MathDB

4

Part of 2019 VJIMC

Problems(2)

VJIMC 2019 P4

Source: VJIMC 2019

3/29/2019
Determine the largest constant K≥0K\geq 0 such that aa(b2+c2)(aa−1)2+bb(c2+a2)(bb−1)2+cc(a2+b2)(cc−1)2≥K(a+b+cabc−1)2\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 holds for all positive real numbers a,b,ca,b,c such that ab+bc+ca=abcab+bc+ca=abc.
Proposed by Orif Ibrogimov (Czech Technical University of Prague).
algebrainequalitiesreal analysisVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic
VJIMC 2019 P4 Category II

Source: VJIMC 2019

3/30/2019
Let D={z∈C:Im⁡(z)>0,Re⁡(z)>0}D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} . Let n≥1n \geq 1 and let a1,a2,…an∈Da_1,a_2,\dots a_n \in D be distinct complex numbers. Define f(z)=z⋅∏j=1nz−ajz−aj‾f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}} Prove that f′f' has at least one root in DD.
Proposed by Géza Kós (Lorand Eotvos University, Budapest)
complex analysisreal analysiscomplex numbersVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic