MathDB

6

Part of 2021 Harvard-MIT Mathematics Tournament.

Problems(4)

2021 Algebra/NT #6: #(2021) > #(2000)

Source:

5/30/2021
Suppose that mm and nn are positive integers with m<nm < n such that the interval [m,n)[m, n) contains more multiples of 20212021 than multiples of 20002000. Compute the maximum possible value of n−mn - m.
number theory
2021 Combo #6: Light pulse partitions

Source:

5/30/2021
A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square’s perimeter nn times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 20212021 regions. Compute the smallest possible value of nn.
Combo
2021 Geo #6: The duo of O and H

Source:

5/30/2021
In triangle ABCABC, let MM be the midpoint of BCBC, HH be the orthocenter, and OO be the circumcenter. Let NN be the reflection of MM over HH. Suppose that OA=ON=11OA = ON = 11 and OH=7.OH = 7. Compute BC2BC^2.
geometrycircumcircle
2021 Team #6

Source:

6/27/2021
Let f(x)=x2+x+1f(x)=x^2+x+1. Determine, with proof, all positive integers nn such that f(k)f(k) divides f(n)f(n) whenever kk is a positive divisor of nn.
number theorypolynomial