MathDB

1

Part of 2007 Romania National Olympiad

Problems(6)

matrices that satisfty A^2+B^2=AB

Source: romanian nmo 2007, grade 11, problem 1

4/15/2007
Let A,B∈M2(R)A,B\in\mathcal{M}_{2}(\mathbb{R}) (real 2×22\times 2 matrices), that satisfy A2+B2=ABA^{2}+B^{2}=AB. Prove that (AB−BA)2=O2(AB-BA)^{2}=O_{2}.
linear algebramatrixalgebrapolynomialvectorlinear algebra unsolved
integral inequality

Source: romanian nmo 2007, grade 12, problem 1

4/15/2007
Let F\mathcal{F} be the set of functions f:[0,1]→Rf: [0,1]\to\mathbb{R} that are differentiable, with continuous derivative, and f(0)=0f(0)=0, f(1)=1f(1)=1. Find the minimum of ∫011+x2⋅(f′(x))2 dx\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx (where f∈Ff\in\mathcal{F}) and find all functions f∈Ff\in\mathcal{F} for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''.
calculusintegrationinequalitiesfunctionderivativelogarithmstrigonometry
complex equation

Source: romanian nmo 2007, grade 10, problem 1

4/15/2007
Show that the equation zn+z+1=0z^{n}+z+1=0 has a solution with ∣z∣=1|z|=1 if and only if n−2n-2 is divisble by 33.
algebra unsolvedalgebra
Grade IX - Problem I

Source: Romanian National Mathematical Olympiad 2007

4/13/2007
Let a,b,c,d∈N∗a, b, c, d \in \mathbb{N^{*}} such that the equation x2−(a2+b2+c2+d2+1)x+ab+bc+cd+da=0x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
algebra
10^10 as the product of two naturals

Source: Romanian NMO 2007, 8th grade, problem nr. 1

7/9/2008
Prove that the number 1010 10^{10} can't be written as the product of two natural numbers which do not contain the digit "0 0" in their decimal representation.
Proving that ABC is an equilateral triangle

Source: Romanian NMO 2007, 7th grade, problem nr. 1

7/11/2008
In a triangle ABC ABC, where a \equal{} BC, b \equal{} CA and c \equal{} AB, it is known that: a \plus{} b \minus{} c \equal{} 2 and 2ab \minus{} c^2 \equal{} 4. Prove that ABC ABC is an equilateral triangle.