MathDB

1

Part of 2012 Romania Team Selection Test

Problems(5)

Writing sqrt{m} as sum of m nth roots

Source: Romania TST 3 2012, Problem 1

5/11/2012
Let mm and nn be two positive integers greater than 11. Prove that there are mm positive integers N1N_1 , …\ldots , NmN_m (some of them may be equal) such that m=∑i=1m(Ni−Ni−1)1n.\sqrt{m}=\sum_{i=1}^m{(\sqrt{N_i}-\sqrt{N_i-1})^{\frac{1}{n}}.}
USAMTSalgebra proposedalgebra
Different sums mod n_1

Source: Romania TST 1 2012, Problem 1

5/3/2012
Let n1,…,nkn_1,\ldots,n_k be positive integers, and define d1=1d_1=1 and di=(n1,…,ni−1)(n1,…,ni)d_i=\frac{(n_1,\ldots,n_{i-1})}{(n_1,\ldots,n_{i})}, for i∈{2,…,k}i\in \{2,\ldots,k\}, where (m1,…,mℓ)(m_1,\ldots,m_{\ell}) denotes the greatest common divisor of the integers m1,…,mℓm_1,\ldots,m_{\ell}. Prove that the sums ∑i=1kaini\sum_{i=1}^k a_in_i with ai∈{1,…,di}a_i\in\{1,\ldots,d_i\} for i∈{1,…,k}i\in\{1,\ldots,k\} are mutually distinct mod  n1\mod n_1.
inductionnumber theorygreatest common divisorgeometrynumber theory proposed
Geometric inequality involving some bisectors

Source: Romania TST 4 2012, Problem 1

5/16/2012
Let ΔABC\Delta ABC be a triangle. The internal bisectors of angles ∠CAB\angle CAB and ∠ABC\angle ABC intersect segments BCBC, respectively ACAC in DD, respectively EE. Prove that DE≤(3−22)(AB+BC+CA).DE\leq (3-2\sqrt{2})(AB+BC+CA).
inequalitiestrigonometryfunctiontrig identitiesLaw of Cosinesinequalities proposed
Identity with kth roots and logarithms

Source: Romania TST 2 2012, Problem 1

5/10/2012
Prove that for any positive integer n≥2n\geq 2 we have that ∑k=2n⌊nk⌋=∑k=2n⌊log⁡kn⌋.\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.
logarithmsfloor functionfunctioninductionalgebra proposedalgebra
n quadratic residue implies an^2+bn+c quadratic residue

Source: Romania TST 5 2012, Problem 1

5/17/2012
Find all triples (a,b,c)(a,b,c) of positive integers with the following property: for every prime pp, if nn is a quadratic residue mod  p\mod p, then an2+bn+can^2+bn+c is a quadratic residue mod  p\mod p.
quadraticsalgebrapolynomialnumber theory proposednumber theory