MathDB

1

Part of 2017 China Team Selection Test

Problems(5)

2017 China TSTST Day 1 Problem 1

Source: 2017 China TSTST Day 1 Problem 1

3/7/2017
Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge ABAB of the regular octahedron from point PP outside if and only if the intersection of non degenerate triangle PABPAB and the solid regular octahedron is exactly edge ABAB.
geometryTststsolid geometry3D geometryoctahedron
Divisors of n are distinct mod m

Source: China TSTST Test 2 Day 1 Q1

3/13/2017
Let nn be a positive integer. Let DnD_n be the set of all divisors of nn and let f(n)f(n) denote the smallest natural mm such that the elements of DnD_n are pairwise distinct in mod mm. Show that there exists a natural NN such that for all n≥Nn \geq N, one has f(n)≤n0.01f(n) \leq n^{0.01}.
number theoryDivisorsmodular arithmetic
Cyclic Inequality

Source: China TSTST 3 Day 1 Problem 1

3/17/2017
Let n≥4n \geq 4 be a natural and let x1,…,xnx_1,\ldots,x_n be non-negative reals such that x1+⋯+xn=1x_1 + \cdots + x_n = 1. Determine the maximum value of x1x2x3+x2x3x4+⋯+xnx1x2x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2.
inequalities
An equality from China TST

Source: China TST 4 Problem 1

3/22/2017
Prove that :∑k=058C2017+k58−kC2075−kk=∑p=029C4091−2p58−2p\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}
TSTcombinatoricsequationChinagenerating functions
NOT-AP triple

Source: 2017 China TST 5 P1

4/8/2017
Given n≥3n\ge 3. consider a sequence a1,a2,...,ana_1,a_2,...,a_n, if (ai,aj,ak)(a_i,a_j,a_k) with i+k=2j (iai+ak≠2aja_i+a_k\ne 2a_j, we call such a triple a NOT−APNOT-AP triple. If a sequence has at least one NOT−APNOT-AP triple, find the least possible number of the NOT−APNOT-AP triple it contains.
combinatorics