Advanced book on Mathematics Olympiad

Number Theory 711 from the(k+ 2 )nd term onward, the terms of the sequence are nontrivial multiples of pk+ 1 , and therefore mus ...

712 Number Theory and bn+n≡b+n≡b−a(modp). It follows thatpdividesan+nbut does not dividebn+n, a contradiction. Hencea=b, as desi ...

Number Theory 713 and the problem is solved. 795.Consider first the casea=0. Sinceby=malways has solutions, it follows that b=±1 ...

714 Number Theory 797.At each cut we add 7 or 11 new pieces. Thus after cuttingxtimes in 8 andytimes in 12 we have 7x+ 11 y+1 pi ...

Number Theory 715 800.Note that for any integerk, we can dissect thed-dimensional cube intokdpieces. If we do this for two integ ...

716 Number Theory 803.The last digit of a perfect square cannot be 3 or 7. This implies thatxmust be even, sayx= 2 x′. The condi ...

Number Theory 717 805.One can verify thatx= 2 m^2 +1 andy= 2 mis a solution. (Diophantus) 806.We will search for numbersxandyfor ...

718 Number Theory 7 z= ( k 1 ) 8 k−^1 · 3 + ( k 3 ) 8 k−^3 · 33 · 7 + ( k 5 ) 8 k−^5 · 35 · 72 +···. Let us compare the power of ...

Number Theory 719 811.It is easy to guess that(x,y,z,t)=( 10 , 10 ,− 1 , 0 )is a solution. Because quadratic Diophantine equatio ...

720 Number Theory = √ p+ √ p− 1 , as desired. This now suggests the path we should follow in the case thatnis odd. Write ( √ m+ ...

Number Theory 721 814.Divide through byx^2 y^2 to obtain the equivalent equation 1 y^2 + 1 xy + 1 x^2 = 1. One of the denominato ...

722 Number Theory find that|U|,|V|,|W|,|T|is even, hence(U 2 ,V 2 ,W 2 ,T 2 )is also a solution, contradicting minimality. Hence ...

Number Theory 723 820.First solution: The solutions are (v+ 1 ,v, 1 , 1 ),for allv; ( 2 , 1 , 1 ,y),for ally; ( 2 , 3 , 2 , 1 ), ...

724 Number Theory For arbitraryq, from what we have proved so far it follows thataq≡± 1 (mod 2t−m). Becauseφ( 2 t−m)= 2 t−m−^1 , ...

Number Theory 725 Combining these, we see that we must have eitherx=1ory=1. Either of these implies the other and gives the solu ...

Combinatorics and Probability 821.The relation from the statement implies (A∩X)∪(B∩X)=A∩B. Applying de Morgan’s law, we obtain ( ...

728 Combinatorics and Probability is a subset ofM.If{ 3 , 12 } ⊂M, it follows again thatMhas at most 10 elements. If { 3 , 12 ...

Combinatorics and Probability 729 person inMBhas exactly one acquaintance inMA. This allows us to establish a bijection betweenM ...

730 Combinatorics and Probability other colors chosen from 1, 2, and 3. Ifx∼x 0 thenfn(x)=fm(x 0 )for some integers mandn. For t ...

Combinatorics and Probability 731 Whenr=4, there are ( 6 4 ) ways to split the numbers into the two cycles (two cycles are neede ...