C. 1031. In the lottery, five out of the numbers 1 to 90 are drawn. Ticket holders mark five numbers on each ticket. A man bought two tickets and marked ten different numbers on them altogether. Given that four out of his ten numbers were drawn, what is the probability that he had a ticket with

C. 1032. Vertices B and C of a triangle are connected to the points that divide the opposite sides into three equal parts. The connecting line segments determine a quadrilateral. Prove that the quadrilateral has a diagonal parallel to side BC.

B. 4262. Given the points P and Q, determine the locus of the intersection of the line e passing through P with the plane S_{e} through Q that is perpendicular to e.

B. 4264. Triangle ABC has a 120^{o} angle at vertex C. The orthocentre of the triangle is M, the centre of the circumscribed circle is O, and the midpoint of arc ACB of the circle is F. Prove that MF=FO.

B. 4265. Find a colouring of positive integers with 7 colours such that the colours of the numbers {a,2a,3a,4a,5a,6a,7a} are pairwise different for every positive integer a.

B. 4266. Let a_{1}, a_{2}, a_{3}, a_{4} denote four consecutive elements in a row of Pascal's triangle. Prove that the numbers , , form an arithmetic progression.

B. 4270. Let ABCDEF be a hexagon inscribed in a circle. Prove that AD^{.}BE^{.}CF=AB^{.}DE^{.}CF+BC^{.}EF^{.}AD+CD^{.}FA^{.}BE+AB^{.}CD^{.}EF+BC^{.}DE^{.}FA.

A. 506. Prove that for every prime p, there exists a colouring of the positive integers with p-1 colours such that the colours of the numbers {a,2a,3a,...,(p-1)a} are pairwise different for every positive integer a.

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 \(\displaystyle {r_0\le 1}\).

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?