MathDB

13

Part of 2018 Hanoi Open Mathematics Competitions

Problems(2)

27 handshakes of m x n students (HOMC 2018 Ind. p13)

Source:

1/31/2020
A competition room of HOMC has m×nm \times n students where m,nm, n are integers larger than 22. Their seats are arranged in mm rows and nn columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally 2727 handshakes. Find the number of students in the room.
combinatorics
n = S(n) + P(n), sum , product digits (HOMC 2018 ind. sen13)

Source:

2/2/2020
For a positive integer nn, let S(n),P(n)S(n), P(n) denote the sum and the product of all the digits of nn respectively. 1) Find all values of n such that n=P(n)n = P(n): 2) Determine all values of n such that n=S(n)+P(n)n = S(n) + P(n).
sum of digitsProductDigitsnumber theorySum