MathDB

2

Part of 2011 Morocco National Olympiad

Problems(7)

Equation of 5th degree

Source:

4/24/2011
Solve in R\mathbb{R} the equation : (x+1)5+(x+1)4(x−1)+(x+1)3(x−1)2+(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 + (x+1)2(x−1)3+(x+1)(x−1)4+(x−1)5= (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 = 0 0.
algebra unsolvedalgebra
Sum with - and +

Source:

4/24/2011
Compute the sum S=1+2+3−4−5+6+7+8−9−10+⋯−2010S=1+2+3-4-5+6+7+8-9-10+\dots-2010 where every three consecutive ++ are followed by two −-.
Four distinct real solutions for a quadratic equation

Source: Morocco 2011

4/24/2011
Prove that the equation x2+p∣x∣=qx−1x^{2}+p|x| = qx - 1 has 4 distinct real solutions if and only if p+∣q∣+2<0p+|q|+2<0 (pp and qq are two real parameters).
quadraticsalgebra unsolvedalgebra
Arithmetic mean of integers of a set

Source:

4/24/2011
One integer was removed from the set S={1,2,3,...,n}S=\left \{ 1,2,3,...,n \right \} of the integers from 11 to nn. The arithmetic mean of the other integers of SS is equal to 1634\frac{163}{4}. What integer was removed ?
Inequality with cot alpha, cot beta, cot gamma

Source: Moroccan MO 11th grade 5th exam 2011

4/24/2011
Let α,β,γ\alpha , \beta ,\gamma be the angles of a triangle ABCABC of perimeter 2p 2p and RR is the radius of its circumscribed circle. (a)(a) Prove that cot⁡2α+cot⁡2β+cot⁡2γ≥3(9⋅R2p2−1).\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right). (b)(b) When do we have equality?
inequalitiesgeometryperimetertrigonometryinequalities proposed
Moroccan MO

Source:

4/30/2011
Solve in (R+∗)4(\mathbb{R}_{+}^{*})^{4} the following system : {x+y+z+t=41x+1y+1z+1t=5−1xyzt\left\{\begin{matrix} x+y+z+t=4\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt} \end{matrix}\right.
linear algebramatrixinequalitiesalgebrapolynomialalgebra unsolved
Moroccan MO

Source:

4/30/2011
Let a,b,ca,b,c be three postive real numbers such that a+b+c=1a+b+c=1. Prove that 9abc≤ab+ac+bc<1/4+3abc9abc\leq ab+ac+bc < 1/4 +3abc.
inequalitiesinequalities unsolved