MathDB

4:

Part of 1994 Romania TST for IMO

Problems(3)

old problem

Source: Romania TST for IMO 1994

9/6/2017
Inscribe an equilateral triangle of minimum side in a given acute-angled triangle ABCABC (one vertex on each side).
geometry
romania tsts

Source: Romania TST for IMO 1994 second exam

9/6/2017
Let be given two concentric circles of radii RR and R1>RR_1 > R. Let quadrilateral ABCDABCD is inscribed in the smaller circle and let the rays CD,DA,AB,BCCD, DA, AB, BC meet the larger circle at A1,B1,C1,D1A_1, B_1, C_1, D_1 respectively. Prove that σ(A1B1C1D1)σ(ABCD)≥R12R2 \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2} where σ(P)\sigma(P) denotes the area of a polygon P.P.
triangle inequalitygeometryinequalities
Romania TST 1994, 3rd P4

Source:

4/30/2010
Find a sequence of positive integer f(n)f(n), n∈Nn \in \mathbb{N} such that
(1)(1) f(n)≤n8f(n) \leq n^8 for any n≥2n \geq 2, (2)(2) for any pairwisely distinct natural numbers a1,a2,⋯ ,aka_1,a_2,\cdots, a_k and nn, we have that f(n)≠f(a1)+f(a2)+⋯+f(ak)f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)
algebra unsolvedalgebra