MathDB

5

Part of 2004 Tournament Of Towns

Problems(7)

Neighbour integers

Source: Tournament of Towns Spring 2004 Junior O #5

5/22/2014
Two 10-digit integers are called neighbours if they differ in exactly one digit (for example, integers 12345678901234567890 and 12345078901234507890 are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.
combinatorics unsolvedcombinatorics
TOT 2004 Spring - Senior O-Level p5 neighbours 10-digit integers

Source:

2/25/2020
Two 1010-digit integers are called neighbours if they differ in exactly one digit (for example, integers 12345678901234567890 and 12345078901234507890 are neighbours). Find the maximal number of elements in the set of 1010-digit integers with no two integers being neighbours.
number theoryDigitsDigit
TOT 2004 Spring - Senior A-Level p5 common tangent circle and y = x^2

Source:

2/25/2020
The parabola y=x2y = x^2 intersects a circle at exactly two points AA and BB. If their tangents at AA coincide, must their tangents at BB also coincide?
parabolacommon tangentcirclegeometrytangentTangents
TOT 2004 Spring - Junior A-Level p5 polygonal billiard table

Source:

2/25/2020
All sides of a polygonal billiard table are in one of two perpendicular directions. A tiny billiard ball rolls out of the vertex AA of an inner 90o90^o angle and moves inside the billiard table, bouncing off its sides according to the law “angle of reflection equals angle of incidence”. If the ball passes a vertex, it will drop in and srays there. Prove that the ball will never return to AA.
combinatorics
Incircles and inequality

Source: Tournament of towns, Junior A-Level paper, Fall 2004

12/25/2004
Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].
geometryinequalitiesinradiustrigonometrygeometry solved
Sum of aproximately equal numbers...

Source: Tournament of towns, Junior B-Level paper, Fall 2004

12/25/2004
How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.
modular arithmeticnumber theory unsolvednumber theory
Arithmetic mean is an integer

Source: Tournament of towns, Senior B-Level paper, Fall 2004

12/25/2004
For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?
inductionnumber theory unsolvednumber theory