Advanced book on Mathematics Olympiad

430 Algebra and so the system has the unique solutiona 0 −(n+ 1 )=a 1 = ··· =an− 1 =0. We obtainR(x)=n+1. Second solution: Note ...

Algebra 431 z+x=by. View this as a homogeneous system in the variablesx, y, z. Because we assume that the system admits nonzero ...

432 Algebra ··· an− 1 − 2 an+an+ 1 =bn in the unknownsa 1 ,a 2 ,...,an. To determineakfor somek, we multiply the first equation ...

Algebra 433 x 1 2 + x 2 3 +···+ xn n+ 1 = 0 , ··· x 1 n + x 2 n+ 1 +···+ xn 2 n− 1 = 0 has only the trivial solution. For a solu ...

434 Algebra 1 + 2 +···+(n− 1 )+ 1 + 2 +···+n= n(n− 1 ) 2 + n(n+ 1 ) 2 =n^2. The problem is solved. Remark.It is interesting to n ...

Algebra 435 rank ⎛ ⎜ ⎜⎜ ⎝ 23 ···n+ 1 34 ···n+ 2 .. . .. . ... .. . n+ 1 n+ 2 ··· 2 n ⎞ ⎟ ⎟⎟ ⎠ =rank ⎛ ⎜ ⎜⎜ ⎝ 23 ···n+ 1 11 ··· 1 ...

436 Algebra is irrational for any nonnegative rational numbersa 1 ,a 2 ,...,annot all equal to zero. Denote the other elements o ...

Algebra 437 247.First solution: Assume first that all numbers are integers. Whenever we choose a number, the sum of the remainin ...

438 Algebra It follows thatλ 1 =λ 2 =0. Change the basis tov, wwithvan eigenvector ofA(which does exist becauseAv=0 has nontrivi ...

Algebra 439 250.First solution: The eigenvalues are the zeros of the polynomial det(λIn−aA−bAt). The matrixλIn−aA−bAtis a circul ...

440 Algebra the kernel, in which case their eigenvalue is 0. BecauseA= 2 B−I, it has the same eigenvectors asB, with eigenvalues ...

Algebra 441 255.There is a more general property, of which the problem is a particular case. Riesz lemma.IfVis a finite-dimensio ...

442 Algebra =‖(U−In)(V−In)y−(V−In)(U−In)y‖ ≤‖(U−In)(V−In)y‖+‖(V−In)(U−In)y‖. The claim follows if we prove that‖(U−In)(V−In)y‖an ...

Algebra 443 The determinant of the system isu^2 +uvtrA+v^2 detA, and an easy algebraic computation shows that this is equal to d ...

444 Algebra X^2 − 5 X+ 6 I 2 = 0 , which can be used to transform the original equation into 4 X− 12 I 2 = ( − 2 − 2 − 2 − 2 ) w ...

Algebra 445 LetPB(λ)=λ^2 +rλ+sbe the characteristic polynomial ofB. By the Cayley–Hamilton Theorem,PB(B)=0. We have O 2 =APB(B)− ...

446 Algebra Consider the pointsP(a 11 ,a 21 ,a 31 ), Q(a 12 ,a 22 ,a 32 ), R(a 13 ,a 23 ,a 33 )in three-di- mensional Euclidean ...

Algebra 447 the trianglePQRwhosex- andy-coordinates are both negative, and if thez-coordinate ofTis positive, chooseSto have the ...

448 Algebra as ( n− 1 k ) = k+ 1 n− 1 ( n− 1 k+ 1 ) + n−k n− 1 ( n− 1 k− 1 ) . To be more explicit, this identity implies that f ...

Algebra 449 269.FixaandcinSand consider the functionfa,c:S{a, c}→S, fa,c(b)=a∗(b∗c). Becausea∗fa,c(b)∗c=(a∗a)∗b∗(c∗c)=b, the fun ...