MathDB

1

Part of 2011 Romania National Olympiad

Problems(6)

p^2 + pq + q^2 = r^2 - 2011 Romania NMO VII p1

Source:

8/15/2024
Find all positive integers rr with the property that there exists positive prime numbers pp and qq so that p2+pq+q2=r2.p^2 + pq + q^2 = r^2 .
number theory
x + y + z<=t,, x^2 + y^2 + z^2>=t , x^3 + y^3 + z^3<=t 2011 Romania NMO VIII p1

Source:

8/15/2024
Find all real numbers x,y,z,t∈[0,∞)x, y,z,t \in [0, \infty) so that x+y+z≤t,   x2+y2+z2≥t   and   x3+y3+z3≤t.x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.
algebrainequalities
Sum of squares >= than sum of cubes

Source: Romanian NO, grade ix, p.1

10/3/2019
Let be a natural number n n and n n real numbers a1,a2,…,an a_1,a_2,\ldots ,a_n such that a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} , \forall m\in\{ 1,2,\ldots ,n \} . Prove that a12+a22+⋯+an2≥n(n+1)(2n+1)6. a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} .
inequalitiesformulasalgebrafactorizationsum of cubes
|f(x+y+sinx+siny)|<=2

Source: Romanian NO 2011, grade x, p.1

10/3/2019
Let f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} a function having the property that ∣f(x+y)+sin⁡x+sin⁡y∣≤2, \left| f(x+y)+\sin x+\sin y \right|\le 2, for all real numbers x,y. x,y.
a) Prove that ∣f(x)∣≤1+cos⁡x, \left| f(x) \right|\le 1+\cos x, for all real numbers x. x. b) Give an example of what f f may be, if the interval (−π,π) \left( -\pi ,\pi \right) is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.
functionalgebra
Romania National Olympiad 2011 - Grade XI - problem 1

Source:

4/19/2011
A row of a matrix belonging to Mn(C)\mathcal{M}_n(\mathbb{C}) is said to be permutable if no matter how we would permute the entries of that row, the value of the determinant doesn't change. Prove that if a matrix has two permutable rows, then its determinant is equal to 00 .
linear algebramatrixlinear algebra unsolved
Necessary and suff. cond. for a ring to possess nonzero nilpotents

Source: Romanian NO 2011, grade xii, p.1

10/3/2019
Prove that a ring that has a prime characteristic admits nonzero nilpotent elements if and only if its characteristic divides the number of its units.
abstract algebraRing Theory