MathDB

5

Part of 2012 Tournament of Towns

Problems(7)

2012 ToT Spring Junior O p5 a little rhinoceros has 17 scratch marks on its body

Source:

3/4/2020
RyNo, a little rhinoceros, has 1717 scratch marks on its body. Some are horizontal and the rest are vertical. Some are on the left side and the rest are on the right side. If RyNo rubs one side of its body against a tree, two scratch marks, either both horizontal or both vertical, will disappear from that side. However, at the same time, two new scratch marks, one horizontal and one vertical, will appear on the other side. If there are less than two horizontal and less than two vertical scratch marks on the side being rubbed, then nothing happens. If RyNo continues to rub its body against trees, is it possible that at some point in time, the numbers of horizontal and vertical scratch marks have interchanged on each side of its body?
combinatorics
2012 ToT Spring Junior A p5 interesting set of p+2 numbers

Source:

3/4/2020
Let pp be a prime number. A set of p+2p + 2 positive integers, not necessarily distinct, is called interesting if the sum of any pp of them is divisible by each of the other two. Determine all interesting sets.
primeSumdivisiblenumber theory
2012 ToT Spring Senior O p5 2 rooks on chessborad, winning strategy

Source:

3/5/2020
In an 8\times 8 chessboard, the rows are numbers from 11 to 88 and the columns are labelled from aa to hh. In a two-player game on this chessboard, the fi rst player has a White Rook which starts on the square b2b2, and the second player has a Black Rook which starts on the square c4c4. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?
ChessboardRookgamecombinatoricsgame strategy
2012 ToT Fall Junior O p5 several field trips for a class of 20 students

Source:

3/22/2020
For a class of 2020 students several field trips were arranged. In each trip at least one student participated. Prove that there was a field trip such that each student who participated in it took part in at least 1/201/20-th of all field trips.
combinatorics
2012 ToT Spring Senior A p5 reflection of angle bisectors, congruent triangles

Source:

3/5/2020
Let ℓ\ell be a tangent to the incircle of triangle ABCABC. Let ℓa,ℓb\ell_a,\ell_b and ℓc\ell_c be the respective images of ℓ\ell under reflection across the exterior bisector of ∠A,∠B\angle A,\angle B and ∠C\angle C. Prove that the triangle formed by these lines is congruent to ABCABC.
geometryreflectionangle bisectorcongruent trianglesincircletangent
2012 ToT Fall Junior A p5 car rides along a circular track in clockwise

Source:

3/22/2020
A car rides along a circular track in the clockwise direction. At noon Peter and Paul took their positions at two different points of the track. Some moment later they simultaneously ended their duties and compared their notes. The car passed each of them at least 3030 times. Peter noticed that each circle was passed by the car 11 second faster than the preceding one while Paul’s observation was opposite: each circle was passed 11 second slower than the preceding one. Prove that their duty was at least an hour and a half long.
combinatorics
2012 ToT Fall Senior O p5 2 counterfeit coins among 239 identical coins

Source:

3/22/2020
Among 239239 coins identical in appearance there are two counterfeit coins. Both counterfeit coins have the same weight different from the weight of a genuine coin. Using a simple balance, determine in three weighings whether the counterfeit coin is heavier or lighter than the genuine coin. A simple balance shows if both sides are in equilibrium or left side is heavier or lighter. It is not required to find the counterfeit coins.
weighingscombinatorics