MathDB

12

Part of 2005 AIME Problems

Problems(2)

Difference of Two Series

Source:

11/13/2005
For positive integers nn, let τ(n)\tau (n) denote the number of positive integer divisors of nn, including 11 and nn. For example, τ(1)=1\tau (1)=1 and τ(6)=4\tau(6) =4. Define S(n)S(n) by S(n)=τ(1)+τ(2)+...+τ(n).S(n)=\tau(1)+ \tau(2) + ... + \tau(n). Let aa denote the number of positive integers n≤2005n \leq 2005 with S(n)S(n) odd, and let bb denote the number of positive integers n≤2005n \leq 2005 with S(n)S(n) even. Find ∣a−b∣|a-b|.
number theoryprime factorization
Square on a Coordinate Plane

Source:

11/13/2005
Square ABCDABCD has center OO, AB=900AB=900, EE and FF are on ABAB with AE<BFAE<BF and EE between AA and FF, m∠EOF=45∘m\angle EOF =45^\circ, and EF=400EF=400. Given that BF=p+qrBF=p+q\sqrt{r}, wherer p,q,p,q, and rr are positive integers and rr is not divisible by the square of any prime, find p+q+rp+q+r.
analytic geometrytrigonometryquadraticsgeometrygeometric transformationrotationreflection