MathDB

3

Part of 2015 Romania Team Selection Tests

Problems(5)

Pythagorean triples

Source: Romania TST 2015 Day 1 Problem 3

4/9/2015
A Pythagorean triple is a solution of the equation x2+y2=z2x^2 + y^2 = z^2 in positive integers such that x<yx < y. Given any non-negative integer nn , show that some positive integer appears in precisely nn distinct Pythagorean triples.
number theoryRomanian TSTinduction
Product set with terms in arithmetic progression !

Source: Romania TST 2015 Day 2 Problem 3

6/4/2015
Given a positive real number tt , determine the sets AA of real numbers containing tt , for which there exists a set BB of real numbers depending on AA , ∣B∣≥4|B| \geq 4 , such that the elements of the set AB={ab∣a∈A,b∈B}AB =\{ ab \mid a\in A , b \in B \} form a finite arithmetic progression .
arithmetic sequenceSetsProductalgebra
Least common multiple function !!!

Source: Romania TST 2015 Day 3 Problem 3

6/4/2015
If kk and nn are positive integers , and k≤nk \leq n , let M(n,k)M(n,k) denote the least common multiple of the numbers n,n−1,…,n−k+1n , n-1 , \ldots , n-k+1.Let f(n)f(n) be the largest positive integer k≤n k \leq n such that M(n,1)<M(n,2)<…<M(n,k)M(n,1)<M(n,2)<\ldots <M(n,k) . Prove that : (a) f(n)<3nf(n)<3\sqrt{n} for all positive integers nn . (b) If NN is a positive integer , then f(n)>Nf(n) > N for all but finitely many positive integers nn.
least common multiplefunctionRomanian TSTnumber theoryRIP
Distinct sums of permuted numbers !!!

Source: Romania TST 2015 Day 4 Problem 3

6/4/2015
Let nn be a positive integer . If σ\sigma is a permutation of the first nn positive integers , let S(σ)S(\sigma) be the set of all distinct sums of the form ∑i=klσ(i)\sum_{i=k}^{l} \sigma(i) where 1≤k≤l≤n1 \leq k \leq l \leq n . (a) Exhibit a permutation σ\sigma of the first nn positive integers such that ∣S(σ)∣≥⌊(n+1)24⌋|S(\sigma)|\geq \left \lfloor{\frac{(n+1)^2}{4}}\right \rfloor . (b) Show that ∣S(σ)∣>nn42|S(\sigma)|>\frac{n\sqrt{n}}{4\sqrt{2}} for all permutations σ\sigma of the first nn positive integers .
permutationsPartial sumsProbabilistic MethodcombinatoricsRomanian TST
Sequence congruence !!!

Source: Romania TST 2015 Day 5 Problem 3

6/4/2015
Define a sequence of integers by a0=1a_0=1 , and an=∑k=0n−1(nk)aka_n=\sum_{k=0}^{n-1} \binom{n}{k}a_k , n≥1n \geq 1 . Let mm be a positive integer , let pp be a prime , and let qq and rr be non-negative integers . Prove that : apmq+r≡apm−1q+r(modpm)a_{p^mq+r} \equiv a_{p^{m-1}q+r} \pmod{p^m}
Sequencebinomial coefficientscongruencenumber theoryRomanian TST