MathDB

3

Part of 2017 Harvard-MIT Mathematics Tournament

Problems(7)

2017 Team #3: Many faces with same number of sides

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2/19/2017
A polyhedron has 7n7n faces. Show that there exist n+1n + 1 of the polyhedron's faces that all have the same number of edges.
combinatorics
2017 Algebra/NT #3: Functional Equation

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2/19/2017
Let f:R→Rf: \mathbb{R}\rightarrow \mathbb{R} be a function satisfying f(x)f(y)=f(x−y)f(x)f(y)=f(x-y). Find all possible values of f(2017)f(2017).
functionalgebrafunctional equation
2017 Geometry #3: set of 2017 points

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2/19/2017
Let SS be a set of 20172017 points in the plane. Let RR be the radius of the smallest circle containing all points in SS on either the interior or boundary. Also, let DD be the longest distance between two of the points in SS. Let aa, bb be real numbers such that a≤DR≤ba\le \frac{D}{R}\le b for all possible sets SS, where aa is as large as possible and bb is as small as possible. Find the pair (a,b)(a, b).
geometry
2017 Combinatorics #3: Nice man depositing coins

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2/19/2017
There are 20172017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that NN of the jars all contain the same positive integer number of coins (i.e. there is an integer d>0d>0 such that NN of the jars have exactly dd coins). What is the maximum possible value of NN?
2017 Theme #3

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5/8/2018
Emilia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). (The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solutions.) Since francium is highly radioactive, its concentration in the final solution should not exceed 2%. What is the highest possible concentration of rubidium in her solution?
algebra
2017 General #3

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5/8/2018
Find the number of integers nn with 1≤n≤20171 \le n \le 2017 so that (n−2)(n−0)(n−1)(n−7)(n-2)(n-0)(n-1)(n-7) is an integer multiple of 10011001.
number theory
2017 Guts #3: Integer solutions to inequality

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2/21/2017
Find the number of pairs of integers (x,y)(x, y) such that x2+2y2<25x^2 + 2y^2 < 25.
inequalities