MathDB

7

Part of 2008 Tournament Of Towns

Problems(4)

TT2008 Junior A-Level - P7

Source:

9/4/2010
In an in finite sequence a1,a2,a3,⋯a_1, a_2, a_3, \cdots, the number a1a_1 equals 11, and each an,n>1a_n, n > 1, is obtained from an−1a_{n-1} as follows:
- if the greatest odd divisor of nn has residue 11 modulo 44, then an=an−1+1,a_n = a_{n-1} + 1,
- and if this residue equals 33, then an=an−1−1.a_n = a_{n-1} - 1.
Prove that in this sequence
(a) the number 11 occurs infi nitely many times;
(b) each positive integer occurs infi nitely many times.
(The initial terms of this sequence are 1,2,1,2,3,2,1,2,3,4,3,⋯1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots )
algebra proposedalgebra
TT2008 Senior A-Level - P7

Source:

9/4/2010
A test consists of 3030 true or false questions. After the test (answering all 3030 questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions ) several times. Can Victor work out a strategy that insure him to get a perfect score after
(a) 3030th attempt?
(b) 2525th attempt?
(Initially, Victor does not know any answer)
combinatorics unsolvedcombinatorics
2008 ToT Spring Junior A P7 angle chasing

Source:

2/26/2020
A convex quadrilateral ABCDABCD has no parallel sides. The angles between the diagonal ACAC and the four sides are 55o,55o,19o55^o, 55^o, 19^o and 16o16^o in some order. Determine all possible values of the acute angle between ACAC and BDBD.
geometryangles
2008 ToT Spring Senior A P7 circumcircle passes through midpoint

Source:

2/26/2020
Each of three lines cuts chords of equal lengths in two given circles. The points of intersection of these lines form a triangle. Prove that its circumcircle passes through the midpoint of the segment joining the centres of the circles.
circumcirclegeometrycircles