MathDB

2

Part of 1986 China Team Selection Test

Problems(2)

China TST 1986 equivalence of two sum inequalities

Source: China TST 1986, problem 2

5/16/2005
Let a1 a_1, a2 a_2, ..., an a_n and b1 b_1, b2 b_2, ..., bn b_n be 2⋅n 2 \cdot n real numbers. Prove that the following two statements are equivalent: i) For any n n real numbers x1 x_1, x2 x_2, ..., xn x_n satisfying x1≤x2≤…≤xn x_1 \leq x_2 \leq \ldots \leq x_ n, we have \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k, ii) We have \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k for every s\in\left\{1,2,...,n\minus{}1\right\} and \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k.
inequalitiesfunctionvectoralgebra unsolvedalgebra
China TST 1986 tetrahedron perimeter max inequality

Source: China TST 1986, problem 6

5/16/2005
Given a tetrahedron ABCDABCD, EE, FF, GG, are on the respectively on the segments ABAB, ACAC and ADAD. Prove that: i) area EFG≤EFG \leq max{area ABCABC,area ABDABD,area ACDACD,area BCDBCD}. ii) The same as above replacing "area" for "perimeter".
geometry3D geometrytetrahedronperimeterinequalitiesfunctiongeometry unsolved