MathDB

6

Part of 1998 Putnam

Problems(2)

1998 Putnam A6

Source:

4/19/2013
Let A,B,CA,B,C denote distinct points with integer coefficients in R2\mathbb{R}^2. Prove that if (∣AB∣+∣BC∣)2<8⋅[ABC]+1(|AB|+|BC|)^2<8\cdot[ABC]+1 then A,B,CA,B,C are three vertices of a square. Here ∣XY∣|XY| is the length of segment XYXY and [ABC][ABC] is the area of triangle ABCABC.
Putnamgeometrycollege contests
1998 Putnam B6

Source:

4/19/2013
Prove that, for any integers a,b,ca,b,c, there exists a positive integer nn such that n3+an2+bn+c\sqrt{n^3+an^2+bn+c} is not an integer.
Putnamgeometry3D geometrymodular arithmeticcollege contests