MathDB

3

Part of 2009 Poland - Second Round

Problems(2)

Two circles and two tangents

Source: Polish MO 2009 Round 2

3/26/2009
Disjoint circles o1,o2 o_1, o_2, with centers I1,I2 I_1, I_2 respectively, are tangent to the line k k at A1,A2 A_1, A_2 respectively and they lie on the same side of this line. Point C C lies on segment I1I2 I_1I_2 and \angle A_1CA_2 \equal{} 90^{\circ}. Let B1 B_1 be the second intersection of A1C A_1C with o1 o_1, and let B2 B_2 be the second intersection of A2C A_2C with o2 o_2. Prove that B1B2 B_1B_2 is tangent to the circles o1,o2 o_1, o_2.
projective geometrygeometryperpendicular bisectorMenelaus
Find sequences of reals with two sums

Source: Polish Second Round 2009

8/4/2011
For every integer n≥3n\ge 3 find all sequences of real numbers (x1,x2,…,xn)(x_1,x_2,\ldots ,x_n) such that ∑i=1nxi=n\sum_{i=1}^{n}x_i=n and ∑i=1n(xi−1−xi+xi+1)2=n\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n, where x0=xnx_0=x_n and xn+1=x1x_{n+1}=x_1.
vectorlinear algebraalgebra unsolvedalgebra