MathDB

A5

Part of 2012 Princeton University Math Competition

Problems(3)

2012 PUMaC Geometry A5

Source:

9/24/2019
Let △ABC\vartriangle ABC be a triangle with ∠BAC=45o,∠BCA=30o\angle BAC = 45^o, \angle BCA = 30^o, and AB=1AB = 1. Point DD lies on segment ACAC such that AB=BDAB = BD. Find the square of the length of the common external tangent to the circumcircles of triangles △BDC\vartriangle BDC and △ABC\vartriangle ABC.
geometry
2012 PUMaC Algebra A5

Source:

10/5/2019
What is the smallest natural number nn greater than 20122012 such that the polynomial f(x)=(x6+x4)n−x4n−x6f(x) =(x^6 + x^4)^n - x^{4n} - x^6 is divisible by g(x)=x4+x2+1g(x) = x^4 + x^2 + 1?
algebra
2012 PUMaC Number Theory A5

Source:

10/5/2019
Call a positive integer xx a leader if there exists a positive integer nn such that the decimal representation of xnx^n starts (not ends) with 20122012. For example, 586586 is a leader since 5863=201230056586^3 =201230056. How many leaders are there in the set {1,2,3,...,2012}\{1, 2, 3, ..., 2012\}?
number theory