MathDB

6

Part of 2006 Tournament of Towns

Problems(4)

Curved Pentagons

Source: Spring 2006 Tournament of Towns Junior A-Level #6

4/15/2015
Let us call a pentagon curved, if all its sides are arcs of some circles. Are there exist a curved pentagon PP and a point AA on its boundary so that any straight line passing through AA divides perimeter of PP into two parts of the same length?
(7 points)
geometry
TOT 2006 Spring - Senior A-Level p6 12 grasshoppers on circumference

Source:

2/25/2020
On a circumference at some points sit 1212 grasshoppers. The points divide the circumference into 1212 arcs. By a signal each grasshopper jumps from its point to the midpoint of its arc (in clockwise direction). In such way new arcs are created. The process repeats for a number of times. Can it happen that at least one of the grasshoppers returns to its initial point after a) 1212 jumps? (4) a) 1313 jumps? (3)
combinatoricsnumber theory
TOT 2006 Fall - Junior A-Level p6 1+1/2+1/3+...+1/n = a_n/b_n

Source:

2/25/2020
Let 1+1/2+1/3+...+1/n=an/bn1 + 1/2 + 1/3 +... + 1/n = a_n/b_n, where ana_n and bnb_n are relatively prime. Show that there exist infinitely many positive integers nn, such that bn+1<bnb_{n+1} < b_n. (8)
number theoryDecreasingSequence
TOT 2006 Fall - Senior A-Level p6 deck of 52 cards divisible by 13!

Source:

2/25/2020
Let us say that a deck of 5252 cards is arranged in a “regular” way if the ace of spades is on the very top of the deck and any two adjacent cards are either of the same value or of the same suit (top and bottom cards regarded adjacent as well). Prove that the number of ways to arrange a deck in regular way is a) divisible by 12!12! (3) b) divisible by 13!13! (5)
number theorycombinatorics