MathDB

Problem 1

Part of 2001 VJIMC

Problems(2)

x-y<=xy/25 if x,y in A, max |A|

Source: VJIMC 2001 1.1

7/23/2021
Let AA be a set of positive integers such that for any x,y∈Ax,y\in A, x>y  ⟹  x−y≥xy25.x>y\implies x-y\ge\frac{xy}{25}.Find the maximal possible number of elements of the set AA.
number theorySetsinequalities
integers uniquely determined by pairwise sums

Source: VJIMC 2001 2.1

7/26/2021
Let n≥2n\ge2 be an integer and let x1,x2,…,xnx_1,x_2,\ldots,x_n be real numbers. Consider N=(n2)N=\binom n2 sums xi+xjx_i+x_j, 1≤i<j≤n1\le i<j\le n, and denote them by y1,y2,…,yNy_1,y_2,\ldots,y_N (in an arbitrary order). For which nn are the numbers x1,x2,…,xnx_1,x_2,\ldots,x_n uniquely determined by the numbers y1,y2,…,yNy_1,y_2,\ldots,y_N?
number theorycombinatorics