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Part of 2009 Tuymaada Olympiad

Problems(3)

Naoki and Richard in turn put chips in free squares

Source: Tuymaada 2009, Junior League, First Day, Problem 1

All squares of a

20\times 20

table are empty. Misha* and Sasha** in turn put chips in free squares (Misha* begins). The player after whose move there are four chips on the intersection of two rows and two columns wins. Which of the players has a winning strategy? Proposed by A. Golovanov US Name Conversions: Misha*: Naoki Sasha**: Richard

geometrygeometric transformationcombinatorics unsolvedcombinatorics

Fractional part of the product of every two of them is 0.5

Source: Tuymaada 2009, Senior League, First Day, Problem 1

Three real numbers are given. Fractional part of the product of every two of them is

1\over 2

. Prove that these numbers are irrational. Proposed by A. Golovanov

symmetryalgebrasystem of equationsalgebra unsolved

Magician asked a spectator to think of a three-digit number

Source: Tuymaada 2009, Senior League, Second Day, Problem 1

A magician asked a spectator to think of a three-digit number

\overline{abc}

and then to tell him the sum of numbers

\overline{acb}

\overline{bac}

\overline{bca}

\overline{cab}

\overline{cba}

. He claims that when he knows this sum he can determine the original number. Is that so?

modular arithmeticalgebra unsolvedalgebra