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

Proposed by: *Balázs Strenner,* Székesfehérvár

(5 points)

**A. 508.** An induced subgraph &tex;\displaystyle S&xet; of the graph &tex;\displaystyle G&xet; is called ``dominant'' if every vertex of &tex;\displaystyle G&xet;, outside &tex;\displaystyle S&xet;, has a neighbor in &tex;\displaystyle S&xet;. 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 &tex;\displaystyle \log_{2010}\, (2009\, x)=\log_{2009}\, (2010\, x)&xet;.

Suggested by *J. Pataki,* Budapest

(5 points)