Advanced book on Mathematics Olympiad

2.2 Polynomials 47 147.LetP(x)be a polynomial of odd degree with real coefficients. Show that the equationP(P(x))=0 has at least ...

48 2 Algebra can be factored over the complex numbers as P(x)=an(x−x 1 )(x−x 2 )···(x−xn). Equating the coefficients ofxin the t ...

2.2 Polynomials 49 Next, a problem from the short list of the 2005 Ibero-American Mathematical Olympiad. Example.Find the larges ...

50 2 Algebra see that the numbers 2 cos^27 π, 2 cos^47 π, and 2 cos^87 πare of the formx+^1 x, withxone of these roots. If we de ...

2.2 Polynomials 51 155.Leta, b, cbe real numbers. Show thata ≥0,b ≥0, andc≥0 if and only if a+b+c≥0,ab+bc+ca≥0, andabc≥0. 156.So ...

52 2 Algebra 2.2.3 The Derivative of a Polynomial............................. This section adds some elements of real analysis. ...

2.2 Polynomials 53 P 1 (xj)P 2 (xj)···Pk(xj)= 1 , forj= 1 , 2 ,...,n. HencePi(xj) =±1, which then implies Pi(x^1 j) = Pi(xj), i ...

54 2 Algebra 168.LetP(z)andQ(z)be polynomials with complex coefficients of degree greater than or equal to 1 with the property t ...

2.2 Polynomials 55 Im P′(w) P(w) = ∑n k= 1 Im 1 w−zk > 0. This shows thatwis not a zero ofP′(z)and so all zeros ofP′(z)lie in ...

56 2 Algebra 2.2.5 Irreducible Polynomials................................... A polynomial is irreducible if it cannot be writte ...

2.2 Polynomials 57 Proof.We argue by contradiction. Suppose thatP(x)=Q(x)R(x), withQ(x)and R(x)not identically equal to±1. Let Q ...

58 2 Algebra 185.Prove that for any distinct integersa 1 ,a 2 ,...,anthe polynomial P(x)=(x−a 1 )(x−a 2 )···(x−an)− 1 cannot be ...

2.2 Polynomials 59 Chebyshev’s theorem.For fixedn≥ 1 , the polynomial 2 −n+^1 Tn(x)is the unique monic nth-degree polynomial sat ...

60 2 Algebra Solution.An inductive argument based on the recurrence relation shows thatTn(x)is a polynomial with integer coeffic ...

2.3 Linear Algebra 61 197.Letx 1 ,x 2 ,...,xn,n≥2, be distinct real numbers in the interval[− 1 , 1 ]. Prove that 1 t 1 + 1 t 2 ...

62 2 Algebra The equalityMm+n=MmMnwritten in explicit form is ( Fm+n+ 1 Fm+n Fm+n Fm+n− 1 ) = ( Fm+ 1 Fm Fm Fm− 1 )( Fn+ 1 Fn Fn ...

2.3 Linear Algebra 63 2.3.2 Determinants............................................ The determinant of ann×nmatrixA= (aij)i,j, ...

64 2 Algebra Solution.The key idea is to viewxnas a variable and think of the determinant as an (n− 1 )st-degree polynomial inxn ...

2.3 Linear Algebra 65 Solution.We have ∣ ∣∣ ∣∣ ∣∣ ∣∣ ∣ ∣∣ ∣ 11 ··· 11 11 ··· 11 a 11 a 12 ··· a 1 ,n− 1 a 1 n a 21 a 22 ··· a 2 ...

66 2 Algebra 206.Prove that ∣ ∣∣ ∣∣ ∣ (x^2 + 1 )^2 (xy+ 1 )^2 (xz+ 1 )^2 (xy+ 1 )^2 (y^2 + 1 )^2 (yz+ 1 )^2 (xz+ 1 )^2 (yz+ 1 )^ ...