MathDB

2

Part of 2013 Greece Team Selection Test

Problems(2)

Concyclic points

Source: Greek IMO TST,2013,Pr.2

8/16/2014
Let ABCABC be a non-isosceles,aqute triangle with AB<ACAB<AC inscribed in circle c(O,R)c(O,R).The circle c1(B,AB)c_{1}(B,AB) crosses ACAC at KK and cc at EE. KEKE crosses cc at FF and BOBO crosses KEKE at LL and ACAC at MM while AEAE crosses BFBF at DD.Prove that: i)D,L,M,FD,L,M,F are concyclic. ii)B,D,K,M,EB,D,K,M,E are concyclic.
geometrycircumcirclegeometry proposed
Find the pairs of integers...

Source: 2013 Greek 2nd TST,Pr.2

5/24/2016
For the several values of the parameter m∈N∗m\in \mathbb{N^{*}},find the pairs of integers (a,b)(a,b) that satisfy the relation
                                                        [a,m]+[b,m](a+b)m=1011\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{[a,m]+[b,m]}{(a+b)m}=\frac{10}{11},
and,moreover,on the Cartesian plane OxyOxy the lie in the square D={(x,y):1≤x≤36,1≤y≤36}D=\{(x,y):1\leq x\leq 36,1\leq y\leq 36\}.
Note:[k,l][k,l] denotes the least common multiple of the positive integers k,lk,l.
parameterizationparametric equationanalytic geometry