MathDB

3

Part of 2003 District Olympiad

Problems(6)

RB coloring of a grid by 2n x 2n lines

Source: 2003 Romania District VII p3

8/15/2024
A grid consists of 2n2n vertical and 2n2n horizontal lines, each group disposed at equal distances. The lines are all painted in red and black, such that exactly nn vertical and nn horizontal lines are red. Find the smallest nn such that for any painting satisfying the above condition, there is a square formed by the intersection of two vertical and two horizontal lines, all of the same colour.
combinatoricsColoring
product of numbers in any row and column is 5 or -5, array n x n

Source: 2003 Romania District VIII p3

8/15/2024
Consider an array n×nn \times n (n≥2n\ge 2) with n2n^2 integers. In how many ways one can complete the array if the product of the numbers on any row and column is 55 or −5-5?
combinatorics
Board

Source: RMO 2003, District Round

4/22/2006
On a board are drawn the points A,B,C,DA,B,C,D. Yetti constructs the points A′,B′,C′,D′A^\prime,B^\prime,C^\prime,D^\prime in the following way: A′A^\prime is the symmetric of AA with respect to BB, B′B^\prime is the symmetric of BB wrt CC, C′C^\prime is the symmetric of CC wrt DD and D′D^\prime is the symmetric of DD wrt AA. Suppose that Armpist erases the points A,B,C,DA,B,C,D. Can Yetti rebuild them? ⋆  ⋆  ⋆\star \, \, \star \, \, \star Note. Any similarity to real persons is purely accidental.
vectoranalytic geometryfunction
Sine inequality

Source: RMO 2003, District Round

5/29/2006
(a) If ABC\displaystyle ABC is a triangle and M\displaystyle M is a point from its plane, then prove that AMsin⁡A≤BMsin⁡B+CMsin⁡C. \displaystyle AM \sin A \leq BM \sin B + CM \sin C . (b) Let A1,B1,C1\displaystyle A_1,B_1,C_1 be points on the sides (BC),(CA),(AB)\displaystyle (BC),(CA),(AB) of the triangle ABC\displaystyle ABC, such that the angles of △A1B1C1\triangle A_1 B_1 C_1 are A1^=α,B1^=β,C1^=γ\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma. Prove that ∑AA1sin⁡α≤∑BCsin⁡α. \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . Dan Ştefan Marinescu, Viorel Cornea
trigonometryinequalitiesgeometry proposedgeometry
Romania District Olympiad 2003 - Grade XI

Source:

3/18/2011
a)Prove that any matrix A∈M4(C)A\in \mathcal{M}_4(\mathbb{C}) can be written as a sum of four matrices B1,B2,B3,B4∈M4(C)B_1,B_2,B_3,B_4\in \mathcal{M}_4(\mathbb{C}) with the rank equal to 11. b)I4I_4 can't be written as a sum of less than four matrices with the rank equal to 11.
Manuela Prajea & Ion Savu
linear algebramatrixlinear algebra unsolved
Finite fields

Source: RMO 2003, District Round

6/3/2006
Let K\displaystyle \mathcal K be a finite field such that the polynomial X2−5\displaystyle X^2-5 is irreducible over K\displaystyle \mathcal K. Prove that: (a) 1+1≠01+1 \neq 0; (b) for all a∈K\displaystyle a \in \mathcal K, the polynomial X5+a\displaystyle X^5+a is reducible over K\displaystyle \mathcal K. Marian Andronache [Edit 1∘1^\circ] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful. [Edit 2∘2^\circ] OK, thanks.
algebrapolynomialquadraticssuperior algebrasuperior algebra unsolved