Revista (format .pdf, 1.2 MB) - Recreaţii Matematice

L149. Determine the position of the point P on the directrix line of the parabolaP, so that the area of the triangle PT 1 T 2 be minimum, where T 1 and T 2 are thecontact points with P of the tangents drawn from P to P.Adrian Corduneanu, IaşiL150. Let us consider the tetrahedron A 1 A 2 A 3 A 4 , and a point P inside it.Denote by A ij ∈ (A i A j ) the orthogonal projections of P on the edge(s) A i A j of thetetrahedron. Prove thatV PA12A 13A 23+ V PA12A 14A 24+ V PA13A 14A 34+ V PA23A 24A 34≤ 1 4 V A 1A 2A 3A 4.When the equality is attained?Marius Olteanu, Rm. Vâlceah ¡2+ √ ¢ 2n+1iL151. Prove than no natural numbers n and k exist such that 3 =h ¡4+ √ ¢ ki15 .Cosmin Manea and Dragoş Petrică, PiteştiL152. For a, b, c ∈ R and x ∈ R + , prove the inequality9a 2 + b 2 + c 2 ≤ 3(x +1) 2 (a + b + c) 4h3(x 2 +1)(a 2 + b 2 + c 2 )+2x(a + b + c) 2i ≤ 1 (ab + bc + ca) 2 a 2 + 1 b 2 + 1 c 2 .I. V. Maftei and Dorel Băiţan, BucureştiL153. Find all functions f : R → R with the property thatf ¡ x 2 + xy + yf (y) ¢ = xf (x + y)+f 2 (y) , ∀x, y ∈ R.Adrian Zahariuc, student, PrincetonL154. Let P ∈ R [X] a polynomial of degree n and p : R → R its associatedpolynomial function. Knowing that the set {x ∈ 6 R | p (x) =0} consists of k (distinct)elements, and the function f : R → R,f(x) =| p (x)| is differentiable h on R, shownithat the maximum number of nonzero complex roots of P equals 2 − 2k.2Vlad Emanuel, student, BucureştiL155. Let A, B ∈ M 2 (C) be two matrices such that the matrix AB − BA isinvertible. Show that the trace of the matrix (I 2 + AB)(AB − BA) −1 is equal to 1.Florentina Cârlan and Marian Tetiva, Bârlad180