MathDB

8

Part of 1993 IMO Shortlist

Problems(2)

Which IMO Shortlist Problem formulation do you prefer?

Source: Imo Shortlist 1993, Romania 3

3/25/2006
Let c1,…,cn∈Rc_1, \ldots, c_n \in \mathbb{R} with n≥2n \geq 2 such that 0≤∑i=1nci≤n. 0 \leq \sum^n_{i=1} c_i \leq n. Show that we can find integers k1,…,knk_1, \ldots, k_n such that ∑i=1nki=0 \sum^n_{i=1} k_i = 0 and 1−n≤ci+n⋅ki≤n 1-n \leq c_i + n \cdot k_i \leq n for every i=1,…,n.i = 1, \ldots, n. [hide="Another formulation:"] Let x1,…,xn,x_1, \ldots, x_n, with n≥2n \geq 2 be real numbers such that ∣x1+…+xn∣≤n. |x_1 + \ldots + x_n| \leq n. Show that there exist integers k1,…,knk_1, \ldots, k_n such that ∣k1+…+kn∣=0. |k_1 + \ldots + k_n| = 0. and ∣xi+2⋅n⋅ki∣≤2⋅n−1 |x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1 for every i=1,…,n.i = 1, \ldots, n. In order to prove this, denote ci=1+xi2c_i = \frac{1+x_i}{2} for i=1,…,n,i = 1, \ldots, n, etc.
algebraSequenceInequalityIMO Shortlist
Highly recommended by the Problem Committee

Source: IMO Shortlist 1993, Indonesia 1

3/25/2006
The vertices D,E,FD,E,F of an equilateral triangle lie on the sides BC,CA,ABBC,CA,AB respectively of a triangle ABC.ABC. If a,b,ca,b,c are the respective lengths of these sides, and SS the area of ABC,ABC, prove that DE≥2⋅2⋅Sa2+b2+c2+4⋅3⋅S. DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}.
geometryratiocircumcircletrigonometryinequalitiesgeometric inequalityIMO Shortlist