Advanced book on Mathematics Olympiad

510 Real Analysis 395.LetLbe the length of the trail andT the total duration of the climb, which is the same as the total durati ...

Real Analysis 511 second day of travel with a homeomorphism (continuous bijection) of the time interval [ 0 , 1 ], we can ensure ...

512 Real Analysis B A C D x y Figure 67 Remark.This result is known as the “pancake theorem.’’ 402.Assume thatfis not continuous ...

Real Analysis 513 fa(x)−f 0 (x)= { 0 forx = 0 , a forx= 0 , does not have the intermediate value property, so it is not the der ...

514 Real Analysis namely,x =^23 andx =−^12. We computeg(− 1 ) =4,g(−^12 )=13,g(^23 ) = 278 , g( 1 )=4. The largest of them is 13 ...

Real Analysis 515 −cos^2 xcos^2 (sinx)cos(sin(sin(sinx)))cos(sin(sin(sin(sinx))))sin(sin(sinx)) −cos^2 xcos^2 (sinx)cos^2 (sin(s ...

516 Real Analysis f(x)=(x+y+z)ln(x+y+z)+xlnx+ylny+zlnz −(x+y)ln(x+y)−(y+z)ln(y+z)−(z+x)ln(z+x). Differentiatingf(x)with respect ...

Real Analysis 517 is monotonic to the left, and to the right ofa, and becauseF(n−^2 )(a)=0,F(n−^2 )(x) = 0 forx =aandxin a ...

518 Real Analysis 416.Replacingfby−fif necessary, we may assumef(b)>f(c), hencef(a)>f(c) as well. Letξbe an absolute minim ...

Real Analysis 519 This implies thatβ=f′(c)(c−α)+f(c), which shows thatM(α, β)lies on the tangent to the graph offat(c, f (c)), a ...

520 Real Analysis Thus the lack of solutions will follow if we show thatFis strictly increasing. Recall that e−t> 1 −tfort ...

Real Analysis 521 424.We first show thatP(x)has rational coefficients. Letkbe the degree ofP(x), and for eachn, letxnbe the rati ...

522 Real Analysis number ofxi’s that are less thanaand the number ofxi’s that are greater thana. The global minimum is attained ...

Real Analysis 523 or AsinB−AsinC−CsinB≥BsinA−CsinA−BsinC. Moving the negative terms to the other side and substituting the sides ...

524 Real Analysis = f(x 1 )+···+f(x 2 k)+f(x 2 k+ 1 )+···+f(x 2 k+ 1 ) 2 k+^1 . Next, we show that f ( x 1 +x 2 +···+xn n ) ≤ f( ...

Real Analysis 525 f (i+ 2 n ) +f (i n ) 2 ≥f ( i+ 1 n ) + 2 n ,i= 1 , 2 ,...,n, or f ( i+ 2 n ) −f ( i+ 1 n ) ≥f ( i+ 1 n ) −f ( ...

526 Real Analysis completing the induction. Hence the conclusion. (USA Mathematical Olympiad, 2000, proposed by B. Poonen) 431.T ...

Real Analysis 527 and the inequality is proved. (T. Popoviciu, solution published by Gh. Eckstein inTimi ̧soara Mathematics Gaze ...

528 Real Analysis 434.We assume thatα ≤β ≤γ, the other cases being similar. The expression is a convex function in each of the v ...

Real Analysis 529 The coefficients in the expansion are positive, so the expression is a convex function inx (being a sum of pow ...