MathDB

5

Part of 2019 South East Mathematical Olympiad

Problems(2)

Red Subset?

Source: 2019 CSMO Grade 10 P5

7/31/2019
Let S={1928,1929,1930,⋯ ,1949}.S=\{1928,1929,1930,\cdots,1949\}. We call one of SS’s subset MM is a red subset, if the sum of any two different elements of MM isn’t divided by 4.4. Let x,yx,y be the number of the red subsets of SS with 44 and 55 elements,respectively. Determine which of x,yx,y is greater and prove that.
combinatoricsCombinatorial Number Theory
One sequence problem

Source: 2019 CSMO Grade 11 P5

7/31/2019
For positive integer n, define ana_n as the number of the triangles with integer length of every side and the length of the longest side being 2n.2n. (1) Find ana_n in terms of n;n; (2)If the sequence {bn}\{ b_n\} satisfying for any positive integer n,n, ∑k=1n(−1)n−k(nk)bk=an.\sum_{k=1}^n(-1)^{n-k}\binom {n}{k} b_k=a_n. Find the number of positive integer nn satisfying that bn≤2019an.b_n\leq 2019a_n.
algebraSequence