MathDB

4

Part of 2023 New Zealand MO

Problems(2)

f(x) = ax^2 + bx + c is divisible by prime p whenever x , 0 < a, b, c <= p

Source: 2023 NZMO - New Zealand Maths Olympiad Round 1 p4

9/2/2023
Let pp be a prime and let f(x)=ax2+bx+cf(x) = ax^2 + bx + c be a quadratic polynomial with integer coefficients such that 0<a,b,c≤p0 < a, b, c \le p. Suppose f(x)f(x) is divisible by pp whenever xx is a positive integer. Find all possible values of a+b+ca + b + c.
number theory
f(m) = f(m + 1), where f(n) is number of subsets of {1,...,n}

Source: 2023 NZMO - New Zealand Maths Olympiad Round 2 p4

9/2/2023
For any positive integer nn, let f(n)f(n) be the number of subsets of {1,2,...,n}\{1, 2, . . . , n\} whose sum is equal to nn. Does there exist infinitely many positive integers mm such that f(m)=f(m+1)f(m) = f(m + 1)? (Note that each element in a subset must be distinct.)
combinatorics