Advanced book on Mathematics Olympiad

Methods of Proof 349 black sequence 2l 1 <l 1 +l 2 <···< 2 lm. By maximal we mean that these sequences cannot be extend ...

350 Methods of Proof of this region except for the verticesBandC. DefineRBandRCanalogously. These regions are illustrated in Fig ...

Methods of Proof 351 the sum of the squares of the numbers in a triple is invariant under the operation. The sum of squares of t ...

352 Methods of Proof e ee ee eee e a aae aaa aaa b b bb b bb b b c c ccc c cc c Figure 55 (i) at least two residue classes are u ...

Methods of Proof 353 When we write this number as a product of two factors, one of the factors is congruent to 1 and the other i ...

354 Methods of Proof This is equivalent to cos α 1 +β 1 +α 2 2 ( sin α 1 +α 2 2 sin β 2 −sin α 1 +β 2 sin α 2 2 ) =sin α 1 2 ( s ...

Methods of Proof 355 =cos∠DCA+cos∠BCA+cos∠CAD+cos∠CAB =cos∠DCA+cos∠CAD+cos∠ADC+cos∠BCA+cos∠CAB+cos∠ABC, and we are done. Remark. ...

356 Methods of Proof (a, 0 , 0 , d), (a, 0 , 0 , a), (a, 0 ,a, 0 ), (a, 0 ,c,a), and their cyclic permutations. At the third ste ...

Methods of Proof 357 = 1 + 4 ∑∞ i= 1 1 2 i ∑∞ j= 0 1 2 j − 1 − 4 ( 1 · 1 2 + 2 · 1 4 + 3 · 1 8 + 4 · 1 16 + 5 · 1 32 ) = 9 − 65 ...

Algebra 81.Assume that both numbers are perfect cubes. Then so is their product (n+ 3 )(n^2 + 3 n+ 3 )=n^3 + 6 n^2 + 12 n+ 9. Ho ...

360 Algebra F(x)=(P 1 (x)^2 +Q 1 (x)^2 )(P 2 (x)^2 +Q 2 (x)^2 )···(Pn(x)^2 +Qn(x)^2 ), where the factorR(x)^2 is incorporated in ...

Algebra 361 wherek=n− 21. Sincen≥5, both factors at the numerator are greater than 5, which shows that after canceling the denom ...

362 Algebra 88.The solution is based on the identity ak+bk=(a+b)(ak−^1 +bk−^1 )−ab(ak−^2 +bk−^2 ). This identity arises naturall ...

Algebra 363 We further change this intox =^3 √ x^3 −x. Raising both sides to the third power, we obtainx^3 =x^3 −x. We conclude ...

364 Algebra = ( x y + y x ) 2 + ( y z + z y ) 2 + ( z x + x z ) 2 − 4. Hence m^2 +n^2 +p^2 =mnp+ 4. Adding 2(mn+np+pm)to both si ...

Algebra 365 The equality holds if and only ifa=b=1, i.e.,x=0. (T. Andreescu, Z. Feng, 101Problems in Algebra, Birkhäuser, 2001) ...

366 Algebra So a good candidate for the minimum is 2n, which is actually attained forx 1 =x 2 = ··· =xn=^12. (Romanian Mathemati ...

Algebra 367 1 2 ±x+x^2 ±x^3 +x^4 ±···±x^2 k−^1 +x^2 k> 0 , for all 2kchoices of the signs+and−. This reduces to ( 1 2 ±x+ 1 2 ...

368 Algebra Remark.In statistics the numbersfiare integers that record the frequency of occurrence of the sampled random variabl ...

Algebra 369 Fora 1 =x 0 −x 1 ,a 2 =x 1 −x 2 ,...,an=xn− 1 −xnthis gives 1 x 0 −x 1 + 1 x 1 −x 2 +···+ 1 xn− 1 −xn ≥ n^2 x 0 −x 1 ...