MathDB

2

Part of 2003 Romania National Olympiad

Problems(6)

Triangle

Source: RMO 2003, Grade 7, Problem 2

10/23/2008
Compute the maximum area of a triangle having a median of length 1 and a median of length 2.
geometry
Meeting

Source: RMO 2003, Grade 8, Problem 2

10/23/2008
In a meeting there are 6 participants. It is known that among them there are seven pairs of friends and in any group of three persons there are at least two friends. Prove that: (a) there exists a person who has at least three friends; (b) there exists three persons who are friends with each other. Valentin Vornicu
pigeonhole principle
Friendly numbers

Source: RMO 2003, Grade 9, Problem 2

10/23/2008
An integer n n, n≥2 n\ge2 is called friendly if there exists a family A1,A2,…,An A_1,A_2,\ldots,A_n of subsets of the set {1,2,…,n} \{1,2,\ldots,n\} such that: (1) i∉Ai i\not\in A_i for every i\equal{}\overline{1,n}; (2) i∈Aj i\in A_j if and only if j∉Ai j\not\in A_i, for every distinct i,j∈{1,2,…,n} i,j\in\{1,2,\ldots,n\}; (3) Ai∩Aj A_i\cap A_j is non-empty, for every i,j∈{1,2,…,n} i,j\in\{1,2,\ldots,n\}. Prove that: (a) 7 is a friendly number; (b) n n is friendly if and only if n≥7 n\ge7. Valentin Vornicu
modular arithmetic
A fatal linear algebra problem by M. Andronache

Source: RomNO 2003, grade xi, p.2

8/27/2019
Let be eight real numbers 1≤a1<a2<a3<a4,x1<x2<x3<x4. 1\le a_1< a_2< a_3< a_4,x_1<x_2<x_3<x_4. Prove that ∣a1x1a1x2a1x3a1x4a2x1a2x2a2x3a2x4a3x1a3x2a3x3a3x4a4x1a4x2a4x3a4x4∣>0. \begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} \\ a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} \\ a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} \\ a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \\ \end{vmatrix} >0.
Marian Andronache, Ion Savu
Matricesandronachelinear algebraalgebra
Complex representation of a pentagon

Source: Romanian National Olympiad 2003, grade x, p.2

8/27/2019
Let be five nonzero complex numbers having the same absolute value and such that zero is equal to their sum, which is equal to the sum of their squares. Prove that the affixes of these numbers in the complex plane form a regular pentagon.
Daniel Jinga
complex numbersabsolute valuealgebrageometrypentagon
Find all count. functions (Titu Andreescu)

Source: RomNO 2003, grade xii, p.2

8/27/2019
Let be an odd natural number n≥3. n\ge 3. Find all continuous functions f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} that satisfy the following equalities. \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n, \forall k\in\{ 1,2,\ldots ,n-1\}
Titu Andreescu
functionreal analysisintegrationDefinite integralTitu Andreescu