MathDB

1

Part of 2013 Vietnam National Olympiad

Problems(2)

system of equations

Source: Problem 1 VMO 2013 day 1

1/11/2013
Solve with full solution: {(sin⁡x)2+1(sin⁡x)2+(cos⁡y)2+1(cos⁡y)2=20yx+y(sin⁡y)2+1(sin⁡y)2+(cos⁡x)2+1(cos⁡x)2=20xx+y\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y} \\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right.
linear algebramatrixtrigonometryalgebra proposedalgebra
function equation

Source: Problem 5 VMO 2013 day 2

1/12/2013
Find all f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} that satisfies f(0)=0,f(1)=2013f(0)=0,f(1)=2013 and (x−y)(f(f2(x))−f(f2(y)))=(f(x)−f(y))(f2(x)−f2(y))(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y)) Note: f2(x)=(f(x))2f^2(x)=(f(x))^2
functionalgebra proposedalgebra