Costa Rica IMO TST 1, Day 2

  1. Consider a point AA in the interior of a circumference, different from its center. Consider all the chords (except the diameter) that pass through AA. Find the locus of all the intersection points of the two tangents to the circumference that intersect it on the chords' extremes.

  2. Let sns_n be a sequence such that s1=1s_1=1 and sn+1=3sn+1s_{n+1}=3s_n+1. Note that s18=193710244s_{18}=193710244 ends with two identical digits. Prove that all of the sequence's terms that end with two or more identical digits come in groups of three, and that these three terms finish with the same number of identical digits.

  3. In a lottery game, a committee picks six different numbers from 11 to 3636 at random. A ticket consists of six numbers (chosen by you) from 11 to 3636. A ticket is said to be a "winning ticket" if all of its numbers are different from the numbers picked by the committee. Prove that with 9 tickets you can guarantee having at least one winning ticket, but that with 8 you cannot.

Note by José Marín Guzmán
6 years, 1 month ago

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Isn't the first one just the polar of A wrt circle. i.e. locus is just a straight line ?

Dinesh Chavan - 6 years, 1 month ago

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Yes. I wonder why such a basic fact is presented as a TST problem.

Jubayer Nirjhor - 6 years, 1 month ago

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