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Part of 2013 Harvard-MIT Mathematics Tournament

Problems(4)

2013 HMMT Algebra #1: System of Equations

Source:

x

y

be real numbers with

x>y

x^2y^2+x^2+y^2+2xy=40

xy+x+y=8

. Find the value of

x

2013 Team #1: ab-Algebra Bash

Source:

a

b

be real numbers such that

\frac {ab}{a^2 + b^2} = \frac {1}{4}

. Find all possible values of

\frac {|a^2-b^2|}{a^2+b^2}

2013 Guts #1: Coordinates and a Marker

Source:

Arpon chooses a positive real number

k

. For each positive integer

n

, he places a marker at the point

(n,nk)

(x,y)

plane. Suppose that two markers whose

x

-coordinates differ by

4

31

. What is the distance between the markers

(7,7k)

(19,19k)

analytic geometrygeometrysimilar triangles

2013 HMMT Geometry # 1

Source:

Jarris the triangle is playing in the

(x, y)

plane. Let his maximum

y

k

. Given that he has side lengths

6

8

10

and that no part of him is below the

x

-axis, find the minimum possible value of

k