MathDB

2

Part of 2015 China Team Selection Test

Problems(4)

China Team Selection Test 2015 TST 1 Day 1 Q2

Source: China Hangzhou

3/13/2015
Let a1,a2,a3,⋯a_1,a_2,a_3, \cdots be distinct positive integers, and 0<c<320<c<\frac{3}{2} . Prove that : There exist infinitely many positive integers kk, such that [ak,ak+1]>ck[a_k,a_{k+1}]>ck .
number theoryinequalitiesChina TST
China Team Selection Test 2015 TST 2 Day 1 Q2

Source: China Hangzhou

3/19/2015
Let a1,a2,a3,⋯ ,ana_1,a_2,a_3, \cdots ,a_n be positive real numbers. For the integers n≥2n\ge 2, prove that(∑j=1n(∏k=1jak)1j∑j=1naj)1n+(∏i=1nai)1n∑j=1n(∏k=1jak)1j≤n+1n \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}
inequalitiesinequalities proposedChina TST
Union of some subsets is original set

Source: CTST TST 3 Day 1 Q2

3/29/2015
Let XX be a non-empty and finite set, A1,...,AkA_1,...,A_k kk subsets of XX, satisying:
(1) ∣Ai∣≤3,i=1,2,...,k|A_i|\leq 3,i=1,2,...,k (2) Any element of XX is an element of at least 44 sets among A1,....,AkA_1,....,A_k.
Show that one can select [3k7][\frac{3k}{7}] sets from A1,...,AkA_1,...,A_k such that their union is XX.
combinatoricsProbabilistic MethodChina TSTChina
Tricoloured Complete Graph

Source: China Team Selection Test 3 Day 2 P2

3/25/2015
Let GG be the complete graph on 20152015 vertices. Each edge of GG is dyed red, blue or white. For a subset VV of vertices of GG, and a pair of vertices (u,v)(u,v), define L(u,v)={u,v}∪{w∣w∈V∋△uvw has exactly 2 red sides} L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}Prove that, for any choice of VV, there exist at least 120120 distinct values of L(u,v)L(u,v).
graph theorycombinatorics