MathDB

13

Part of 2017 AIME Problems

Problems(2)

How many integers does it take to screw in a cube?

Source: 2017 AIME I #13

3/8/2017
For every m≥2m \geq 2, let Q(m)Q(m) be the least positive integer with the following property: For every n≥Q(m)n \geq Q(m), there is always a perfect cube k3k^3 in the range n<k3≤m⋅nn < k^3 \leq m \cdot n. Find the remainder when ∑m=22017Q(m) \sum_{m = 2}^{2017} Q(m) is divided by 1000.
2017 AIME I
Life of Pisosceles Triangles

Source: 2017 AIME II #13

3/23/2017
For each integer n≥3n\ge 3, let f(n)f(n) be the number of 3-element subsets of the vertices of a regular nn-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of nn such that f(n+1)=f(n)+78f(n+1)=f(n)+78.
AMCAIMEAIME II