CONCURS #009
15.02.2022, ora 20:00Problema 1 [246 puncte]
$In\ clasa\ a\ VI-a\ A\ numarul\ baietilor\ este\ cu\ 3\ mai\ mare\ decat\ numarul\ fetelor.$
$Aflati\ cati\ elevi\ sunt\ in\ clasa,\ daca\ raportul\ dintre\ numarul\ fetelor\ si\ al\ baietilor\ este\ \frac{5}{6}.$
Problema 2 [292 puncte]
$Un\ telefon\ costa\ 800\ de\ euro.\ El\ se\ scumpeste\ cu\ 25\%,\ iar\ de\ black\ friday\ se\ ieftineste\ cu$$25\%.\ Care\ va\ fi\ pretul\ telefonului\ dupa\ ieftinire?$
Problema 3 [338 puncte]
$Fie\ multimea\ A\ cu\ elementele\ 1,2,3...,12 .\ Aflati\ numarul\ submultimilor\ cu\ doua\ elemente\ ale\ lui\ A,\ ce\ contin\ $
$ale\ lui\ A,\ ce\ contin\ doua\ elemente\ a\ si\ b,\ cu\ a < b,\ cu\ proprietatea\ ca\ a\ divide\ pe\ b.$
Problema 4 [384 puncte]
$Aflati\ cate\ perechi\ de\ numere\ naturale\ ( a,b),\ cu\ a < b < 50,\ au\ c.m.m.d.c(a,b) =7.$
Problema 5 [430 puncte]
$Fie\ punctele\ A,B,C\ coliniare,\ in\ aceasta\ ordine,\ iar\ M,N\ mijloacele\ segmentelor\ AB,$
$ respectiv\ BC.\ Daca\ MN=5\cdot BC,\ aflati\ raportul\ dintre\ AN\ si\ NC.$
Problema 6 [476 puncte]
$Se\ considera\ n\ unghiuri\ in\ jurul\ unui\ punct,\ avand\ masurile\ exprimate\ in\ grade\ prin$
$numere\ naturale\ consecutive.\ Aflati\ valoarea\ maxima\ a\ lui\ n.$
Problema 7 [522 puncte]
$Aflati\ numarul\ solutiilor\ ecuatiei\ (x,y)+[x,y] = 20,\ unde\ x\ si\ y\ sunt\ naturale\ nenule,$
$(x,y)\ este\ cel\ mai\ mare\ divizor\ comun,\ iar\ [x,y]\ este\ cel\ mai\ mic\ multiplu\ comun$
$al\ numerelor\ x\ si\ y.$
Problema 8 [568 puncte]
$Numerele\ x\ si\ y\ sunt\ direct\ proportionale\ cu\ 3\ si\ 4,\ iar\ produsul\ lor\ este\ 3468.$
$Calculati\ x^{2}+y^{2}.$