Advanced book on Mathematics Olympiad

3.4 Equations with Functions as Unknowns 187 Example.Letx 1 ,x 2 ,...,xnbe arbitrary real numbers. Prove the inequality x 1 1 +x ...

188 3 Real Analysis Increase the left-hand side tox+ √ k; then square both sides. We obtain x^2 +k+ 2 x √ k≤k+kx^2 + 1 +x^2 , wh ...

3.4 Equations with Functions as Unknowns 189 543.Find all functionsf:( 0 ,∞)→( 0 ,∞)subject to the conditions (i)f(f(f(x)))+ 2 x ...

190 3 Real Analysis choice (Zorn’s lemma) implies the existence of a basis for this vector space. If(ei)i∈Iis this basis, then a ...

3.4 Equations with Functions as Unknowns 191 We conclude our discussion about functional equations with another instance in whic ...

192 3 Real Analysis Example.Find all continuous functionsf:R→Rsatisfying the equation f(x)=λ( 1 +x^2 ) [ 1 + ∫x 0 f(t) 1 +t^2 dt ...

3.4 Equations with Functions as Unknowns 193 is equal to zero. For functions defined on the entire two-dimensional plane, the fi ...

194 3 Real Analysis 553.A not uncommon mistake is to believe that the product rule for derivatives says that (f g)′=f′g′.Iff(x)= ...

3.4 Equations with Functions as Unknowns 195 3.4.3 Ordinary Differential Equations of Higher Order The field of higher-order ord ...

196 3 Real Analysis z(t)=(a+ib)exp ( − 1 + √ 5 2 it ) +(c+id)exp ( − 1 − √ 5 2 it ) , for arbitrary real numbersa, b, c, d. Sinc ...

3.4 Equations with Functions as Unknowns 197 with general solutionu=C 1 y^2 +C 2 y−^2. Remember thatu=p^2 =(y′)^2 , from which w ...

198 3 Real Analysis 564.Letnbe a positive integer. Show that the equation ( 1 −x^2 )y′′−xy′+n^2 y= 0 admits as a particular solu ...

3.4 Equations with Functions as Unknowns 199 Iterating, we eventually obtain yiv(x)=− ∫∞ 0 e−t (^2) / 2 cos x^2 2 t^2 dt=−y(x), ...

200 3 Real Analysis 571.Determine allnth-degree polynomialsP(x), with real zeros, for which the equality ∑n i= 1 1 P(x)−xi = n^2 ...

4 Geometry and Trigonometry...................................... Geometry is the oldest of the mathematical sciences. Its age-o ...

202 4 Geometry and Trigonometry product of the magnitudes of the vectors and the cosine of the angle between them. A dot product ...

4.1 Geometry 203 establishes an isomorphism between(R^3 ,×)and(so( 3 ),[·,·]). Proof.The proof is straightforward if we write th ...

204 4 Geometry and Trigonometry 579.Does there exist a bijectionf of (a) a plane with itself or (b) three-dimensional space with ...

4.1 Geometry 205 A M N B CP Q D Figure 24 a problem in plane geometry there is no danger in identifying the areas with the cross ...

206 4 Geometry and Trigonometry 584.On the sides of the triangleABCconstruct in the exterior the rectanglesABB 1 A 2 , BCC 1 B 2 ...