MathDB

1

Part of 2006 Moldova Team Selection Test

Problems(3)

Divisors and finding n

Source: Moldavian TST_1, Problem 1

3/6/2006
Determine all even numbers nn, n∈Nn \in \mathbb N such that 1d1+1d2+⋯+1dk=16201003,{ \frac{1}{d_{1}}+\frac{1}{d_{2}}+ \cdots +\frac{1}{d_{k}}=\frac{1620}{1003}}, where d1,d2,…,dkd_1, d_2, \ldots, d_k are all different divisors of nn.
number theory proposednumber theory
Lucas divisibility

Source: Moldova TST 2006, Test II problem 1

3/25/2006
Let (an)(a_n) be the Lucas sequence: a0=2,a1=1,an+1=an+an−1a_0=2,a_1=1, a_{n+1}=a_n+a_{n-1} for n≥1n\geq 1. Show that a59a_{59} divides (a30)59−1(a_{30})^{59}-1.
number theory solvednumber theory
Modlova 3rd tst, problem 1

Source: Moldova TST III

3/26/2006
Let the point PP in the interior of the triangle ABCABC. (AP,(BP,(CP(AP, (BP, (CP intersect the circumcircle of ABCABC at A1,B1,C1A_{1}, B_{1}, C_{1}. Prove that the maximal value of the sum of the areas A1BCA_{1}BC, B1ACB_{1}AC, C1ABC_{1}AB is p(R−r)p(R-r), where p,r,Rp, r, R are the usual notations for the triangle ABCABC.
geometrycircumcircletrigonometryinequalitiesincentergeometry proposed