MathDB

5

Part of 2002 Estonia National Olympiad

Problems(4)

n aborigines living on island, each of them telling only the truth or only lying

Source: 2002 Estonia National Olympiad Final Round grade 9 p5

3/16/2020
There were n>1n> 1 aborigines living on an island, each of them telling only the truth or only lying, and each having at least one friend among the others. The new governor asked each aborigine whether there are more truthful aborigines or liars among his friends, or an equal number of both. Each aborigine answered that there are more liars than truthful aborigines among his friends. The governor then ordered one of the aborigines to be executed for being a liar and asked each of the remaining n−1n- 1 aborigines the same question again. This time each aborigine answered that there are more truthful aborigines than liars among his friends. Determine whether the executed aborigine was truthful or a liar, and whether there are more truthful aborigines or liars remaining on the island.
combinatorics
1 --> 1, 3, 2, 3, 1 -> 1, 4, 3, 5, 2, 5, 3, 4, 1 sequence

Source: 2002 Estonia National Olympiad Final Round grade 10 p5

3/16/2020
The teacher writes numbers 11 at both ends of the blackboard. The first student adds a 22 in the middle between them, each next student adds the sum of each two adjacent numbers already on the blackboard between them (hence there are numbers 1,3,2,3,11, 3, 2, 3, 1 on the blackboard after the second student, 1,4,3,5,2,5,3,4,11, 4, 3, 5, 2, 5, 3, 4, 1 after the third student etc.) Find the sum of all numbers on the blackboard after the nn-th student.
SumSequencealgebra
a robot that moves along the border of a regular octagon

Source: 2002 Estonia National Olympiad Final Round grade 11 p5

3/14/2020
Juku built a robot that moves along the border of a regular octagon, passing each side in exactly 11 minute. The robot starts in some vertex AA and upon reaching each vertex can either continue in the same direction, or turn around and continue in the opposite direction. In how many different ways can the robot move so that after nn minutes it will be in the vertex BB opposite to AA?
combinatoricsoctagon
no of possible distributions of prizes in a lottery

Source: 2002 Estonia National Olympiad Final Round grade 12 p5

3/14/2020
There is a lottery at Juku’s birthday party with a number of identical prizes, where each guest can win at most one prize. It is known that if there was one prize less, then the number of possible distributions of the prizes among the guests would be 50%50\% less than it actually is, while if there was one prize more, then the number of possible distributions of the prizes would be 50%50\% more than it actually is. Find the number of possible distributions of the prizes.
combinatorics