MathDB

3

Part of 2002 Romania National Olympiad

Problems(6)

Set of interior points P on two bisecting lines

Source: Romanian MO 2002

12/8/2010
Let ABCDABCD be a trapezium and ABAB and CDCD be it's parallel edges. Find, with proof, the set of interior points PP of the trapezium which have the property that PP belongs to at least two lines each intersecting the segments ABAB and CDCD and each dividing the trapezium in two other trapezoids with equal areas.
geometrytrapezoidgeometry proposed
Power sums of a,b,c,d,e in [-2,2]

Source: Romanian MO 2002

12/8/2010
Find all real numbers a,b,c,d,ea,b,c,d,e in the interval [−2,2][-2,2], that satisfy: \begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 \end{align*}
trigonometryalgebra proposedalgebra
Considering the planes of the frustum ABCDEF

Source: Romanian MO 2002

12/8/2010
Let [ABCDEF][ABCDEF] be a frustum of a regular pyramid. Let GG and G′G' be the centroids of bases ABCABC and DEFDEF respectively. It is known that AB=36,DE=12AB=36,DE=12 and GG′=35GG'=35. a)a) Prove that the planes (ABF),(BCD),(CAE)(ABF),(BCD),(CAE) have a common point PP, and the planes (DEC),(EFA),(FDB)(DEC),(EFA),(FDB) have a common point P′P', both situated on GG′GG'. b)b) Find the length of the segment [PP′][PP'].
geometry3D geometryfrustumgeometry proposed
x^n-y^n-2^k has no positive integer solutions

Source: Romanian MO 2002

12/8/2010
Let kk and nn be positive integers with n>2n>2. Show that the equation: xn−yn=2kx^n-y^n=2^k has no positive integer solutions.
modular arithmeticnumber theory proposednumber theory
Prove the two invertible matrices U,V exist

Source:

12/8/2010
Let A∈M4(C)A\in M_4(C) be a non-zero matrix. a)a) If rank(A)=r<4\text{rank}(A)=r<4, prove the existence of two invertible matrices U,V∈M4(C)U,V\in M_4(C), such that: UAV=(Ir000)UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix} where IrI_r is the rr-unit matrix. b)b) Show that if AA and A2A^2 have the same rank kk, then the matrix AnA^n has rank kk, for any n≥3n\ge 3.
linear algebramatrixvectorinductionlinear algebra unsolved
f must be a constant function

Source:

12/8/2010
Let f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} be a continuous and bounded function such that x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t, \text{for any}\ x\in\mathbb{R}. Prove that ff is a constant function.
functionintegrationreal analysisreal analysis unsolved