MathDB

3

Part of 2009 Tournament Of Towns

Problems(8)

Cardboard circular disk - TT 2009 Junior-O3

Source:

9/3/2010
A cardboard circular disk of radius 55 centimeters is placed on the table. While it is possible, Peter puts cardboard squares with side 55 centimeters outside the disk so that:
(1) one vertex of each square lies on the boundary of the disk; (2) the squares do not overlap; (3) each square has a common vertex with the preceding one.
Find how many squares Peter can put on the table, and prove that the fi rst and the last of them must also have a common vertex.
(4 points)
Solve the equation - TT 2009 Junior-A3

Source:

9/3/2010
Find all positive integers aa and bb such that (a+b2)(b+a2)=2m(a + b^2)(b + a^2) = 2^m for some integer m.m.
(6 points)
symmetrynumber theory proposednumber theory
Tetrahedron and sphere - TT 2009 Senior-A3

Source:

9/3/2010
Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent.
(7 points)
geometry3D geometrytetrahedronspheregeometry unsolved
Four positive integers - TT 2009 Senior-O3

Source:

9/3/2010
Are there positive integers a;b;ca; b; c and dd such that a3+b3+c3+d3=100100a^3 + b^3 + c^3 + d^3 =100^{100} ?
(4 points)
number theory proposednumber theory
2009 ToT Spring Junior O P3 30digit integer with cubical blocks

Source:

3/7/2020
Alex is going to make a set of cubical blocks of the same size and to write a digit on each of their faces so that it would be possible to form every 3030-digit integer with these blocks. What is the minimal number of blocks in a set with this property? (The digits 66 and 99 do not turn one into another.)
minimumDigitsnumber theorycombinatorics
2009 ToT Spring Junior A P3 chess piece car in 101x101 board

Source:

3/7/2020
In each square of a 101×101101\times 101 board, except the central one, is placed either a sign " turn" or a sign " straight". The chess piece " car" can enter any square on the boundary of the board from outside (perpendicularly to the boundary). If the car enters a square with the sign " straight" then it moves to the next square in the same direction, otherwise (in case it enters a square with the sign " turn") it turns either to the right or to the left ( its choice). Can one place the signs in such a way that the car never enter the central square?
combinatorics
2009 ToT Spring Senior O P3 x_n = O(x_{n-1}+x_{n-2}), O(n) greatest odd divisor

Source:

3/7/2020
For each positive integer nn, denote by O(n)O(n) its greatest odd divisor. Given any positive integers x1=ax_1 = a and x2=bx_2 = b, construct an in nite sequence of positive integers as follows: xn=O(xn−1+xn−2)x_n = O(x_{n-1} + x_{n-2}), where n=3,4,...n = 3,4,... (a) Prove that starting from some place, all terms of the sequence are equal to the same integer. (b) Express this integer in terms of aa and bb.
divisorrecurrence relationnumber theory
2009 ToT Spring Senior A P3 max no of chips can be removed from board

Source:

3/7/2020
Each square of a 10×1010\times 10 board contains a chip. One may choose a diagonal containing an even number of chips and remove any chip from it. Find the maximal number of chips that can be removed from the board by these operations.
combinatorics