MathDB

1

Part of 1998 Turkey MO (2nd round)

Problems(2)

Turkish NMO 1998, 1. Problem, points in a isosceles triangle

Source:

7/31/2011
Let DD be the point on the base BCBC of an isosceles △ABC\vartriangle ABC triangle such that ∣BD∣∣DC∣= 2\frac{\left| BD \right|}{\left| DC \right|}=\text{ }2, and let PP be the point on the segment [AD]\left[ AD \right] such that ∠BAC=∠BPD\angle BAC=\angle BPD. Prove that ∠DPC=12∠BAC\angle DPC=\frac{1}{2}\angle BAC.
geometry proposedgeometry
solving equation x^3+3367=2^n

Source: Turkish NMO 1998, 4. Problem

7/31/2011
Find all positive integers xx and nn such that x3+3367=2n{{x}^{3}}+3367={{2}^{n}}.
modular arithmeticquadraticsnumber theory proposednumber theory