Advanced book on Mathematics Olympiad

550 Real Analysis 488.Expand the cosine in a Taylor series, cosax= 1 − (ax)^2 2! + (ax)^4 4! − (ax)^6 6! +···. Let us forget for ...

Real Analysis 551 P(x)= 1 4 + 1 16 x+ 1 64 x^2. By the residue formula for Taylor series we have ∣ ∣∣ ∣P(x)+ 1 x− 4 ∣ ∣∣ ∣= x^3 ...

552 Real Analysis = ∑∞ k= 0 (− 1 )k 1 · 3 ···( 2 k− 1 ) 2 k·k! xk= ∑∞ k= 0 (− 1 )k ( 2 k)! 22 k·k!·k! xk = ∑∞ k= 0 (− 1 )k 1 22 ...

Real Analysis 553 bm= 1 π ∫ 2 π 0 xsinmxdx=− xcosmx mπ ∣∣ ∣ 2 π 0 + 1 mπ ∫ 2 π 0 cosmxdx=− 2 m , form≥ 1. Therefore, x=π− 2 1 si ...

554 Real Analysis 1 T ∫T 0 |f(x)|^2 dx=a 02 + 2 ∑∞ n= 1 (an^2 +b^2 n). Our particular function has the Fourier series expansion ...

Real Analysis 555 which allows us to write the function as an expression with no fractions: f(x)=(cos 2x+cos 4x+···+cos 2nx)^2 + ...

556 Real Analysis ∫π 0 f′′(x)(nsinx−sinnx)dx= π 2 (na 1 −n^2 an). Therefore, 0 ≤ ∫π 0 (f (x)−f′′(x))(nsinx−sinnx)dx= π 2 (na 1 − ...

Real Analysis 557 499.We switch to polar coordinates, where the homogeneity condition becomes the simpler u(r, θ )=rng(θ), where ...

558 Real Analysis 501.Using the Leibniz–Newton fundamental theorem of calculus, we can write f (x, y)−f( 0 , 0 )= ∫x 0 ∂f ∂x (s, ...

Real Analysis 559 showing thatg 1 is continuous. This concludes the solution. 502.First, observe that if|x|+|y|→∞thenf (x, y)→∞, ...

560 Real Analysis ∂f ∂A (A, B)=−sinA+sin(A+B)= 0 , ∂f ∂B (A, B)=−sinB+sin(A+B)= 0. From here we obtain sinA=sinB=sin(A+B), which ...

Real Analysis 561 be the maximum once we check that no value on the boundary of the domain exceeds this number. But when one of ...

562 Real Analysis u= 8 v v^2 + 3 , v= 8 u u^2 + 3 . This we transform into uv^2 + 3 u= 8 v, u^2 v+ 3 v= 8 u, then subtract the s ...

Real Analysis 563 line passing through the origin that is the clockwise rotation oflby 90◦. The origin of the coordinate system ...

564 Real Analysis that the second derivative isstrictlynegative; the case in which it is zero makes the points collinear, in whi ...

Real Analysis 565 which is equivalent tocoscosBPDAP C=vv^12. Snell’s law follows once we note that the angles of incidence and r ...

566 Real Analysis We want to maximize the expressionab √ 1 −x^2 +cd √ 1 −y^2 , which is twice the area of the rectangle. Let f ( ...

Real Analysis 567 The method of Lagrange multipliers gives rise to the system of equations (λ− 1 ) −a^3 +a(b^2 +c^2 ) 4 = a+b+c ...

568 Real Analysis One possible solution to this system isa=b=c=d=^14 , in which casef(^14 ,^14 ,^14 ,^14 )= Otherwise, let us a ...

Real Analysis 569 513.Fixα, β, γand consider the function f (x, y, z)= cosx sinα + cosy sinβ + cosz sinγ with the constraintsx+y ...