Subcontests
(6)Lima sequances, gcd {a_i - a_j with a_i> a_j} =1, a'=2a_k-a_l
Let n≥2 be an integer. A sequence α=(a1,a2,...,an) of n integers is called Lima if gcd{ai−aj such that ai>aj and 1≤i,j≤n}=1, that is, if the greatest common divisor of all the differences ai−aj with ai>aj is 1. One operation consists of choosing two elements ak and aℓ from a sequence, with k=ℓ , and replacing aℓ by aℓ′=2ak−aℓ .
Show that, given a collection of 2n−1 Lima sequences, each one formed by n integers, there are two of them, say β and γ, such that it is possible to transform β into γ through a finite number of operations.Notes.
The sequences (1,2,2,7) and (2,7,2,1) have the same elements but are different.
If all the elements of a sequence are equal, then that sequence is not Lima. concurrent wanted, <CBP =ACB, <QCB = <CBA., 2 circles
Let ABC be an acute scalene triangle such that AB<AC. The midpoints of sides AB and AC are M and N, respectively. Let P and Q be points on the line MN such that ∠CBP=∠ACB and ∠QCB=∠CBA. The circumscribed circle of triangle ABP intersects line AC at D (D=A) and the circumscribed circle of triangle AQC intersects line AB at E (E=A). Show that lines BC,DP, and EQ are concurrent.Nicolás De la Hoz, Colombia