MathDB

P3

Part of 2023 District Olympiad

Problems(4)

Inequality

Source: Romanian District Olympiad 2023 9.3

3/11/2023
Let x,yx,y{} and zz{} be positive real numbers satisfying x+y+z=1x+y+z=1. Prove that
[*]1−x2−yzx2+x=(1−y)(1−z)x2+x;1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x}; [*]x2−yzx2+x+y2−zxy2+y+z2−xyz2+z⩽0.\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.
algebrainequalities
Complex number inequality

Source: Romanian District Olympiad 2023 10.3

3/11/2023
Let n⩾2n\geqslant 2 be an integer. Determine all complex numbers zz{} which satisfy ∣zn+1−zn∣⩾∣zn+1−1∣+∣zn+1−z∣.|z^{n+1}-z^n|\geqslant|z^{n+1}-1|+|z^{n+1}-z|.
complex numbersinequalities
Function has at least three fixed points

Source: Romanian District Olympiad 2023 11.3

3/11/2023
Let f:[a,b]→[a,b]f:[a,b]\to[a,b] be a continuous function. It is known that there exist α,β∈(a,b)\alpha,\beta\in (a,b) such that f(α)=af(\alpha)=a and f(β)=bf(\beta)=b. Prove that the function f∘ff\circ f has at least three fixed points.
real analysiscontinuityfunction
Limits of integrals

Source: Romanian District Olympiad 2023 12.3

3/11/2023
Let f:[0,1]→Rf:[0,1]\to\mathbb{R} be a continuous function. Prove that lim⁡n→∞∫01f(xn) dx=f(0).\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).Furthermore, if f(0)=0f(0)=0 and ff is right-differentiable in 00{}, prove that the limits \lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx \text{and} \lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)exist, are finite and are equal.
real analysisIntegrallimit