Revista (format .pdf, 2.3 MB) - RECREAÅ¢II MATEMATICE

L132. Pentru a, b, c numere reale pozitive, demonstraţi inegalitateaa1b 2 + 1 c 2‹+b1c 2 + 1 a 2‹+c1a 2 + 1 b 2‹≥ 18a + b + c .XL133. Dacă a, b, c sunt numere reale pozitive cu a + b + c = 3, arătaţi căabab + a + b + 1 − b)9X(a 2ab + a + b ≤ 1.Florin Stănescu, GăeştiTitu Zvonaru, ComăneştiL134. Fie n şi k numere întregi pozitive. Demonstraţi identităţile:a)n−1Xj=1•jk˜=nhnki− k 2hnkihnb)ki+1;nXj=1•kÉnj˜=[ k√ n]Xq=1•nq k˜.Marian Tetiva, BârladL135. Fie A, B ∈ M n (R) şi X un vector nenul din R n astfel încât AX = O şiexistă Y ∈ R n pentru care AY = BX. Notăm cu A j matricea obţinută înlocuindcoloana j a matricei A cu coloana j a matricei B. Arătaţi că nPj=1 detA j = 0.Adrian Reisner, ParisTraining problems for mathematical contestsA. Junior highschool levelG216. The numbers from 1 to 9 are arranged on a square with 3 × 3 cells sothat the product of the numbers on row k or column k be a perfect square, for anyk ∈ {1, 2, 3}. It is possible to set a odd number in the central cell of the square?Marius Mâinea, GăeştiG217. A number of p small squares are drawn on the blackboard. Johnny coloursa square, Ann colours three squares, Johnny coulours five of them, Ann colours sevenones and so on. The child who has not sufficient squares to coulour when his/her turncomes on loses. Determine the number p for which the winner of the game is Johnnyand establish how many small squares would remain for Ann|Pto colour (as a functionof p ) .Gheorghe Iurea, IaşiG218. The real numbers a 1 , a 2 , . . . , a n (n ∈ N, n ≥ 2) are considered. Show thata subset A ⊆ {1, 2, . . . , n} exists with the property that a i | ≥ 1 | a i | .i∈A 4nPi=1Radu Miron, elev, IaşiG219. Let a, b, c be nonzero numbers, a odd and b > c, such that a = 2 b cb − c and(a, b, c) = 1. Show that a b c is a perfect square.Neculai Stanciu, Buzău and Titu Zvonaru, Comăneşti78