MathDB

3

Part of 2004 CentroAmerican

Problems(2)

Cevian triangle of ABC is similar to ABC

Source: CentroAmerican 2004

12/10/2010
ABCABC is a triangle, and EE and FF are points on the segments BCBC and CACA respectively, such that CECB+CFCA=1\frac{CE}{CB}+\frac{CF}{CA}=1 and ∠CEF=∠CAB\angle CEF=\angle CAB. Suppose that MM is the midpoint of EFEF and GG is the point of intersection between CMCM and ABAB. Prove that triangle FEGFEG is similar to triangle ABCABC.
geometryparallelogramgeometry proposed
Lots of pearls and necklaces

Source: CentroAmerican 2004

12/10/2010
With pearls of different colours form necklaces, it is said that a necklace is prime if it cannot be decomposed into strings of pearls of the same length, and equal to each other. Let nn and qq be positive integers. Prove that the number of prime necklaces with nn beads, each of which has one of the qnq^n possible colours, is equal to nn times the number of prime necklaces with n2n^2 pearls, each of which has one of qq possible colours.
Note: two necklaces are considered equal if they have the same number of pearls and you can get the same colour on both necklaces, rotating one of them to match it to the other.
combinatorics proposedcombinatorics