MathDB

6

Part of 2015 Tournament of Towns

Problems(3)

Sequence with + , - and x

Source: Tournament of Towns 2015 Junior A-Level Question 6

10/27/2023
Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs ++ , −- and ×\times. He may also use a special sign ±\pm: computing the values of the resulting expression, he chooses values ++ or −- for every ±\pm in all possible combinations. For instance, the expression 5±15 \pm 1 results in {4,6}\{4, 6 \}, and (2±0.5)±0.5(2 \pm 0.5) \pm 0.5 results in {1,2,3}\{1, 2, 3 \}. Can Pete construct such an expression: a)a) If the numbers on the blackboard are 1,2,41, 2, 4; b)b) For any collection of 100100 distinct real numbers on a blackboard?
combinatoricsTournament of Towns
Expelling Binary Wizards

Source: Tournament of Towns Spring 2015 Senior A-level

2/24/2017
An Emperor invited 20152015 wizards to a festival. Each of the wizards knows who of them is good and who is evil, however the Emperor doesn’t know this. A good wizard always tells the truth, while an evil wizard can tell the truth or lie at any moment. The Emperor gives each wizard a card with a single question, maybe different for different wizards, and after that listens to the answers of all wizards which are either “yes” or “no”. Having listened to all the answers, the Emperor expels a single wizard through a magic door which shows if this wizard is good or evil. Then the Emperor makes new cards with questions and repeats the procedure with the remaining wizards, and so on. The Emperor may stop at any moment, and after this the Emperor may expel or not expel a wizard. Prove that the Emperor can expel all the evil wizards having expelled at most one good wizard. (1010 points)
combinatorics
Cutting a Melon

Source: Tournament of Towns Fall 2015 Senior A - level

2/23/2017
Basil has a melon in a shape of a ball, 2020 in diameter. Using a long knife, Basil makes three mutually perpendicular cuts. Each cut carves a circular segment in a plane of the cut, hh deep (hh is a height of the segment). Does it necessarily follow that the melon breaks into two or more pieces if (a) h=17h = 17 ? (6 points) (b) h=18h = 18 ? (6 points)
combinatorics