Advanced book on Mathematics Olympiad

1.2 Mathematical Induction 7 18.Prove that for anyn≥1, a 2n× 2 ncheckerboard with 1×1 corner square removed can be tiled by piec ...

8 1 Methods of Proof Example.Letf :N→Nbe a strictly increasing function such thatf( 2 )=2 and f (mn)=f (m)f (n)for every relativ ...

1.2 Mathematical Induction 9 27.Show that for alln>3 there exists ann-gon whose sides are not all equal and such that the sum ...

10 1 Methods of Proof The inequality is now obvious sincea 1 a 2 ≥1 anda 1 +a 2 ≥ 2 √ a 1 a 2. Now instead of exhausting all pos ...

1.3 The Pigeonhole Principle 11 1.3 The Pigeonhole Principle........................................ Thepigeonhole principle(orD ...

12 1 Methods of Proof The third example comes from the 67th W.L. Putnam Mathematical Competition, 2006. Example.Prove that for e ...

1.3 The Pigeonhole Principle 13 40.A chess player trains by playing at least one game per day, but, to avoid exhaustion, no more ...

14 1 Methods of Proof 45.Each of nine straight lines divides a square into two quadrilaterals with the ratio of their areas equa ...

1.4 Ordered Sets and Extremal Elements 15 Solution.The guess is that a tight way of arranging the small squares inside the big s ...

16 1 Methods of Proof This is equivalent to 2 x^2 − 2 √ 2 x+ 1 ≥ 0 , or(x √ 2 − 1 )^2 ≥0, which is obvious and we are done.  W ...

1.4 Ordered Sets and Extremal Elements 17 Solution.For the solution, assume that such a dissection exists, and look at the botto ...

18 1 Methods of Proof Now consider a sphere of the same radius as the planets. Remove from it all north poles defined by directi ...

1.5 Invariants and Semi-Invariants 19 65.The positive integers are colored by two colors. Prove that there exists an infinite se ...

20 1 Methods of Proof A deep theorem of Reidemeister states that two diagrams represent the same knot if they can be transformed ...

1.5 Invariants and Semi-Invariants 21 less well-known mathematician Gerwin proved that given two polygons of equal area, the fir ...

22 1 Methods of Proof 67.An ordered triple of numbers is given. It is permitted to perform the following operation on the triple ...

1.5 Invariants and Semi-Invariants 23 73.In the squares of a 3×3 chessboard are written the signs+and−as described in Figure 11( ...

24 1 Methods of Proof Notice that the sum of the five numbers on the pentagon is preserved by the operation, so it is natural to ...

2 Algebra........................................................ It is now time to split mathematics into branches. First, alge ...

26 2 Algebra and the last written by the second author of the book, and the second given at a Soviet Union college entrance exam ...