MathDB

2

Part of 1992 China Team Selection Test

Problems(2)

sum equality implies product-sum inequality

Source: China TST 1992, problem 2

6/27/2005
Let n≥2,n∈N,n \geq 2, n \in \mathbb{N}, find the least positive real number λ\lambda such that for arbitrary ai∈Ra_i \in \mathbb{R} with i=1,2,…,ni = 1, 2, \ldots, n and bi∈[0,12]b_i \in \left[0, \frac{1}{2}\right] with i=1,2,…,ni = 1, 2, \ldots, n, the following holds: ∑i=1nai=∑i=1nbi=1⇒∏i=1nai≤λ∑i=1naibi.\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.
inequalitiesalgebra unsolvedalgebra
(3n + 1) * (3n + 1) table

Source: China TST 1992, problem 5

6/27/2005
A (3n+1)×(3n+1)(3n + 1) \times (3n + 1) table (n∈N)(n \in \mathbb{N}) is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a 2×22 \times 2 squares.
geometrygeometric transformationrotationinductionrectanglecombinatorics unsolvedcombinatorics