MathDB

Problem 1

Part of 2004 VJIMC

Problems(2)

tangents form equilateral triangle

Source: VJIMC 2004 1.1

7/2/2021
Suppose that f:[0,1]→Rf:[0,1]\to\mathbb R is a continuously differentiable function such that f(0)=f(1)=0f(0)=f(1)=0 and f(a)=3f(a)=\sqrt3 for some a∈(0,1)a\in(0,1). Prove that there exist two tangents to the graph of ff that form an equilateral triangle with an appropriate segment of the xx-axis.
geometryfunctioncalculus
(Q,+) and (Q+,*) isomorphic?

Source: VJIMC 2004 2.1

7/12/2021
Are the groups (Q,+)(\mathbb Q,+) and (Q+,⋅)(\mathbb Q^+,\cdot) isomorphic?
group theoryabstract algebra