CONCURS #018
01.03.2022, ora 20:00Problema 1 [241 puncte]
$Calculati\ (2^{50}\cdot 17+2^{51}+2^{52}):4^{25}.$
Problema 2 [282 puncte]
$Doua\ caiete\ si\ trei\ creioane\ costa\ 24\ lei,\ iar\ 5\ caiete\ si\ 7\ creioane\ costa\ 59\ lei.$
$Cat\ costa\ 7\ caiete\ si\ 4\ creioane?$
Problema 3 [323 puncte]
$Fie\ p\ si\ q\ numere\ prime\ astfel\ incat\ \frac{p}{2257}=\frac{1}{q}. Calculati\ p+q.$
Problema 4 [364 puncte]
$Aflati\ cate\ numere\ de\ patru\ cifre\ dau\ restul\ 3\ la\ impartirea\ cu\ 19.$
Problema 5 [405 puncte]
$Aflati\ suma\ numerelor\ de\ forma\ \overline{abc},\ care\ verifica\ relatia\ \overline{abc}+\overline{ab}^{2}-9=\overline{75c}.$
Problema 6 [446 puncte]
$Avem\ sirul\ de\ fractii\ \frac{1}{1}, \frac{1}{2} ,..., \frac{1}{100} ,\frac{2}{2}, \frac{2}{3} ,..., \frac{2}{100} , \frac{3}{3}, \frac{3}{4} ,...,\frac{3}{100} ,... ,\frac{99}{99} , \frac{99}{100} , \frac{100}{100}.$ $ Aflati\ suma\ dintre\ numarator\ si\ numitor\ pentru\ a\ 1272-a\ fractie\ din\ sir.$
Problema 7 [487 puncte]
$Numarul\ A\ are\ 47\ de\ divizori.\ Aflati\ numarul\ minim\ de\ divizori\ ai\ numarului\ 135\cdot A.$
Problema 8 [528 puncte]
$Fie\ a,\ b,\ c\ numere\ naturale\ astfel\ incat\ a+a\cdot b+a\cdot c+a\cdot b \cdot\ c = 2022.$
$Aflati\ numarul\ solutiilor\ ecuatiei.$