MathDB

4

Part of 2009 Tournament Of Towns

Problems(8)

Password - TT 2009 Junior-O4 and Senior-O1

Source:

9/3/2010
We only know that the password of a safe consists of 77 different digits. The safe will open if we enter 77 different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than 77 attempts?
(5 points for Juniors and 4 points for Seniors)
Easy geometry problem - TT 2009 Junior-A4

Source:

9/3/2010
Let ABCDABCD be a rhombus. PP is a point on side BC BC and QQ is a point on side CDCD such that BP=CQBP = CQ. Prove that centroid of triangle APQAPQ lies on the segment BD.BD.
(6 points)
geometry
New factorial function - TT 2009 Senior-A4

Source:

9/3/2010
Denote by [n]![n]! the product 1⋅11⋅111⋅...⋅111...1⏟n ones 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}.(nn factors in total). Prove that [n+m]![n + m]! is divisible by [n]!×[m]! [n]! \times [m]!
(8 points)
factorialfunctionfloor functionnumber theory proposednumber theory
Regular 2009-gon - TT 2009 Senior-O4

Source:

9/3/2010
A point is chosen on each side of a regular 20092009-gon. Let SS be the area of the 20092009-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the 20092009-gon with vertices at the images of these reflections also has area S.S.
(4 points)
geometrygeometric transformationreflectiontrigonometrygeometry unsolved
2009 ToT Spring Junior O P4 10% increase integer and decrease digits

Source:

3/7/2020
We increased some positive integer by 10%10\% and obtained a positive integer. Is it possible that in doing so we decreased the sum of digits exactly by 10%10\% ?
Digitsnumber theory
2009 ToT Spring Junior A P4 progression arithmetic and geometric

Source:

3/7/2020
Consider an in finite sequence consisting of distinct positive integers such that each term (except the rst one) is either an arithmetic mean or a geometric mean of two neighboring terms. Does it necessarily imply that starting at some point the sequence becomes either arithmetic progression or a geometric progression?
geometric progressionArithmetic Progressionalgebra
2009 ToT Spring Senior O P4 zeros and ones written in a row

Source:

3/7/2020
Several zeros and ones are written down in a row. Consider all pairs of digits (not necessarily adjacent) such that the left digit is 11 while the right digit is 00. Let MM be the number of the pairs in which 11 and 00 are separated by an even number of digits (possibly zero), and let NN be the number of the pairs in which 11 and 00 are separated by an odd number of digits. Prove that M≥NM \ge N.
combinatorics
2009 ToT Spring Senior A P4 3 planes cut parallelepiped into 8 hexahedrons

Source:

2/26/2020
Three planes dissect a parallelepiped into eight hexahedrons such that all of their faces are quadrilaterals (each plane intersects two corresponding pairs of opposite faces of the parallelepiped and does not intersect the remaining two faces). One of the hexahedrons has a circumscribed sphere. Prove that each of these hexahedrons has a circumscribed sphere.
hexahedronparallelepiped3D geometrygeometrysphere