MathDB

3

Part of 2011 Morocco National Olympiad

Problems(8)

System

Source:

4/24/2011
Solve in R3\mathbb{R}^{3} the following system {x2−y=z−1y2−z=x−1z2−x=y−1\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.
linear algebramatrixalgebra unsolvedalgebra
Max and min

Source: Morocco 2011

4/24/2011
Let aa and bb be two real numbers and letM(a,b)=max⁡{3a2+2b;3b2+2a}M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}. Find the values of aa and bb for which M(a,b)M(a,b) is minimal.
inequalities unsolvedinequalities
Functional equation

Source: Morocco 2011

4/24/2011
Find all functions f:R→Rf : \mathbb{R} \to \mathbb{R} which verify the relation (x−2)f(y)+f(y+2f(x))=f(x+yf(x)),∀x,y∈R.(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.
functionsearchalgebra unsolvedalgebra
A mistake in dividing

Source:

4/24/2011
When dividing an integer mm by a positive integer nn, (0<n≤100)(0< n\leq 100), a student finds mn=0,167a1a2...\frac{m}{n}= 0,167a_{1}a_{2}.... Prove that the student made a mistake while computing.
number theory unsolvednumber theory
Singapore TST 2009 Day 1 Problem 1

Source:

8/2/2010
Two circles are tangent to each other internally at a point  T\ T . Let the chord  AB\ AB of the larger circle be tangent to the smaller circle at a point  P\ P . Prove that the line  TP\ TP bisects  ∠ATB\ \angle ATB .
geometrygeometric transformationhomothetysearchfunctioncircumcirclepower of a point
Functional equation Moroccan MO

Source:

4/30/2011
Find all functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} such that ∀x∈R  f(x)=max(2xy−f(y))\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y)) where y∈Ry\in \mathbb{R}.
functionalgebra unsolvedalgebra
Functional equation Moroccan MO

Source:

4/30/2011
Find all functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} such that for all x,y,∈Rx,y, \in \mathbb{R}, xf(x+xy)=xf(x)+f(x2)⋅f(y).xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).
functionalgebra unsolvedalgebra
Numbers and digits

Source:

4/28/2011
Problem 3 (MAR CP 1992) : From the digits 1,2,...,91,2,...,9, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : 123456789123456789, 123456798123456798, ..., 987654321987654321. What is the 100000th100000th number ?