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Costa Rica IMO TST 1, Day 2

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

  2. Let \(s_n\) be a sequence such that \(s_1=1\) and \(s_{n+1}=3s_n+1\). Note that \(s_{18}=193710244\) 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 \(1\) to \(36\) at random. A ticket consists of six numbers (chosen by you) from \(1\) to \(36\). 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
2 years 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 · 2 years ago

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@Dinesh Chavan Yes. I wonder why such a basic fact is presented as a TST problem. Jubayer Nirjhor · 2 years ago

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