MathDB

4

Part of 2015 China Team Selection Test

Problems(2)

China Team Selection Test 2015 TST 1 Day 2 Q1

Source: China Hangzhou

3/14/2015
Prove that : For each integer n≥3n \ge 3, there exists the positive integers a1<a2<⋯<ana_1<a_2< \cdots <a_n , such that for i=1,2,⋯ ,n−2 i=1,2,\cdots,n-2 , With ai,ai+1,ai+2a_{i},a_{i+1},a_{i+2} may be formed as a triangle side length , and the area of the triangle is a positive integer.
algebraTrianglesgeometryChina TST
Bounded functions on reals

Source: 2015 China TST 2 Day 2 Q1

3/19/2015
Let nn be a positive integer, let f1(x),…,fn(x)f_1(x),\ldots,f_n(x) be nn bounded real functions, and let a1,…,ana_1,\ldots,a_n be nn distinct reals. Show that there exists a real number xx such that ∑i=1nfi(x)−∑i=1nfi(x−ai)<1\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1.
functionalgebraalgebra proposedcombinatorics