MathDB

2

Part of 2009 Romania National Olympiad

Problems(4)

For any 8 certain points, 4 terms dependent on these are not al positive

Source: Romanian NO, Grade IX, P.2

10/2/2019
Show that for any four positive real numbers a,b,c,d a,b,c,d and four negative real numbers e,f,g,h, e,f,g,h, the terms ae+bc,ef+cg,fd+gh,da+hb ae+bc,ef+cg,fd+gh,da+hb are not all positive.
algebra
inf|z²-az+a|

Source: Romanian NO, grade x, p.2

10/2/2019
Let be a real number a∈[2+2,4]. a\in \left[ 2+\sqrt 2,4 \right] . Find inf⁡∣z∣≤1z∈C∣z2−az+a∣. \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| .
complex numbersminimainequalitiescomplex disc
Romania National Olympiad 2009 - Grade XI

Source:

4/10/2011
Let f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} a continuous function such that for any x∈Rx\in \mathbb{R}, the limit lim⁡h→0∣f(x+h)−f(x)h∣\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right| exists and it is finite. Prove that in any real point, ff is differentiable or it has finite one-side derivates, of the same modul, but different signs.
functionlimitreal analysisreal analysis unsolved
nr. of nilpotents divides nr. of div. of 0

Source: Romanian NO, grde xii, p.2

10/2/2019
a) Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring. b) Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.
Ring Theoryabstract algebraNilpotentalgebraCommutative Algebra