MathDB

A1

Part of 2016 Costa Rica - Final Round

Problems(2)

sum ...+1/(\sqrt{2015}+\sqrt{2016}) OLCOMA Costa Rica Final 2016 Shortlist A1 D1

Source:

9/23/2021
Prove that (11+2+12+3+13+4+...+12015+2016)2(2017+2414)=20152\left( \frac{1}{\sqrt1+\sqrt2}+\frac{1}{\sqrt2+\sqrt3}+\frac{1}{\sqrt3+\sqrt4}+...+\frac{1}{\sqrt{2015}+\sqrt{2016}}\right)^2(2017+24\sqrt{14})=2015^2
algebra
4x4 non linear system of equations with cubic radicals

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist A1 day2

9/24/2021
Find all solutions of the system
yz4x23+2wx=0\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 xz4y3+5wy=0\sqrt[3]{\frac{xz^4}{y}}+5wy=0 xyx3+7wz−1/3=0\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0 x12+1254y5+3432z4=16x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16
where x,y,z≥0x, y, z \ge 0 and w∈Rw \in R
I attached the system, in case I have any typos
algebrasystem of equations