MathDB

Problem 5

Part of 2024 Kyiv City MO Round 1

Problems(4)

Answer to this NT might surprise you...

Source: Kyiv City MO 2024 Round 1, Problem 8.5

1/28/2024
Find the smallest positive integer nn that has at least 77 positive divisors 1=d1<d2<…<dk=n1 = d_1 < d_2 < \ldots < d_k = n, k≥7k \geq 7, and for which the following equalities hold:
d7=2d5+1 and d7=3d4−1d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1
Proposed by Mykyta Kharin
number theoryDivisors
Is this algebra statement not nice?

Source: Kyiv City MO 2024 Round 1, Problem 10.5/11.4

1/28/2024
Find the smallest real number MM, for which {a}+{b}+{c}≤M\{a\}+\{b\}+\{c\}\leq M for any real positive numbers a,b,ca, b, c with abc=2024abc = 2024. Here {a}\{a\} denotes the fractional part of number aa.
Proposed by Fedir Yudin, Anton Trygub
algebra
Smallest value of |253^m - 40^n|

Source: Kyiv City MO 2024 Round 1, Problem 9.5

1/28/2024
Find the smallest value of the expression ∣253m−40n∣|253^m - 40^n| over all pairs of positive integers (m,n)(m, n).
Proposed by Oleksii Masalitin
number theoryDiophantine equation
Funky number theory

Source: Kyiv City MO 2024 Round 1, Problem 11.5

1/28/2024
Find all functions f:N→Nf : \mathbb{N} \to \mathbb{N} such that for any positive integers m,nm, n the number (f(m))2+2mf(n)+f(n2)(f(m))^2+ 2mf(n) + f(n^2) is the square of an integer.
Proposed by Fedir Yudin
functional equationnumber theoryFunctional Equations