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2

Part of 1994 Cono Sur Olympiad

Problems(2)

Proof in a circle

Source: Cono Sur 1994-problem 2

6/1/2006
Consider a circle CC with diameter AB=1AB=1. A point P0P_0 is chosen on CC, P0≠AP_0 \ne A, and starting in P0P_0 a sequence of points P1,P2,…,Pn,…P_1, P_2, \dots, P_n, \dots is constructed on CC, in the following way: QnQ_n is the symmetrical point of AA with respect of PnP_n and the straight line that joins BB and QnQ_n cuts CC at BB and Pn+1P_{n+1} (not necessary different). Prove that it is possible to choose P0P_0 such that: i ∠P0AB<1\angle {P_0AB} < 1. ii In the sequence that starts with P0P_0 there are 22 points, PkP_k and PjP_j, such that △APkPj\triangle {AP_kP_j} is equilateral.
geometry unsolvedgeometry
Quadratic eqution

Source:

2/2/2006
Solve the following equation in integers with gcd (x, y) = 1 x2+y2=2z2x^2 + y^2 = 2 z^2
quadraticsnumber theorygreatest common divisorcalculusintegrationanalytic geometryconics