CONCURS #011
17.02.2022, ora 20:00A fost o eroare la Pb.2, se vor anula punctajele egale cu 0
Problema 1 [255 puncte]
$Calculati\ A=[\sqrt{2022}]+(\sqrt{23}-4)\cdot (\sqrt{23}+4),\ unde\ [x]\ este\ partea\ intreaga\ a\ lui\ x.$
Problema 2 [310 puncte]
$Fie\ x\ si\ y\ numere\ reale\ astfel\ incat\ x^{2}+4\cdot y^{2}+3364=84\cdot x+160\cdot y.$
$Aflati\ media\ aritmetica\ a\ numerelor\ x\ si\ y.$
Problema 3 [365 puncte]
$Fie\ x>0\ astfel\ incat\ x+\frac{1}{x}=13.\ Calculati\ E=x^{2}+\frac{1}{x^{2}}+x^{3}+\frac{1}{x^{3}}. $
Problema 4 [420 puncte]
$Aflati\ valoarea\ maxima\ a\ expresiei\ E(x)=-3\cdot x^{2}+24\cdot x-17,\ unde\ x\in \mathbb{R}.$
Problema 5 [475 puncte]
$Fie\ ABCDA'B'C'D'\ un\ cub\ cu\ AB=20,\ iar\ M\ mijlocul\ lui\ BC.$
$Aflati\ distanta\ de\ la\ punctul\ D'\ la\ dreapta\ AM,\ cu\ doua\ zecimale\ exacte.$
Problema 6 [530 puncte]
$In\ piramida\ regulata\ VABCD,\ cu\ baza\ ABCD,\ avem\ AB=\sqrt{1152},$
$ M\ mijlocul\ lui\ VC\ si\ m(\widehat{BM,VA})=30^{0}.\ Aflati\ lungimea\ lui\ BM.$
Problema 7 [585 puncte]
$Avem\ 3\ cifre\ egale\ cu\ 2,\ 4\ cifre\ egale\ cu\ 3,\ si\ 2\ cifre\ egale\ cu\ 7.\ Cu\ aceste\ cifre\ se\ $$formeaza\ numere\ de\ 9\ cifre.\ Cate\ dintre\ numerele\ formate\ sunt$$ palindromuri?$
Problema 8 [640 puncte]
$Fie\ p,q,r,s\ numere\ prime\ astfel\ incat\ p\cdot\ q\cdot\ r\cdot\ s=97097.$
$Calculati\ A=p+q+r+s.$