**A. 507.** The circles \(\displaystyle K_1,\dots,K_6\) are externally tangent to the circle \(\displaystyle K_0\) in this order. For each \(\displaystyle 1\le i\le 5\), the circles \(\displaystyle K_i\) and \(\displaystyle K_{i+1}\) are externally tangent to each other, and \(\displaystyle K_1\) and \(\displaystyle K_6\) are externally tangent to each other as well, according to the *Figure.* Denote by \(\displaystyle r_i\) the radius of \(\displaystyle K_i\) (\(\displaystyle 0\le i\le6\)). Prove that if \(\displaystyle r_1r_4=r_2r_5=r_3r_6=1\) then \(\displaystyle {r_0\le 1}\).

Proposed by: *Balázs Strenner,* Székesfehérvár

(5 points)

**A. 508.** An induced subgraph \(\displaystyle S\) of the graph \(\displaystyle G\) is called ``dominant'' if every vertex of \(\displaystyle G\), outside \(\displaystyle S\), has a neighbor in \(\displaystyle S\). Does there exist such a graph which has an even number of dominant subgraphs?

Proposed by: *László Miklós Lovász,* Budapest

(5 points)

**C. 1033.** Solve the equation \(\displaystyle \log_{2010}\, (2009\, x)=\log_{2009}\, (2010\, x)\).

Suggested by *J. Pataki,* Budapest

(5 points)