MathDB

8

Part of 2018 IFYM, Sozopol

Problems(5)

Problem 8 of Second round

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

9/22/2018
Find all positive integers nn for which a square n x n can be covered with rectangles k x 1 and one square 1 x 1 when:
a) k=4k = 4 b) k=8k = 8
tablecombinatoricscombinatorics unsolved
Problem 8 of Third round

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

9/22/2018
Some of the towns in a country are connected with bidirectional paths, where each town can be reached by any other by going through these paths. From each town there are at least n≥3n \geq 3 paths. In the country there is no such route that includes all towns exactly once. Find the least possible number of towns in this country (Answer depends from nn).
graph theorycombinatorics
Problem 8 of First round

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

9/21/2018
The row x1,x2,…x_1, x_2,… is defined by the following recursion
x1=1x_1=1 and xn+1=xn+xnx_{n+1}=x_n+\sqrt{x_n}
Prove that
∑n=120181xn<3\sum_{n=1}^{2018}{\frac{1}{x_n}}<3.
algebrarecurrence relationrecursion
Problem 8 of Fourth round

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

9/22/2018
Are there infinitely many positive integers that can’t be presented as a sum of no more than fifteen fourth degrees of positive integers. (For example 15 isn’t such number as it can be presented as the sum of 15.1415.1^4)
number theorynumber theory unsolved
Problem 8 of Finals

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

9/22/2018
Prove that for every positive integer n≥2n \geq 2 the following inequality holds:
en−1n!<nn+12e^{n-1}n!<n^{n+\frac{1}{2}}
algebraInequalityinequalities