MathDB

4

Part of 2004 Federal Math Competition of S&M

Problems(3)

Combinatorics

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
A set SS of 100100 points, no four in a plane, is given in space. Prove that there are no more than 4.10124 .101^2 tetrahedra with the vertices in SS, such that any two of them have at most two vertices in common.
combinatorics
Combinatorics problem

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
Baron Minchausen talked to a mathematician. Baron said that in his country from any town one can reach any other town by a road. Also, if one makes a circular trip from any town, one passes through an odd number of other towns. By this, as an answer to the mathematician’s question, Baron said that each town is counted as many times as it is passed through. Baron also added that the same number of roads start at each town in his country, except for the town where he was born, at which a smaller number of roads start. Then the mathematician said that baron lied. How did he conclude that?
combinatorics
Problem on sequences

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
The sequence (an)(a_n) is given by a1=x∈Ra_1 = x \in \mathbb{R} and 3an+1=an+13a_{n+1} = a_n+1 for n≥1n \geq 1. Set
A=∑n=1∞[an−16]A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big], B=∑n=1∞[an+16]B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big].
Compute the sum A+BA+B in terms of xx.
Sequences