MathDB

4

Part of 2015 Korea National Olympiad

Problems(2)

Partitions with special conditions

Source: 2015 Korean Mathematical Olympiad P4

11/1/2015
For positive integers n,k,ln, k, l, we define the number of ll-tuples of positive integers (a1,a2,⋯al)(a_1,a_2,\cdots a_l) satisfying the following as Q(n,k,l)Q(n,k,l).
(i): n=a1+a2+⋯+aln=a_1+a_2+\cdots +a_l
(ii): a1>a2>⋯>al>0a_1>a_2>\cdots > a_l > 0.
(iii): ala_l is an odd number.
(iv): There are kk odd numbers out of aia_i.
For example, from 9=8+1=6+3=6+2+19=8+1=6+3=6+2+1, we have Q(9,1,1)=1Q(9,1,1)=1, Q(9,1,2)=2Q(9,1,2)=2, Q(9,1,3)=1Q(9,1,3)=1.
Prove that if n>k2n>k^2, ∑l=1nQ(n,k,l)\sum_{l=1}^n Q(n,k,l) is 00 or an even number.
combinatoricspartitionsbijection
Inequality of relatively prime numbers

Source: 2015 Korean Mathematical Olympiad P8

11/1/2015
For a positive integer nn, a1,a2,⋯aka_1, a_2, \cdots a_k are all positive integers without repetition that are not greater than nn and relatively prime to nn. If k>8k>8, prove the following. ∑i=1k∣ai−n2∣<n(k−4)2\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}
inequalitiesnumber theoryrelatively primeprime numbersCoprime integers