MathDB

4

Part of 2012 IMC

Problems(2)

IMC 2012 Day 1, Problem 4

Source:

7/28/2012
Let f:  R→Rf:\;\mathbb{R}\to\mathbb{R} be a continuously differentiable function that satisfies f′(t)>f(f(t))f'(t)>f(f(t)) for all t∈Rt\in\mathbb{R}. Prove that f(f(f(t)))≤0f(f(f(t)))\le0 for all t≥0t\ge0.
Proposed by Tomáš Bárta, Charles University, Prague.
functionlimitintegrationIMCcollege contests
Cyclic equations

Source: IMC 2012, Day 2, Problem 4

7/29/2012
Let n≥2n \ge 2 be an integer. Find all real numbers aa such that there exist real numbers x1,x2,…,xnx_1,x_2,\dots,x_n satisfying x1(1−x2)=x2(1−x3)=⋯=xn(1−x1)=a.x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.
Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.
functionlinear algebramatrixalgebrapolynomialtrigonometryfloor function