MathDB

3

Part of 1987 China Team Selection Test

Problems(2)

China TST 1987 sequence challenge

Source: China TST 1987, problem 3

5/16/2005
Let r1=2r_1=2 and rn=∏k=1n−1ri+1r_n = \prod^{n-1}_{k=1} r_i + 1, n≥2.n \geq 2. Prove that among all sets of positive integers such that ∑k=1n1ai<1,\sum^{n}_{k=1} \frac{1}{a_i} < 1, the partial sequences r1,r2,...,rnr_1,r_2, ... , r_n are the one that gets nearer to 1.
inductionalgebra unsolvedalgebra
two triangles with a common edge [variation on Turan]

Source: China TST 1987, problem 6

5/16/2005
Let G G be a simple graph with 2⋅n 2 \cdot n vertices and n2+1 n^{2}+1 edges. Show that this graph G G contains a K4−one edge K_{4}-\text{one edge}, that is, two triangles with a common edge.
inductioncombinatorics unsolvedcombinatorics