MathDB

4

Part of 2003 Romania National Olympiad

Problems(6)

Function

Source: RMO 2003, Grade 9, Problem 4

10/23/2008
Let P P be a plane. Prove that there exists no function f:P→P f: P\rightarrow P such that for every convex quadrilateral ABCD ABCD, the points f(A),f(B),f(C),f(D) f(A),f(B),f(C),f(D) are the vertices of a concave quadrilateral. Dinu Şerbănescu
functioncombinatorial geometry
Triangle

Source: RMO 2003, Grade 7, Problem 4

10/23/2008
In triangle ABC ABC, P P is the midpoint of side BC BC. Let M∈(AB) M\in(AB), N∈(AC) N\in(AC) be such that MN∥BC MN\parallel BC and {Q} \{Q\} be the common point of MP MP and BN BN. The perpendicular from Q Q on AC AC intersects AC AC in R R and the parallel from B B to AC AC in T T. Prove that: (a) TP∥MR TP\parallel MR; (b) \angle MRQ\equal{}\angle PRQ. Mircea Fianu
geometryparallelogram
Tetrahedron

Source: RMO 2003, Grade 8, Problem 4

10/23/2008
In tetrahedron ABCD ABCD, G1,G2 G_1,G_2 and G3 G_3 are barycenters of the faces ACD,ABD ACD,ABD and BCD BCD respectively. (a) Prove that the straight lines BG1,CG2 BG_1,CG_2 and AG3 AG_3 are concurrent. (b) Knowing that AG_3\equal{}8,BG_1\equal{}12 and CG_2\equal{}20 compute the maximum possible value of the volume of ABCD ABCD.
geometry3D geometrytetrahedron
Char. of Z as being ident. as a countab. reun. of fin. reun. of fin. cyc. subgr.

Source: Romanian National Olympiad 2003, grade x, p.4

8/27/2019
a) Prove that the sum of all the elements of a finite union of sets of elements of finite cyclic subgroups of the group of complex numbers, is an integer number. b) Show that there are finite union of sets of elements of finite cyclic subgroups of the group of complex numbers such that the sum of all its elements is equal to any given integer.
Paltin Ionescu
complex numbersgroup theoryalgebracyclic group
3x3 adjugate matrix

Source: RomNO 2003, grade xi, p.4

8/27/2019
Let be a 3×3 3\times 3 real matrix A. A. Prove the following statements. a) f(A)≠O3, f(A)\neq O_3, for any polynomials f∈R[X] f\in\mathbb{R} [X] whose roots are not real. b) \exists n\in\mathbb{N}  \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0
Laurențiu Panaitopol
linear algebramatrixalgebrapolynomialadjugate
Too complicated: nr. of bin. oper. & sequences of groups

Source: RomNO 2003, grade xii, p.4

8/27/2019
i(L) i(L) denotes the number of multiplicative binary operations over the set of elements of the finite additive group L L such that the set of elements of L, L, along with these additive and multiplicative operations, form a ring. Prove that
a) i(Z12)=4. i\left( \mathbb{Z}_{12} \right) =4. b) i(A×B)≥i(A)i(B), i(A\times B)\ge i(A)i(B) , for any two finite commutative groups B B and A. A. c) there exist two sequences (Gk)k≥1,(Hk)k≥1 \left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1} of finite commutative groups such that lim⁡k→∞#Gki(Gk)=0 \lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0 and lim⁡k→∞#Hki(Hk)=∞. \lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty.
Barbu Berceanu
abstract algebragroup theoryRing Theorylimits