4x+5\in M. Show that MM is infinite.
limitcalculusintegrationalgebra unsolvedalgebra
Three quadratic trinomials all with an integer root

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

8/1/2011
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by 11, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by 11, the new trinomial, too, has an integral root. Can this be true?
quadraticscalculusintegrationalgebrapolynomialalgebra unsolvedBaron Munchausen