MathDB

6

Part of 1995 IMO Shortlist

Problems(3)

Inequality with n(n-1)/2

Source: IMO Shortlist 1995, A6

8/10/2008
Let n n be an integer,n≥3. n \geq 3. Let x1,x2,…,xn x_1, x_2, \ldots, x_n be real numbers such that x_i < x_{i\plus{}1} for 1 \leq i \leq n \minus{} 1. Prove that \frac{n(n\minus{}1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n\minus{}1}_{i\equal{}1} (n\minus{}i)\cdot x_i \right) \cdot \left(\sum^{n}_{j\equal{}2} (j\minus{}1) \cdot x_j \right)
inequalitiesn-variable inequalityIMO Shortlist
Tetrahedron and centroid inequality

Source: IMO Shortlist 1995, G6

8/10/2008
Let A1A2A3A4 A_1A_2A_3A_4 be a tetrahedron, G G its centroid, and A1′,A2′,A3′, A'_1, A'_2, A'_3, and A4′ A'_4 the points where the circumsphere of A1A2A3A4 A_1A_2A_3A_4 intersects GA1,GA2,GA3, GA_1,GA_2,GA_3, and GA4, GA_4, respectively. Prove that GA1⋅GA2⋅GA3⋅GA⋅4≤GA1′⋅GA2′⋅GA3′⋅GA4′ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4 and \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.
geometry3D geometrytetrahedrongeometric inequalitysphereIMO Shortlist
f(m + f(n)) = n + f(m + 95)

Source: IMO Shortlist 1995, S6

8/10/2008
Let N \mathbb{N} denote the set of all positive integers. Prove that there exists a unique function f:N↦N f: \mathbb{N} \mapsto \mathbb{N} satisfying f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95) for all m m and n n in N. \mathbb{N}. What is the value of \sum^{19}_{k \equal{} 1} f(k)?
functionalgebrafunctional equationIMO Shortlist