MathDB

1

Part of ICMC 7

Problems(2)

Divisibility of Fibonacci sum

Source: ICMC 7 Round 1 Problem 1

1/8/2024
Let FnF_n{} denote the nn{}-th Fibonacci number. Prove that 320233^{2023} divides 32⋅F4+33⋅F6+34⋅F8+⋯+32023F4046.3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.Proposed by Dylan Toh
number theoryFibonacciICMC
Divisibility with 2024 integers

Source: ICMC 7 Round 2 Problem 1

3/12/2024
Prove that there exist distinct positive integers a1,a2,…,a2024a_1, a_2,\ldots , a_{2024} such that for each i∈{1,2,…,2024}i\in\{1,2,\ldots,2024\}ai∣a1a2⋯ai−1ai+1⋯a2024+k,a_i\mid a_1a_2\cdots a_{i-1}a_{i+1}\cdots a_{2024}+k,where a) k=1k=1 and b) k=2024.k=2024.
Proposed by Ishan Nath
number theory