MathDB

2016 Taiwan TST Round 3

Part of TST Round 3

Subcontests

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Sequence and Euler Function

Let kk be a positive integer. A sequence a0,a1,...,an,n>0a_0,a_1,...,a_n,n>0 of positive integers satisfies the following conditions: (i)(i) a0=an=1a_0=a_n=1; (ii)(ii) 2≤ai≤k2\leq a_i\leq k for each i=1,2,...,n−1i=1,2,...,n-1; (iii)(iii)For each j=2,3,...,kj=2,3,...,k, the number jj appears ϕ(j)\phi(j) times in the sequence a0,a1,...,ana_0,a_1,...,a_n, where ϕ(j)\phi(j) is the number of positive integers that do not exceed jj and are coprime to jj; (iv)(iv)For any i=1,2,...,n−1i=1,2,...,n-1, gcd⁡(ai,ai−1)=1=gcd⁡(ai,ai+1)\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1}), and aia_i divides ai−1+ai+1a_{i-1}+a_{i+1}. Suppose there is another sequence b0,b1,...,bnb_0,b_1,...,b_n of integers such that bi+1ai+1>biai\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i} for all i=0,1,...,n−1i=0,1,...,n-1. Find the minimum value of bn−b0b_n-b_0.

Super robots and overpower laser beam

There's a convex 3n3n-polygon on the plane with a robot on each of it's vertices. Each robot fires a laser beam toward another robot. On each of your move,you select a robot to rotate counter clockwise until it's laser point a new robot. Three robots AA, BB and CC form a triangle if AA's laser points at BB, BB's laser points at CC, and CC's laser points at AA. Find the minimum number of moves that can guarantee nn triangles on the plane.

Function Equation

Determine all functions f:R+→R+f:\mathbb{R}^+\rightarrow \mathbb{R}^+ satisfying f(x+y+f(y))=4030x−f(x)+f(2016y),∀x,y∈R+f(x+y+f(y))=4030x-f(x)+f(2016y), \forall x,y \in \mathbb{R}^+.