MathDB

3

Part of 2014 China Team Selection Test

Problems(3)

China Team Selection Test 2014 TST 1 Day 1 Q3

Source: China Nanjing , 12 Mar 2014

3/12/2014
Let the function f:N∗→N∗f:N^*\to N^* such that (1) (f(m),f(n))≤(m,n)2014,∀m,n∈N∗(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*; (2) n≤f(n)≤n+2014,∀n∈N∗n\le f(n)\le n+2014 , \forall n\in N^* Show that: there exists the positive integers NN such that f(n)=n f(n)=n , for each integer n≥Nn \ge N. (High School Affiliated to Nanjing Normal University )
functionalgebrapolynomialnumber theory proposednumber theoryChina TST
Pairs of points with same distance in convex n-gon

Source: 2014 China TST 2 Day 1 Q3

3/20/2014
AA is the set of points of a convex nn-gon on a plane. The distinct pairwise distances between any 22 points in AA arranged in descending order is d1>d2>...>dm>0d_1>d_2>...>d_m>0. Let the number of unordered pairs of points in AA such that their distance is did_i be exactly μi\mu _i, for i=1,2,...,mi=1, 2,..., m. Prove: For any positive integer k≤mk\leq m, μ1+μ2+...+μk≤(3k−1)n\mu _1+\mu _2+...+\mu _k\leq (3k-1)n.
combinatorics proposedcombinatorics
China Team Selection Test 2014 TST 3 Day 1 Q3

Source: China Nanjing , 23 Mar 2014

3/23/2014
Show that there are no 2-tuples (x,y) (x,y) of positive integers satisfying the equation (x+1)(x+2)⋯(x+2014)=(y+1)(y+2)⋯(y+4028). (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).
number theory proposednumber theoryDiophantine equationChina TST