CONCURS #019
09.03.2022, ora 20:00Problema 1 [296 puncte]
$Calculati\ A+B,\ unde\ A=u(2^{57})+u(6^{129}),\ iar\ B=2022\cdot 2021-2022\cdot 2019.$
$Am\ notat\ cu\ u(N)\ ultima\ cifra\ a \ lui\ N.$
Problema 2 [392 puncte]
$Numarul\ A\ are\ 3\ divizori,\ iar\ numarul\ B\ are\ 5\ divizori.\ $
$Aflati\ numarul\ minim\ de\ divizori\ ai\ numarului\ A\cdot B.$
Problema 3 [488 puncte]
$Intr-o\ urna\ sunt\ 100\ de\ bile\ numerotate\ de\ la\ 1\ la\ 100.\ Care\ este\ numarul\ minim$
$de\ bile\ ce\ trebuie\ scose\ din\ urna\ pentru\ a\ fi\ siguri\ ca\ am\ scos\ cel\ putin\ una\ divizibila\ cu\ 7?$
Problema 4 [584 puncte]
$Cate\ fractii\ ireductibile\ contine\ sirul:\ \frac{1}{4},\ \frac{3}{9},\ \frac{5}{14},\ ...,\ \frac{201}{504}?$
Problema 5 [680 puncte]
$Aflati\ suma\ numerelor\ de\ forma\ \overline{abc},\ stiind\ ca\ a=2\cdot b\ si\ c=3\cdot b.$
Problema 6 [776 puncte]
$Fie\ p\ numar\ prim\ si\ n\ natural\ astfel\ incat\ p+n^{2}=3\cdot n+552.$ $ Aflati\ valoarea\ maxima\ a \ sumei\ p+n.$
Problema 7 [872 puncte]
$Fie\ numarul\ A=2^{73}+3^{89}. Aflati\ cel\ mai\ mic\ divizor\ propriu\ al\ lui\ A.$
Problema 8 [968 puncte]
$Daca\ \frac{a}{b+c+d}=\frac{b}{a+c+d}=\frac{c}{a+b+d}=\frac{d}{a+b+c},\ calculati\ E=\frac{a+b}{c+d}+\frac{a+c}{b+d}+\frac{a+d}{b+c}+\frac{b+c}{a+d}+\frac{b+d}{a+c}+\frac{c+d}{a+b}.$