MathDB

2

Part of 2011 Tournament of Towns

Problems(7)

2011 ToT Spring Junior O p2 111 of 121 rectangles with integer perimeters

Source:

3/4/2020
A rectangle is divided by 1010 horizontal and 1010 vertical lines into 121121 rectangular cells. If 111111 of them have integer perimeters, prove that they all have integer perimeters.
geometryrectangleperimeterIntegercombinatorial geometry
Coordinate problem

Source: Tournament of towns 2011

10/3/2016
Passing through the origin of the coordinate plane are 180180 lines, including the coordinate axes, which form 11 degree angles with one another at the origin. Determine the sum of the x-coordinates of the points of intersection of these lines with the line y=100−xy = 100-x
analytic geometrygeometrycombinatorics
2011 ToT Spring Senior A p2 vertices of rectangle have integer coordinates

Source:

3/4/2020
In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is 20112011, prove that the sides of the rectangular box are parallel to the coordinate axes.
analytic geometrygeometryrectangleparallellattice pointscombinatorial geometry
2011 ToT Fall Junior O p2 Several guests at a round table eat from 2011 berries

Source:

3/22/2020
Several guests at a round table are eating from a basket containing 20112011 berries. Going in clockwise direction, each guest has eaten either twice as many berries as or six fewer berries than the next guest. Prove that not all the berries have been eaten.
combinatorics
2011 ToT Fall Junior A p2 right triangle wanted, AP = 2PB, CP = 2PQ

Source:

3/22/2020
On side ABAB of triangle ABCABC a point PP is taken such that AP=2PBAP = 2PB. It is known that CP=2PQCP = 2PQ where QQ is the midpoint of ACAC. Prove that ABCABC is a right triangle.
geometryright triangle
2011 ToT Fall Senior O p2 winning ticket on lottery

Source:

3/22/2020
Peter buys a lottery ticket on which he enters an nn-digit number, none of the digits being 00. On the draw date, the lottery administrators will reveal an n×nn\times n table, each cell containing one of the digits from 11 to 99. A ticket wins a prize if it does not match any row or column of this table, read in either direction. Peter wants to bribe the administrators to reveal the digits on some cells chosen by Peter, so that Peter can guarantee to have a winning ticket. What is the minimum number of digits Peter has to know?
combinatoricsgame
49 natural with different pairwise sums, max no > 600

Source: Tournament of Towns 2011 oral p2

5/19/2020
4949 natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than 600600.
[hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600
sumscombinatoricsalgebramax