MathDB

6

Part of 2011 Postal Coaching

Problems(5)

Separating people into two groups

Source:

12/31/2011
In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups AA and BB such that in AA everybody knows everybody else and in BB nobody knows anybody else.
inductiongraph theorycombinatorics unsolvedcombinatorics
Prove that centre of circle divides altitude in golden ratio

Source:

12/31/2011
Let TT be an isosceles right triangle. Let SS be the circle such that the difference in the areas of T∪ST \cup S and T∩ST \cap S is the minimal. Prove that the centre of SS divides the altitude drawn on the hypotenuse of TT in the golden ratio (i.e., (1+5)2\frac{(1 + \sqrt{5})}{2})
ratiogeometrygeometry unsolved
Existence of monotonic perfect square

Source:

12/31/2011
A positive integer is called monotonic if when written in base 1010, the digits are weakly increasing. Thus 1222677812226778 is monotonic. Note that a positive integer cannot have first digit 00. Prove that for every positive integer nn, there is an nn-digit monotonic number which is a perfect square.
number theory unsolvednumber theory
Prove that one point is contained by at least n arcs

Source:

12/31/2011
On a circle there are nn red and nn blue arcs given in such a way that each red arc intersects each blue one. Prove that some point is contained by at least nn of the given coloured arcs.
combinatorics unsolvedcombinatorics
Existence of integers

Source:

12/31/2011
Prove that there exist integers a,b,ca, b, c all greater than 20112011 such that (a+b)c=…2010⋅2011…(a+\sqrt{b})^c=\ldots 2010 \cdot 2011\ldots [Decimal point separates an integer ending in 20102010 and a decimal part beginning with 20112011.]
algebrapolynomialalgebra unsolved