MathDB

5

Part of 2006 Tournament of Towns

Problems(7)

Defensively Painting Cubes

Source: Spring 2006 Tournament of Towns Junior O-Level #5

4/15/2015
Pete has n3n^3 white cubes of the size 1×1×11\times 1\times 1. He wants to construct a n×n×nn\times n\times n cube with all its faces being completely white. Find the minimal number of the faces of small cubes that Basil must paint (in black colour) in order to prevent Pete from fulfilling his task. Consider the cases: a) n=2n = 2; (2 points) b) n=3n = 3. (4 points)
How Many Ones?

Source: Spring 2006 Tournament of Towns Junior A-Level #5

4/15/2015
Numbers 0,10, 1 and 22 are placed in a table 2005×20062005 \times 2006 so that total sums of the numbers in each row and in each column are factors of 33. Find the maximal possible number of 11's that can be placed in the table.
(6 points)
How Many Faces Must be Painted?

Source: Spring 2006 Tournament of Towns Senior O-Level #5

9/9/2015
Pete has n3n^3 white cubes of the size 1×1×11\times1\times1. He wants to construct a n×n×nn\times n\times n cube with all its faces being completely white. Find the minimal number of the faces of small cubes that Basil must paint (in black colour) in order to prevent Pete from fulfilling his task. Consider the cases: a) n=3n = 3; (3 points) b) n=1000n = 1000. (3 points)
TOT 2006 Spring - Senior A-Level p5 product of infitie numbers

Source:

2/25/2020
Prove that one can find infinite number of distinct pairs of integers such that every digit of each number is no less than 77 and the product of two numbers in each pair is also a number with all its digits being no less than 77. (6)
number theorycombinatorics
TOT 2006 Fall - Junior O-Level p5 square is n congruent non-convex polygons

Source:

2/25/2020
A square is dissected into nn congruent non-convex polygons whose sides are parallel to the sides of the square, and no two of these polygons are parallel translates of each other. What is the maximum value of nn? (4)
combinatorial geometrycombinatoricsmaxpolygonnon-convex
TOT 2006 Fall - Senior O-Level p5 regular octahedron inscribed in cube

Source:

2/25/2020
Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)
3D geometryoctahedroncubeinscribedgeometry
TOT 2006 Fall - Junior A-Level p5 encelope of area 1 of a square of area 1

Source:

2/25/2020
Consider a square painting of size 1×11 \times 1. A rectangular sheet of paper of area 22 is called its “envelope” if one can wrap the painting with it without cutting the paper. (For instance, a 2×12 \times 1 rectangle and a square with side 2\sqrt2 are envelopes.) a) Show that there exist other envelopes. (4) b) Show that there exist infinitely many envelopes. (3)
geometry