MathDB

6

Part of 2012 JBMO ShortLists

Problems(2)

Find area!

Source:

2/4/2015
Let O1O_1 be a point in the exterior of the circle ω\omega of center OO and radius RR , and let O1NO_1N , O1DO_1D be the tangent segments from O1O_1 to the circle. On the segment O1NO_1N consider the point BB such that BN=RBN=R .Let the line from BB parallel to ONON intersect the segment O1DO_1D at CC . If AA is a point on the segment O1DO_1D other than CC so that BC=BA=aBC=BA=a , and if the incircle of the triangle ABCABC has radius rr , then find the area of △ABC\triangle ABC in terms of a,R,ra ,R ,r.
geometry
Isn't a perfect square!

Source:

2/4/2015
If aa , bb , cc , dd are integers and A=2(a−2b+c)4+2(b−2c+a)4+2(c−2a+b)4A=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4 , B=d(d+1)(d+2)(d+3)+1B=d(d+1)(d+2)(d+3)+1 , then prove that (A+1)2+B\left (\sqrt{A}+1 \right )^2 +B cannot be a perfect square.
quadratics