MathDB

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Part of 2021 Harvard-MIT Mathematics Tournament.

Problems(4)

2021 Algebra/NT #1: Who is an integer?

Source:

5/30/2021
Compute the sum of all positive integers nn for which the expression n+7n−1\frac{n+7}{\sqrt{n-1}} is an integer.
algebranumber theory
2021 Combo #1: colored squares with numbers

Source:

5/30/2021
Leo the fox has a 55 by 55 checkerboard grid with alternating red and black squares. He fills in the grid with the numbers 1,2,3,…,251, 2, 3, \dots, 25 such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the 1313 squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.
Combo
2021 Geo #1: Circle on the coordinate plane

Source:

5/30/2021
A circle contains the points (0,11)(0, 11) and (0,−11)(0, -11) on its circumference and contains all points (x,y)(x, y) with x2+y2<1x^2+y^2<1 in its interior. Compute the largest possible radius of the circle.
analytic geometry
2021 Team #1

Source:

6/27/2021
Let aa and bb be positive integers with a>ba>b. Suppose that a+b+a−b\sqrt{\sqrt{a}+\sqrt{b}}+\sqrt{\sqrt{a}-\sqrt{b}} is an integer. (a) Must a\sqrt{a} be an integer? (b) Must b\sqrt{b} be an integer?
algebra