MathDB

2013 Tournament of Towns

Part of Tournament Of Towns

Subcontests

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2013 ToT Spring Senior A p6 sums related to 5 real numbers

There are fi ve distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal. (a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers. (b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).

2013 ToT Spring Junior A p6 peprendicular wanted and given, incircle

Let ABCABC be a right-angled triangle, II its incenter and B0,A0B_0, A_0 points of tangency of the incircle with the legs ACAC and BCBC respectively. Let the perpendicular dropped to AIAI from A0A_0 and the perpendicular dropped to BIBI from B0B_0 meet at point PP. Prove that the lines CPCP and ABAB are perpendicular.

2013 ToT Fall Junior A p6 3n+1 is a prime number

The number 1−12+13−14+...+12n−1−12n1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n} is represented as an irreducible fraction. If 3n+13n+1 is a prime number, prove that the numerator of this fraction is a multiple of 3n+13n + 1.
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