MathDB

3

Part of 2012 Poland - Second Round

Problems(2)

Divisor of a product

Source: Polish MO second round 2012

2/19/2012
Let m,n∈Z+m,n\in\mathbb{Z_{+}} be such numbers that set {1,2,…,n}\{1,2,\ldots,n\} contains exactly mm different prime numbers. Prove that if we choose any m+1m+1 different numbers from {1,2,…,n}\{1,2,\ldots,n\} then we can find number from m+1m+1 choosen numbers, which divide product of other mm numbers.
number theoryleast common multiplevectorabsolute valuenumber theory unsolved
Sum of digits

Source: Polish MO second round 2012

2/19/2012
Denote by S(k)S(k) the sum of the digits in the decimal representation of kk. Prove that there are infinitely many n∈Z+n\in \mathbb{Z_{+}} such that: S(2n+n)<S(2n){S(2^{n}+n})<S(2^{n}).
inductionnumber theory unsolvednumber theory