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

) Show that we cannot find a numbering for the upper basis so that i + a. i 9,∀i ∈ {1, 2,...,2008}.Gabriel Popa and Gheorghe Iurea, IaşiG152. In the isosceles triangle ABC (AB = AC), B 0 , C 0 denote the feet ofthe altitudes from B, respectively C. If AB =2B 0 C 0 , determine the angles of thetriangle.Nela Ciceu, BacăuandTituZvonaru,ComăneştiG153. In the triangle ABC, M is the midpoint of the side [BC], m(\ABC) =30 ◦and m(\ACB) = 105 ◦ . The perpendicular from C on AM cuts AB at Q. Calculatrethe value of the ratio QAQB .Neculai Roman, Mirceşti (Iaşi)G154. Let D be the midpoint of the side [BC] in the equilateral triangle ABCof side length 1, andletP beamovingpointon[CD]. Denote by M and N theprojections of the points B, respectively C on AP Find the area of the geometriclocus described by the segment [MN].Mariu Olteanu, Rm. VâlceaG155. Let C be the circumcircle of the acute-angled triangle 4ABC. DenotebyP the intersection point of the tangents to the circle at B and C, {D} = AP ∩ C,while M and N are the midpoints of the small arc BC, respectively of the big arcBC. Show that the straight lines AM, DN and BC meet at a point.Gabriel Popa, IaşiB. Highschool levelL146. The straight lines d 1 , d 2 ,..., d n+1 , are considered in the plane such thatany two lines are not parallel. We denote by α k = m( d k\,d k+1 ), α k ≤ 90 ◦ , k = 1,n.Asegmentoflength2 is cosidered on d 1 that is projected on d 2 , then the obtainedsegment is projected on d 3 and so on, until a segment of length 1 is obtained ond n+1 . Knowing that tan ¡ min © α i | i = 1,n ª¢ = p √ n4 − 1, determine the angles α k ,k = 1,n.Cristian Săvescu, student, BucureştiL147. A convex polygon with n sides, n ≥ 4, isconsideredsuchthatanypairofdiagonals are not parallel and and any three diagonals do not meet at other pointsexcept the vertices of the polygon. Let us denote by n i the number of intersectionpoints of the diagonals inside the polygon and by n e the number of intersection pointsof the diagonals outside the polygon.a) Show that exactly eight polygons exist such that the inequality n i >n e issatisfied.b) Show that exactly three polygons exist such that n i + n e kn 2 , k ∈ N ∗ .Mihai Haivas, IaşiL148. ApointD is considered on the side (AB) of the triangle ABC such thatAB =4AD. In the same halfplane as point C with respect to the side AB, wetakeapointP such that \PDA ≡ \ACB and PB =2PD. Prove that the quadrilateralABCP is inscriptible, that si it admits a circumscribed circle.Nela Ciceu, BacăuandTituZvonaru,Comăneşti179