MathDB

6

Part of 2017 China Team Selection Test

Problems(5)

2017 China TSTST 1 Day 2 Problem 6

Source: 2017 China TSTST 1 Day 2 Problem 6

3/7/2017
For a given positive integer nn and prime number pp, find the minimum value of positive integer mm that satisfies the following property: for any polynomial f(x)=(x+a1)(x+a2)…(x+an)f(x)=(x+a_1)(x+a_2)\ldots(x+a_n) (a1,a2,…,ana_1,a_2,\ldots,a_n are positive integers), and for any non-negative integer kk, there exists a non-negative integer k′k' such that vp(f(k))<vp(f(k′))≤vp(f(k))+m.v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m. Note: for non-zero integer NN,vp(N)v_p(N) is the largest non-zero integer tt that satisfies pt∣Np^t\mid N.
number theoryPolynomialsprime numbersalgebrapolynomial
Subset of reals (mod 1) with positive measure

Source: China TSTST 2017 Test 2 Day 2 Q6

3/13/2017
Let MM be a subset of R\mathbb{R} such that the following conditions are satisfied:
a) For any x∈M,n∈Zx \in M, n \in \mathbb{Z}, one has that x+n∈Mx+n \in \mathbb{M}. b) For any x∈Mx \in M, one has that −x∈M-x \in M. c) Both MM and R\mathbb{R} \ MM contain an interval of length larger than 00.
For any real xx, let M(x)={n∈Z+∣nx∈M}M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}. Show that if α,β\alpha,\beta are reals such that M(α)=M(β)M(\alpha) = M(\beta), then we must have one of α+β\alpha + \beta and α−β\alpha - \beta to be rational.
algebracombinatoricsnumber theory
Center of grid is endpoint of path

Source: China TSTST 3 Day 2 Q3

3/18/2017
Every cell of a 2017×20172017\times 2017 grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let V1V_1 be the set of all black cells, V2V_2 be the set of all white cells. For set Vi(i=1,2)V_i (i=1,2), if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs GiG_i. If both G1G_1 and G2G_2 are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of G1G_1 or G2G_2.
combinatorics
3D geometry from China TST

Source: China TST 4 Problem 6

3/23/2017
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
geometry3D geometrydodecahedron
k flowing chromatic

Source: 2017 China TST 5 P6

4/14/2017
We call a graph with n vertices k−flowing−chromatick-flowing-chromatic if: 1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors. 2. we can choose a hamilton cycle v1,v2,⋯ ,vnv_1,v_2,\cdots , v_n, and move the chess on viv_i to vi+1v_{i+1} with i=1,2,⋯ ,ni=1,2,\cdots ,n and vn+1=v1v_{n+1}=v_1, such that any two neighboring chess also have different colors. 3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote χ(G)\chi (G) the chromatic number of G. Find all the positive number m such that there is a graph G with χ(G)≤m\chi (G)\le m and T(G)≥2mT(G)\ge 2^m without a cycle of length small than 2017.
graph theorycombinatorics