MathDB

2

Part of 1993 Cono Sur Olympiad

Problems(2)

Proof in a circle

Source: Cono Sur 1993-problem 2

5/30/2006
Consider a circle with centre OO, and 33 points on it, A,BA,B and CC, such that ∠AOB<∠BOC\angle {AOB}< \angle {BOC}. Let DD be the midpoint on the arc ACAC that contains the point BB. Consider a point KK on BCBC such that DK⊥BCDK \perp BC. Prove that AB+BK=KCAB+BK=KC.
geometry unsolvedgeometry
Prove succession

Source: Cono Sur 1993-problem 5

5/30/2006
Prove that there exists a succession a1,a2,...,ak,...a_1, a_2, ... , a_k, ..., where each aia_i is a digit (ai∈(0,1,2,3,4,5,6,7,8,9)a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ) and a0=6a_0=6, such that, for each positive integrer nn, the number xn=a0+10a1+100a2+...+10n−1an−1x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1} verify that xn2−xnx_n^2-x_n is divisible by 10n10^n.
inductionmodular arithmeticnumber theory unsolvednumber theory