Advanced book on Mathematics Olympiad

450 Algebra u, v∈S. Hence the existence and uniqueness of the solution to the equationa∗x=b is equivalent to the existence and u ...

Algebra 451 For an example of such a binary operation consider any setSendowed with the operationa∗b=afor anya, b∈S. 276.Using t ...

452 Algebra Using this and the associativity and commutativity of∗, we obtain 2 nr(y)= 0 ∗y∗ 2 y∗···∗ 2 ny =( 0 ∗ny )∗(y∗(n+ 1 ) ...

Algebra 453 and cancelingawe obtaina⊥a=e, for alla∈G. Using this fact, we obtain a∗b=(a⊥a)⊥(a⊥b)=e⊥(a⊥b)=a⊥b, which shows that t ...

454 Algebra xy^2 x−^1 =x(yx−^1 )x(yx−^1 )=(yx−^1 )x(yx−^1 )x=y^2. Thus for anyx, y, we havexy^2 =y^2 x. This means that squares ...

Algebra 455 M^2 =(G 1 +G 2 +···+Gk)^2 = ∑k i= 1 Gi ⎛ ⎝ ∑k j= 1 Gj ⎞ ⎠= ∑k i= 1 Gi ( ∑ G∈ G−i^1 G ) = ∑ G∈ ∑k i= 1 Gi(G−i^1 G)= ...

456 Algebra Canceling the powers of 2, this amounts to showing that{^2 n 5 m|m, nintegers}is dense. We further simplify the prob ...

Algebra 457 is true ifBCis replaced byAB. It follows thatis preserved both by the translations in the groupGBCand in the analog ...

458 Algebra ( 1 −(xy)n)v =1; hencev(xy)n =(xy)nv =v−1. We claim that the inverse of 1 −(yx)nis 1+(yx)n−^1 yvx. Indeed, we comput ...

Real Analysis 297.Examining the sequence, we see that themth term of the sequence is equal ton exactly for thosemthat satisfy n^ ...

460 Real Analysis 298.If we were given the recurrence relationxn=xn− 1 +n, for alln, the terms of the sequence would be the tria ...

Real Analysis 461 an= 1 2 (a 2 n− 2 +a 2 )−an− 2 = 2 an− 1 + 2 a 1 −an− 2 = 2 (n^2 − 2 n+ 1 )+ 2 −(n^2 − 4 n+ 4 )=n^2. This comp ...

462 Real Analysis Remark.A solution based on the Binet formula is possible if we note the factorization λ^4 − 3 λ^3 − 6 λ^2 + 3 ...

Real Analysis 463 304.Assume that we have found such numbers for everyn. Thenqn+ 1 (x)−xqn(x)must be divisible byp(x). But qn+ 1 ...

464 Real Analysis Remark.In the particular casea=b=1, we obtain the well-known identity for the Fibonacci sequenceFn+ 1 Fn− 1 −F ...

Real Analysis 465 309.Denote the vertices of the octagon byA 1 =A, A 2 ,A 3 ,A 4 ,A 5 =E, A 6 ,A 7 ,A 8 in successive order. Any ...

466 Real Analysis λ^3 − 3 λ^2 + 6 λ− 4 = 0. An easy check shows thatλ 1 =1 is a solution to this equation. Sinceλ^3 − 3 λ^2 + 6 ...

Real Analysis 467 Hencexnandynsatisfy the same recurrence. This implies thatxn =ynfor alln. The conclusion now follows by induct ...

468 Real Analysis =(Fn+Fn− 1 )^2 +(Fn−Fn− 1 )^2 −(Fn− 2 )^2 =(Fn+ 1 )^2 +(Fn− 2 )^2 −(Fn− 2 )^2 =(Fn+ 1 )^2. With this the probl ...

Real Analysis 469 P(k)≈ μk ek·k! . The exercise we just solved shows that this approximation is good. 315.Let us assume that the ...