MathDB

12

Part of 2019 India PRMO

Problems(2)

Find sum of natural numbers satisfying the following condition

Source:

8/11/2019
A natural number k>1k > 1 is called good if there exist natural numbers
a1<a2<⋯<aka_1 < a_2 < \cdots < a_k
such that
1a1+1a2+⋯+1ak=1\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1.
Let f(n)f(n) be the sum of the first nn [good numbers, n≥n \geq 1. Find the sum of all values of nn for which f(n+5)/f(n)f(n+5)/f(n) is an integer.
Natural Numbersfunction
Easy counting

Source: PRMO 2019 Leg 2 P12

8/25/2019
Let NN be the number of ways of choosing a subset of 55 distinct numbers from the set 10a+b:1≤a≤5,1≤b≤5{10a+b:1\leq a\leq 5, 1\leq b\leq 5} where a,ba,b are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when NN is divided by 7373?
combinatoricscountingEnumerationEnumerative CombinatoricsPRMO