MathDB

1

Part of 2003 Romania National Olympiad

Problems(6)

Set

Source: RMO 2003, Grade 7, Problem 1

10/23/2008
Find the maximum number of elements which can be chosen from the set {1,2,3,…,2003} \{1,2,3,\ldots,2003\} such that the sum of any two chosen elements is not divisible by 3.
Representable as sum of two perfect squares

Source: RMO 2003, Grade 8, Problem 1

10/23/2008
Let m,n m,n be positive integers. Prove that the number 5^n\plus{}5^m can be represented as sum of two perfect squares if and only if n\minus{}m is even. Vasile Zidaru
Positive integers

Source: RMO 2003, Grade 9, Problem 1

10/23/2008
Find positive integers a,b a,b if for every x,y∈[a,b] x,y\in[a,b], \frac1x\plus{}\frac1y\in[a,b].
3d geometric inequality

Source: Romanian National Olympiad, grade x, p.1

8/27/2019
Let be a tetahedron OABC OABC with OA⊥OB⊥OC⊥OA. OA\perp OB\perp OC\perp OA. Show that OH≤r(1+3), OH\le r\left( 1+\sqrt 3 \right) , where H H is the orthocenter of ABC ABC and r r is radius of the inscribed spere of OABC. OABC.
Valentin Vornicu
geometry3D geometrytetrahedrongeometric inequalityinequalities
Locus of a point in a certain relation to a rhombus

Source: RomNO 2003, grade xi, p.1

8/27/2019
Find the locus of the points M M that are situated on the plane where a rhombus ABCD ABCD lies, and satisfy: MA⋅MC+MB⋅MD=AB2 MA\cdot MC+MB\cdot MD=AB^2
Ovidiu Pop
geometryrhombusanalytic geometryPure geometry
Ring of matrices over a field is isomorphic to the field

Source: RomNO 2003, grade xii. p. 1

8/27/2019
a) Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field. b) Prove that (Mn(R),+,⋅)≇(Mn(C),+,⋅), \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , for any natural number n≥2. n\ge 2.
Marian Andronache, Ion Sava
linear algebraabstract algebraRing TheoryIsomorphisms