MathDB

4

Part of 2017 China Team Selection Test

Problems(5)

2017 China TSTST Day 2 Polynomials

Source: 2017 China TSTST Day 2 Problem 4

3/7/2017
Find out all the integer pairs (m,n)(m,n) such that there exist two monic polynomials P(x)P(x) and Q(x)Q(x) ,with deg⁡P=m\deg{P}=m and deg⁡Q=n\deg{Q}=n,satisfy that P(Q(t))≠Q(P(t))P(Q(t))\not=Q(P(t)) holds for any real number tt.
TststPolynomialsalgebrapolynomialChina
China Team Selection Test 2017 TST 2 Day 2 Q4

Source: China Shanghai ,Mar 12, 2017

3/12/2017
An integer n>1n>1 is given . Find the smallest positive number mm satisfying the following conditions： for any set {a,b}\{a,b\} ⊂{1,2,⋯ ,2n−1}\subset \{1,2,\cdots,2n-1\} ,there are non-negative integers x,y x, y ( not all zero) such that 2n∣ax+by2n|ax+by and x+y≤m.x+y\leq m.
number theoryalgebraChina TST
number theory from China TST

Source: China TST 4 Problem 4

3/22/2017
Given integer d>1,md>1,m,prove that there exists integer k>l>0k>l>0, such that (22k+d,22l+d)>m.(2^{2^k}+d,2^{2^l}+d)>m.
number theoryChina TSTkobayashi
Existence of polynomial satisfying conditions

Source: China TSTST 3 Day 2 Problem 1

3/17/2017
Show that there exists a degree 5858 monic polynomial P(x)=x58+a1x57+⋯+a58P(x) = x^{58} + a_1x^{57} + \cdots + a_{58} such that P(x)P(x) has exactly 2929 positive real roots and 2929 negative real roots and that log⁡2017∣ai∣\log_{2017} |a_i| is a positive integer for all 1≤i≤581 \leq i \leq 58.
algebrapolynomial
Loci of circumcenters

Source: 2017 China TST 5 P4

4/14/2017
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with 0<l≤20<l\le 2. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
geometrycircumcircle