MathDB

1

Part of 2010 Tournament Of Towns

Problems(8)

Number of fruits in baskets...

Source:

2/7/2011
Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of 3131.
number theory unsolvednumber theory
Cutting a piece of cheese in a given ratio...

Source:

2/7/2011
Alex has a piece of cheese. He chooses a positive number a≠1a\neq 1 and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio 1:a.1:a. His goal is to divide the cheese into two piles of equal masses. Can he do it?
rationumber theory unsolvednumber theory
2010 ships delivering three kind of fruits.

Source:

2/9/2011
20102010 ships deliver bananas, lemons and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Prove that the total number of fruits is a multiple of 3131.
number theory unsolvednumber theory
Splitting all lines in a plane in pairs...

Source:

2/9/2011
Is it possible to split all straight lines in a plane into the pairs of perpendicular lines, so that every line belongs to a single pair?
analytic geometrygeometry unsolvedgeometry
Sum of black squares is equal to sum of white squares.

Source:

2/12/2011
In a multiplication table, the entry in the ii-th row and the jj-th column is the product ijij From an m×nm\times n subtable with both mm and nn odd, the interior (m−2)(n−2)(m-2) (n-2) rectangle is removed, leaving behind a frame of width 11. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.
geometryrectanglenumber theory proposednumber theory
Constructing a perpendicular line to a given line...

Source:

2/12/2011
A round coin may be used to construct a circle passing through one or two given points on the plane. Given a line on the plane, show how to use this coin to construct two points such that they de ne a line perpendicular to the given line. Note that the coin may not be used to construct a circle tangent to the given line.
geometry unsolvedgeometry
Achieving gains in a exchange machine...

Source:

2/19/2011
The exchange rate in a Funny-Money machine is ss McLoonies for a Loonie or 1s\frac{1}{s} Loonies for a McLoonie, where ss is a positive real number. The number of coins returned is rounded off to the nearest integer. If it is exactly in between two integers, then it is rounded up to the greater integer. (a)(a) Is it possible to achieve a one-time gain by changing some Loonies into McLoonies and changing all the McLoonies back to Loonies? (b)(b) Assuming that the answer to (a)(a) is "yes", is it possible to achieve multiple gains by repeating this procedure, changing all the coins in hand and back again each time?
Finding the erased distance with the other distances...

Source:

2/19/2011
There are 100100 points on the plane. All 49504950 pairwise distances between two points have been recorded. (a)(a) A single record has been erased. Is it always possible to restore it using the remaining records? (b)(b) Suppose no three points are on a line, and kk records were erased. What is the maximum value of kk such that restoration of all the erased records is always possible?
geometry unsolvedgeometry