MathDB

2

Part of 1996 Polish MO Finals

Problems(2)

Area apef/area abp does not depend on p

Source:

10/30/2005
Let PP be a point inside a triangle ABCABC such that ∠PBC=∠PCA<∠PAB\angle PBC = \angle PCA < \angle PAB. The line PBPB meets the circumcircle of triangle ABCABC at a point EE (apart from BB). The line CECE meets the circumcircle of triangle APEAPE at a point FF (apart from EE). Show that the ratio ∣APEF∣∣ABP∣\frac{\left|APEF\right|}{\left|ABP\right|} does not depend on the point PP, where the notation ∣P1P2...Pn∣\left|P_1P_2...P_n\right| stands for the area of an arbitrary polygon P1P2...PnP_1P_2...P_n.
geometrycircumcircle
Find all n such that x_n = 111111

Source:

10/30/2005
Let p(k)p(k) be the smallest prime not dividing kk. Put q(k)=1q(k) = 1 if p(k)=2p(k) = 2, or the product of all primes <p(k)< p(k) if p(k)>2p(k) > 2. Define the sequence x0,x1,x2,...x_0, x_1, x_2, ... by x0=1x_0 = 1, xn+1=xnp(xn)q(xn)x_{n+1} = \frac{x_np(x_n)}{q(x_n)}. Find all nn such that xn=111111x_n = 111111
inductionnumber theory unsolvednumber theory