MathDB

2

Part of 2010 Romania National Olympiad

Problems(6)

SM and CL are parallel

Source: Romanian MO 2010 Grade 7

8/6/2012
Let ABCDABCD be a rectangle of centre OO, such that ∠DAC=60∘\angle DAC=60^{\circ}. The angle bisector of ∠DAC\angle DAC meets DCDC at SS. Lines OSOS and ADAD meet at LL, and lines BLBL and ACAC meet at MM. Prove that lines SMSM and CLCL are parallel.
geometryrectangleratioangle bisectorgeometry proposed
a+b=c+d and a^2+b^2=c^2+d^2

Source: Romanian MO 2010 Grade 8

8/6/2012
How many four digit numbers abcd‾\overline{abcd} simultaneously satisfy the equalities a+b=c+da+b=c+d and a2+b2=c2+d2a^2+b^2=c^2+d^2?
number theory proposednumber theory
Lengths of sides of ABC form an arithmetic progression

Source: Romanian MO 2010 Grade 9

8/6/2012
Prove that there is a similarity between a triangle ABCABC and the triangle having as sides the medians of the triangle ABCABC if and only if the squares of the lengths of the sides of triangle ABCABC form an arithmetic sequence.
Marian Teler & Marin Ionescu
geometry unsolvedgeometry
Inequality iff w=kv

Source: Romanian MO 2010 Grade 10

8/6/2012
Consider v,wv,w two distinct non-zero complex numbers. Prove that ∣zw+wˉ∣≤∣zv+vˉ∣,|zw+\bar{w}|\le |zv+\bar{v}|, for any z∈C,∣z∣=1z\in\mathbb{C},|z|=1, if and only if there exists k∈[−1,1]k\in [-1,1] such that w=kvw=kv.
Dan Marinescu
inequalitiesalgebra unsolvedalgebra
Romania National Olympiad 2010 - Grade XI

Source:

4/10/2011
Let A,B,C∈Mn(R)A,B,C\in \mathcal{M}_n(\mathbb{R}) such that ABC=OnABC=O_n and rank B=1\text{rank}\ B=1. Prove that AB=OnAB=O_n or BC=OnBC=O_n.
linear algebralinear algebra unsolved
A has property P iff A is a field

Source: Romanian MO 2010 Grade 12

8/6/2012
We say that a ring AA has property (P)(P) if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element. a) If in AA, 1+1=01+1=0, prove that AA has property (P)(P) if and only if AA is a field. b) Give an example of a ring that is not a field, containing at least two elements, and having property (P)(P).
Dan Schwarz
algebrapolynomialRing Theorysuperior algebrasuperior algebra unsolved