MathDB

3

Part of 2008 Vietnam Team Selection Test

Problems(2)

wanting set

Source: Vietnam TST 2008, Problem 3

9/5/2008
Let an integer n>3 n > 3. Denote the set T\equal{}\{1,2, \ldots,n\}. A subset S of T is called wanting set if S has the property: There exists a positive integer c c which is not greater than n2 \frac {n}{2} such that |s_1 \minus{} s_2|\ne c for every pairs of arbitrary elements s1,s2∈S s_1,s_2\in S. How many does a wanting set have at most are there ?
combinatorics proposedcombinatorics
two sets

Source: Vietnam TST 2008, Problem 6

9/5/2008
Consider the set M={1,2,…,2008} M = \{1,2, \ldots ,2008\}. Paint every number in the set M M with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets: S1={(x,y,z)∈M3 ∣ x,y,z have the same color and 2008∣(x+y+z)} S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}; S2={(x,y,z)∈M3 ∣ x,y,z have three pairwisely different colors and 2008∣(x+y+z)} S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}. Prove that 2∣S1∣>∣S2∣ 2|S_1| > |S_2| (where ∣X∣ |X| denotes the number of elements in a set X X).
functionalgebrapolynomialgroup theorycombinatorics proposedcombinatorics