MathDB

4

Part of 2002 Moldova National Olympiad

Problems(10)

Soccer tournament

Source: Moldova NMO 2002 grade 7 problem nr.4

10/31/2008
Twelve teams participated in a soccer tournament. According to the rules, one team gets 2 2 points for a victory, 1 1 point for a draw and 0 0 points for a defeat. When the tournament was over, all teams had distinct numbers of points, and the team ranked second had as many points as the teams ranked on the last five places in total. Who won the match between the fourth and the eighth place teams?
combinatorics
Infinitely many solutions (a,b,c)

Source: Moldova NMO 2002 grade 7 problem nr.8

10/31/2008
Prove that there are infinitely many triplets (a,b,c) (a,b,c) that satisfy the following equalities: \dfrac{2a\minus{}b\plus{}6}{4a\plus{}c\plus{}2}\equal{}\dfrac{b\minus{}2c}{a\minus{}c}\equal{}\dfrac{2a\plus{}b\plus{}2c\minus{}2}{6a\plus{}2c\minus{}2}
Maximum number of phone numbers in a company

Source: Moldova NMO 2002 grade 8 problem nr.4

11/2/2008
All the internal phone numbers in a certain company have four digits. The director wants the phone numbers of the administration offices to consist of digits 1 1, 2 2, 3 3 only, and that any of these phone numbers coincide in at most one position. What is the maximum number of distinct phone numbers that these offices can have ?
Minimum and maximum value of a sum

Source: Moldova NMO 2002 grade 8 problem nr.8

11/2/2008
Let x∈R x\in \mathbb R. Find the minimum and maximum values of the expresion: E\equal{}\dfrac{(1\plus{}x)^8\plus{}16x^4}{(1\plus{}x^2)^4}
trigonometry
s[ABP]+s[CDQ]=s[MNPQ]

Source: Moldova NMO 2002 grade 9 problem nr.4

11/2/2008
Let ABCD ABCD be a convex quadrilateral and let N N on side AD AD and M M on side BC BC be points such that \dfrac{AN}{ND}\equal{}\dfrac{BM}{MC}. The lines AM AM and BN BN intersect at P P, while the lines CN CN and DM DM intersect at Q Q. Prove that if S_{ABP}\plus{}S_{CDQ}\equal{}S_{MNPQ}, then either AD∥BC AD\parallel BC or N N is the midpoint of DA DA.
Circles and tangents

Source: Moldova NMO 2002 grade 9 problem nr.8

11/3/2008
The circles C1 C_1 and C2 C_2 with centers O1 O_1 and O2 O_2 respectively are externally tangent. Their common tangent not intersecting the segment O1O2 O_1O_2 touches C1 C_1 at A A and C2 C_2 at B B. Let C C be the reflection of A A in O1O2 O_1O_2 and P P be the intersection of AC AC and O1O2 O_1O_2. Line BP BP meets C2 C_2 again at L L. Prove that line CL CL is tangent to the circle C2 C_2.
geometrygeometric transformationreflectionsymmetry
Table 2n+1 x 2n+1

Source: Moldova NMO 2002 grade 10 problem nr.4

11/3/2008
In each line and column of a table (2n \plus{} 1)\times (2n \plus{} 1) are written arbitrarly the numbers 1,2,...,2n \plus{} 1. It was constated that the repartition of the numbers is symmetric to the main diagonal of this table. Prove that all the numbers on the main diagonal are distinct.
Circumcircles with a common point

Source: Moldova NMO 2002 grade 10 problem nr.8

11/3/2008
Let the triangle ADB1 ADB_1 s.t. m(∠DAB1)≠90∘ m(\angle DAB_1)\ne 90^\circ.On the sides of this triangle externally are constructed the squares ABCD ABCD and AB1C1D1 AB_1C_1D_1 with centers O1 O_1 and O2 O_2, respectively.Prove that the circumcircles of the triangles BAB1 BAB_1, DAD1 DAD_1 and O1AO2 O_1AO_2 share a common point, that differs from A A.
geometrycircumcircle
Fairly easy inequality

Source: Moldova NMO 2002 grade 11 problem nr.4

11/3/2008
At least two of the nonnegative real numbers a1,a2,...,an a_1,a_2,...,a_n aer nonzero. Decide whether a a or b b is larger if a\equal{}\sqrt[2002]{a_1^{2002}\plus{}a_2^{2002}\plus{}\ldots\plus{}a_n^{2002}} and b\equal{}\sqrt[2003]{a_1^{2003}\plus{}a_2^{2003}\plus{}\ldots\plus{}a_n^{2003} }
inequalitiesAMCUSA(J)MOUSAMO
Inequality in a tetrahedron

Source: Moldova NMO 2002 grade 11 problem nr.8

11/3/2008
The circumradius of a tetrahedron ABCD ABCD is R R, and the lenghts of the segments connecting the vertices A,B,C,D A,B,C,D with the centroids of the opposite faces are equal to ma,mb,mc m_a,m_b,m_c and md m_d, respectively. Prove that: m_a\plus{}m_b\plus{}m_c\plus{}m_d\leq \dfrac{16}{3}R
inequalitiesgeometry3D geometrytetrahedroncircumcircle