Advanced book on Mathematics Olympiad

590 Real Analysis It follows thatf(x+xn)−f(x)is a polynomial of degreen−2 for allxn.In particular, there exist polynomialsP 1 (x ...

Real Analysis 591 the functional equation, we can findx 0 such thatf(x 0 ) =x 0. Rewrite the functional equation asf(f(x))−f( ...

592 Real Analysis If on any of these intervalsf−g=f′+g′, then sincef+g=g′−f′it follows thatf =g′on that interval, and sog′+g=g′− ...

Real Analysis 593 for some real constantC. In particular,fis bounded. (R. Gelca) 556.The idea is to write the equation as Bydx+A ...

594 Real Analysis Let us focus on the functionf(t)=tlnt−t. Its derivative isf′(t)=lnt, which is negative ift1. The minimum offis ...

Real Analysis 595 c· a b · a ab− 1 x ab− 1 = x cx ab, which is equivalent to c^2 a ab =b. By eliminatingc, we obtain the family ...

596 Real Analysis 562.The relation from the statement implies right away thatfis differentiable. Differ- entiating f(x)+x ∫x 0 f ...

Real Analysis 597 564.Consider the change of variablex=cost. Then, by the chain rule, dy dx = dy dt dx dt =− dy dt sint and d^2 ...

598 Real Analysis This splits into d^2 y dx^2 =0 and ( dy dx ) 3 = 1. The first of these has the solutionsy=ax+b, witha =0, be ...

Real Analysis 599 Returning to the problem, we see that there existsc∈[ 0 ,x]such that ∫x 0 e−ty′y′′dt= ∫c 0 y′y′′dt= 1 2 [ ((y′ ...

600 Real Analysis which implies thatA=B=0, and thereforey 1 =y 2 , as desired. (M. Ghermanescu, ̆ Ecua ̧tii Diferen ̧tiale(Diffe ...

Real Analysis 601 ∑n i= 1 ln|P(x)−xi|=n^2 lnC|x|, whereCis some positive constant. After adding the logarithms on the left we ha ...

Geometry and Trigonometry 573.This is the famous Jacobi identity. Identifying vectors with so( 3 )matrices, we compute −→u×(−→v ...

604 Geometry and Trigonometry Hence the solution to the equation is −→x = m −→a ·−→b −→ b + 1 −→a ·−→b −→a ×−→c. (C. Co ̧sni ̧ta ...

Geometry and Trigonometry 605 =f^2 (−→v ×−→v′)×(−→v ×−→v′′). By thecab-bacformula this is further equal to f^2 (−→v′′·(−→v ×−→v′ ...

606 Geometry and Trigonometry n(−→a ·f(−→c)+−→c ·f(−→a))= 0 , a·f(m −→ b +n−→c)+(m −→ b +n−→c)·f(−→a)= 0. Adding the first two e ...

Geometry and Trigonometry 607 And this is thecab-bacidentity once we notice that−→a · −→ b =−^12 tr(AB). 581.An easy computation ...

608 Geometry and Trigonometry and the analogous formulas for−→v 2 ,−→v 3 , and−→v 4. Since the rational multiple of a vector and ...

Geometry and Trigonometry 609 A B C D H P Q H Figure 73 Proof.Let −−→ MN=−→v 1 , −→ NP =−→v 2 , −→ PQ=−→v 3 , −−→ QM =−→v 4 , an ...

610 Geometry and Trigonometry The lemma is proved. Remark.A. Dang gave an alternative solution by observing that trianglesAH C a ...