MathDB

3

Part of 2011 Tournament of Towns

Problems(9)

2011 ToT Spring Junior O p3 worms grow at the rate of 1 m per hour

Source:

3/4/2020
Worms grow at the rate of 11 metre per hour. When they reach their maximal length of 11 metre, they stop growing. A full-grown worm may be dissected into two not necessarily equal parts. Each new worm grows at the rate of 11 metre per hour. Starting with 11 full-grown worm, can one obtain 1010 full-grown worms in less than 11 hour?
combinatoricstime
Combinatorics

Source: Tournament of towns 2011

10/3/2016
Baron Munchausen has a set of 5050 coins. The mass of each is a distinct positive integer not exceeding 100100, and the total mass is even. The Baron claims that it is not possible to divide the coins into two piles with equal total mass. Can the Baron be right?
combinatoricsBaron Munchausen
2011 ToT Spring Senior O p3 along a circle are 100 white points

Source:

3/4/2020
Along a circle are 100100 white points. An integer kk is given, where 2≤k≤502 \le k \le 50. In each move, we choose a block of kk adjacent points such that the first and the last are white, and we paint both of them black. For which values of kk is it possible for us to paint all 100100 points black after 5050 moves?
combinatoricsColoring
2011 ToT Spring Senior A p3 does there exist an in nite triangular beam

Source:

3/4/2020
(a) Does there exist an in nite triangular beam such that two of its cross-sections are similar but not congruent triangles? (b) Does there exist an in nite triangular beam such that two of its cross-sections are equilateral triangles of sides 11 and 22 respectively?
similar trianglesEquilateralcombinatorial geometrygeometrycongruent triangles
2011 ToT Fall Junior O p3 9x9 chessboard minus central 4x4

Source:

3/22/2020
From the 9×99 \times 9 chessboard, all 1616 unit squares whose row numbers and column numbers are both even have been removed. Disect the punctured board into rectangular pieces, with as few of them being unit squares as possible.
combinatorics
2011 ToT Fall Junior A p3 a balance and a set of pairwise different weights

Source:

3/22/2020
A balance and a set of pairwise different weights are given. It is known that for any pair of weights from this set put on the left pan of the balance, one can counterbalance them by one or several of the remaining weights put on the right pan. Find the least possible number of weights in the set.
weighingscombinatorics
2011 ToT Fall Senior O p3 AB = 10, BC = 14, CD = 11,DA = 5, angle

Source:

3/22/2020
In a convex quadrilateral ABCD,AB=10,BC=14,CD=11ABCD, AB = 10, BC = 14, CD = 11 and DA=5DA = 5. Determine the angle between its diagonals.
geometryanglesconvexquadrilateral
2011 ToT Fall Senior A p3 line bisects two segments

Source:

3/22/2020
In triangle ABCABC, points A1,B1,C1A_1,B_1,C_1 are bases of altitudes from vertices A,B,CA,B,C, and points CA,CBC_A,C_B are the projections of C1C_1 to ACAC and BCBC respectively. Prove that line CACBC_AC_B bisects the segments C1A1C_1A_1 and C1B1C_1B_1.
bisects segmentequal segmentsgeometryaltitudeprojection
3 points start moving, forming a triangle, circumcenter on a straight line

Source: Tournament of Towns 2011 oral p3

5/19/2020
Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?
geometryCircumcenterLocussimilar trianglessimilarity