MathDB

2

Part of 2014 Romania Team Selection Test

Problems(5)

Fractionary parts of cubes in Q/Z

Source: Romania,2014,First TST,Problem 2

7/18/2014
Let n≥2n \ge 2 be an integer. Show that there exist n+1n+1 numbers x1,x2,…,xn+1∈Q∖Zx_1, x_2, \ldots, x_{n+1} \in \mathbb{Q} \setminus \mathbb{Z}, so that {x13}+{x23}+⋯+{xn3}={xn+13}\{ x_1^3 \} + \{ x_2^3 \} + \cdots + \{ x_n^3 \}=\{ x_{n+1}^3 \}, where {x}\{ x \} is the fractionary part of xx.
inductionmodular arithmeticnumber theoryalgebra
Bounds on the iterations of a function!

Source: Romania TST 2014 Day 2 Problem 2

1/21/2015
Let aa be a real number in the open interval (0,1)(0,1). Let n≥2n\geq 2 be a positive integer and let fn ⁣:R→Rf_n\colon\mathbb{R}\to\mathbb{R} be defined by fn(x)=x+x2nf_n(x) = x+\frac{x^2}{n}. Show that a(1−a)n2+2a2n+a3(1−a)2n2+a(2−a)n+a2<(fn∘ ⋯ ∘fn)(a)<an+a2(1−a)n+a\frac{a(1-a)n^2+2a^2n+a^3}{(1-a)^2n^2+a(2-a)n+a^2}<(f_n \circ\ \cdots\ \circ f_n)(a)<\frac{an+a^2}{(1-a)n+a} where there are nn functions in the composition.
functionalgebra unsolvedalgebra
Sum of divisors!

Source: Romania TST 2014 Day 3 Problem 2

1/21/2015
For every positive integer nn, let σ(n)\sigma(n) denote the sum of all positive divisors of nn (11 and nn, inclusive). Show that a positive integer nn, which has at most two distinct prime factors, satisfies the condition σ(n)=2n−2\sigma(n)=2n-2 if and only if n=2k(2k+1+1)n=2^k(2^{k+1}+1), where kk is a non-negative integer and 2k+1+12^{k+1}+1 is prime.
number theory unsolvednumber theory
Cyclotomic polynomial as a composition!

Source: Romania TST 2014 Day 4 Problem 2

1/21/2015
Let pp be an[color=#FF0000] odd prime number. Determine all pairs of polynomials ff and gg from Z[X]\mathbb{Z}[X] such that f(g(X))=∑k=0p−1Xk=Φp(X).f(g(X))=\sum_{k=0}^{p-1} X^k = \Phi_p(X).
algebrapolynomialalgebra unsolved
Counting words!

Source: Romania TST 2014 Day 5 Problem 2

1/21/2015
Let mm be a positive integer and let AA, respectively BB, be two alphabets with mm, respectively 2m2m letters. Let also nn be an even integer which is at least 2m2m. Let ana_n be the number of words of length nn, formed with letters from AA, in which appear all the letters from AA, each an even number of times. Let bnb_n be the number of words of length nn, formed with letters from BB, in which appear all the letters from BB, each an odd number of times. Compute bnan\frac{b_n}{a_n}.
functioncombinatorics unsolvedcombinatorics