CONCURS #020
10.03.2022, ora 20:00Problema 1 [291 puncte]
$Calculati\ [\sqrt{2022}]+[\frac{2022}{17}]+2022\cdot (\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+...+\frac{1}{336\cdot 337})$
$Am\ notat\ cu\ [x]\ partea\ intreaga\ a \ lui\ x.$
Problema 2 [382 puncte]
$Aflati\ solutia\ ecuatiei\ \frac{2x-1}{3}-\frac{3x-5}{2}=2x+5. $
Problema 3 [473 puncte]
$Aflati\ media\ geometrica\ a \ numerelor:\ \sqrt{(23-\sqrt{33})^{2}}+\sqrt{(5-\sqrt{33})^{2}}\ si\ 50.$
Problema 4 [564 puncte]
$Fie\ multimea\ A=\{ n\in \mathbb{N}\ | \sqrt{n+13}\ si\ \sqrt{n+109}\ sunt\ simultan\ numere\ naturale \}.$
$Aflati\ cardinalul\ multimii\ A. $
Problema 5 [655 puncte]
$Pentru\ n\ natural\ definim\ pp(n)\ ca\ fiind\ cel\ mai\ mic\ patrat\ perfect,\ mai\ mare$
$sau\ egal\ cu\ n.\ Daca\ pp(n)\cdot pp(n+1)=1764,\ aflati\ n.$
Problema 6 [746 puncte]
$Aflati\ cate\ numere\ naturale\ mai\ mici\ decat\ 96\ sunt\ prime\ cu\ 96.$
Problema 7 [837 puncte]
$Fie\ ABCD\ patrat\ cu\ AB=48,\ M\ mijlocul\ lui\ CD\ si\ N\ mijlocul\ lui\ AM.\ $$ Aflati\ aria\ patrulaterului\ BCMN.$
Problema 8 [928 puncte]
$In\ triunghiul\ ABC,\ M\ e\ mijlocul\ lui\ AB,\ N\ mijlocul\ lui\ AC,\ P\ mijlocul\ lui\ NC,\ si\ $$MP\bigcap BC=\{Q\}.\ Aflati\ valoarea\ raportului\ \frac{BQ}{CQ}.$