MathDB

2

Part of 2013 European Mathematical Cup

Problems(2)

An area equality

Source: European Mathematical Cup 2013, Junior Division, P2

7/3/2014
Let PP be a point inside a triangle ABCABC. A line through PP parallel to ABAB meets BCBC and CACA at points LL and FF, respectively. A line through PP parallel to BCBC meets CACA and BABA at points MM and DD respectively, and a line through PP parallel to CACA meets ABAB and BCBC at points NN and EE respectively. Prove \begin{align*} [PDBL] \cdot [PECM] \cdot [PFAN]=8\cdot [PFM] \cdot [PEL] \cdot [PDN] \\ \end{align*}
Proposed by Steve Dinh
geometrygeometry proposed
Problem 2

Source: 2nd European Mathematical Cup

12/16/2013
Palindrome is a sequence of digits which doesn't change if we reverse the order of its digits. Prove that a sequence (xn)n=0∞(x_n)^{\infty}_{n=0} defined as
xn=2013+317nx_n=2013+317n
contains infinitely many numbers with their decimal expansions being palindromes.
inductionnumber theory opennumber theory