MathDB

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Part of 2003 District Olympiad

Problems(6)

disjoint subests of {1,2,..., 10} 2003 Romania District VII p1

Source:

8/15/2024
Find the disjoint sets BB and CC such that B∪C={1,2,...,10}B \cup C = \{1,2,..., 10\} and the product of the elements of CC equals the sum of elements of BB.
algebranumber theory
angle between planes, starting with an equilateral

Source: 2003 Romania District VIII P2

5/24/2020
Let ABCABC be an equilateral triangle. On the plane (ABC)(ABC) rise the perpendiculars AA′AA' and BB′BB' on the same side of the plane, so that AA′=ABAA' = AB and BB′=12ABBB' =\frac12 AB. Determine the measure the angle between the planes (ABC)(ABC) and (A′B′C′)(A'B'C').
3D geometrygeometryanglesplanesEquilateral
Natural functions

Source: RMO 2003, District Round

5/29/2006
Find all functions f:N∗→N∗\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast (N∗={1,2,3,…}\displaystyle N^\ast = \{ 1,2,3,\ldots \}) with the property that, for all n≥1\displaystyle n \geq 1, f(1)+f(2)+…+f(n) f(1) + f(2) + \ldots + f(n) is a perfect cube ≤n3\leq n^3. Dinu Teodorescu
functiongeometry3D geometryalgebra proposedalgebra
Cube

Source: RMO 2003, District Round

5/29/2006
In the interior of a cube we consider 2003\displaystyle 2003 points. Prove that one can divide the cube in more than 20033\displaystyle 2003^3 cubes such that any point lies in the interior of one of the small cubes and not on the faces.
geometry3D geometryanalytic geometryleast common multiplealgebra proposedalgebra
Romania District Olympiad 2003 - Grade XI

Source:

3/18/2011
In the xOyxOy system, consider the collinear points Ai(xi,yi), 1≤i≤4A_i(x_i,y_i),\ 1\le i\le 4, such that there are invertible matrices M∈M4(C)M\in \mathcal{M}_4(\mathbb{C}) such that (x1,x2,x3,x4)(x_1,x_2,x_3,x_4) and (y1,y2,y3,y4)(y_1,y_2,y_3,y_4) are their first two lines. Prove that the sum of the entries of M−1M^{-1} doesn't depend of MM.
Marian Andronache
linear algebralinear algebra unsolved
groups * matrices

Source: RMO 2003, District Round

6/21/2006
Let (G,⋅)(G,\cdot) be a finite group with the identity element, ee. The smallest positive integer nn with the property that xn=ex^{n}= e, for all x∈Gx \in G, is called the exponent of GG. (a) For all primes p≥3p \geq 3, prove that the multiplicative group Gp\mathcal G_{p} of the matrices of the form (1^a^b^0^1^c^0^0^1^)\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}, with \hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z, is not commutative and has exponent pp. (b) Prove that if (G,∘)\left( G, \circ \right) and (H,∙)\left( H, \bullet \right) are finite groups of exponents mm and nn, respectively, then the group (G×H,⊙)\left( G \times H, \odot \right) with the operation given by (g,h)⊙(g′,h′)=(g∘g′,h∙h′)(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right), for all (g,h), (g′,h′)∈G×H\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H, has the exponent equal to lcm(m,n)\textrm{lcm}(m,n). (c) Prove that any n≥3n \geq 3 is the exponent of a finite, non-commutative group. Ion Savu
linear algebramatrixnumber theoryleast common multipleinductionabstract algebrasuperior algebra