Subcontests
(3)equal segments wanted, tangents form endpoints of a diameter related
Through the endpoints A and B of a diameter AB of a given circle, the tangents ℓ and m have been drawn. Let C=A be a point on ℓ and let q1,q2 be two rays from C. Ray qi cuts the circle in Di and Ei with Di between C and Ei,i=1,2. Rays AD1,AD2,AE1,AE2 meet m in the respective points M1,M2,N1,N2. Prove that M1M2=N1N2. Absolute Valued Sequence Inequality
Let a1,a2,a3,… be a sequence of real numbers satisfying the inequality |a_{k+m}-a_k-a_m| \leq 1 \text{for all} \ k,m \in \mathbb{Z}_{>0}. Show that the following inequality holds for all positive integers k,m kak−mam<k1+m1.