MathDB

2

Part of 2009 District Olympiad

Problems(6)

(MB - MS)(NC - NS)<=0 2009 Romania District VII p2

Source:

8/16/2024
Hiven an acute triangle ABCABC, consider the midpoints MM and NN of the sides ABAB and ACAC, respectively. If point SS is variable on side BCBC, prove that (MB−MS)(NC−NS)≤0(MB - MS)(NC - NS) \le 0
geometrygeometric inequality
|a - b| = 2|b -c| = 3|c - d| = 4|d- e| = 5|e - a| 2009 Romania District VIII p2

Source:

8/16/2024
Real numbers a,b,c,d,ea, b, c, d, e, have the property ∣a−b∣=2∣b−c∣=3∣c−d∣=4∣d−e∣=5∣e−a∣.|a - b| = 2|b -c| = 3|c - d| = 4|d- e| = 5|e - a|. Prove they are all equal.
algebrasystem of equations
Arithmetic progression

Source:

9/25/2013
Numbers from 11 to 100100 are written on the board. Is it possible to cross 1010 numbers in such way, that we couldn't select 10 numbers from rest which would form arithmetic progression?
arithmetic sequence
Constrained complex equation of three variables

Source: Romanian District Olympiad 2009, Grade X, Problem 2

10/8/2018
Find the complex numbers z1,z2,z3 z_1,z_2,z_3 of same absolute value having the property that: 1=z1z2z3=z1+z2+z3. 1=z_1z_2z_3=z_1+z_2+z_3.
complex numbersabsolute valuealgebra
Romania District Olympiad 2009 - Grade XI

Source:

4/10/2011
Let n∈N∗n\in \mathbb{N}^* and a matrix A∈Mn(C), A=(aij)1≤i,j≤nA\in \mathcal{M}_n(\mathbb{C}),\ A=(a_{ij})_{1\le i, j\le n} such that:
aij+ajk+aki=0, (∀)i,j,k∈{1,2,…,n}a_{ij}+a_{jk}+a_{ki}=0,\ (\forall)i,j,k\in \{1,2,\ldots,n\}
Prove that rank A≤2\text{rank}\ A\le 2.
linear algebramatrixlinear algebra unsolved
A characterization of rings of idempotent elements

Source: Romanian District Olympiad 2009, Grade XII, Problem 2

10/8/2018
Prove that in an abelian ring A A in which 1≠0, 1\neq 0, every element is idempotent if and only if the number of polynomial functions from A A to A A is equal to the square of the cardinal of A. A.
abstract algebraalgebrapolynomialfunctionsuperior algebra