CONCURS #010
16.02.2022, ora 20:00Problema 1 [272 puncte]
$Determinati\ partea\ intreaga\ a\ numarului\ A=\sqrt{2022^{2}+2023}.$
Problema 2 [344 puncte]
$Calculati\ \sqrt{784}\cdot (\frac{\sqrt{2}-\sqrt{1}}{\sqrt{2}}+\frac{\sqrt{3}-\sqrt{2}}{\sqrt{6}}+\frac{\sqrt{4}-\sqrt{3}}{\sqrt{12}}).$
Problema 3 [416 puncte]
$Fie\ x\ si\ y\ numere\ reale\ astfel\ incat\ |x-18\sqrt{3}|+|y-24\sqrt{3}|=0. $
$Calculati\ media\ geometrica\ a\ numerelor\ x\ si\ y.$
Problema 4 [488 puncte]
$Aflati\ numarul\ solutiilor\ in\ Z\ ale\ ecuatiei\ x\cdot y+2\cdot x+3\cdot y=2022.$
Problema 5 [560 puncte]
$Fie\ ABCD\ un\ patrat\ cu\ AB=40,\ M\ si\ N\ mijloacele\ laturilor\ BC,\ respectiv\ CD,\ iar$
$AM\bigcap BN=\{ O\}.\ Calculati\ aria\ patrulaterului\ MONC.$
Problema 6 [632 puncte]
$Se\ dau\ trei\ cercuri\ tangente\ exterior\ doua\ cate\ doua,\ de\ centre\ A,B,C\ si\ raze\ 12,21,44.$
$Calculati\ aria\ triunghiului\ ABC.$
Problema 7 [704 puncte]
$Aflati\ numarul\ solutiilor\ naturale\ ale\ ecuatiei\ x\cdot y\cdot z=1001.$
Problema 8 [776 puncte]
$Fie\ a,b,c\ numere\ reale\ pozitive\ astfel\ incat\ a\cdot b\cdot c=1.\ Calculati\ valoarea$
$expresiei\ E=\frac{1}{1+b+b\cdot c}+\frac{1}{1+c+a\cdot c}+\frac{1}{1+a+a\cdot b}.$