MathDB

3

Part of 2017 VJIMC

Problems(2)

Convex polyhedron with numbers at its vertices

Source: VJIMC 2017, Category I, Problem 3

4/2/2017
Let PP be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of PP in such a way that the sum of these numbers is 11. Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most 38\frac{3}{8}.
college contestsgeometry
Cyclic system of equations with many solutions

Source: VJIMC 2017, Category II, Problem 3

4/2/2017
Let n≥2n \ge 2 be an integer. Consider the system of equations \begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align} 1. Prove that (1)(1) has infinitely many real solutions (x1,…,xn)(x_1,\dotsc,x_n) such that the numbers x1,…,xnx_1,\dotsc,x_n are distinct. 2. Prove that every solution of (1)(1), such that the numbers x1,…,xnx_1,\dotsc,x_n are not all equal, satisfies ∣x1x2⋯xn∣=2n/2\vert x_1x_2\cdots x_n\vert=2^{n/2}.
algebrasystem of equations