MathDB

1

Part of 2004 Romania National Olympiad

Problems(6)

Rhombus

Source: RMO 2004, Grade 7, Problem 1

2/26/2006
On the sides AB,ADAB,AD of the rhombus ABCDABCD are the points E,FE,F such that AE=DFAE=DF. The lines BC,DEBC,DE intersect at PP and CD,BFCD,BF intersect at QQ. Prove that: (a) PEPD+QFQB=1\frac{PE}{PD} + \frac{QF}{QB} = 1; (b) P,A,QP,A,Q are collinear. Virginia Tica, Vasile Tica
geometryrhombusparallelogramvector
Natural numbers

Source: RMO 2004, Grade 8, Problem 1

2/26/2006
Find all non-negative integers nn such that there are a,b∈Za,b \in \mathbb Z satisfying n2=a+bn^2=a+b and n3=a2+b2n^3=a^2+b^2. Lucian Dragomir
Increasing functions

Source: Romanian MO 2004, 9th grade, Problem 1

2/26/2006
Find the strictly increasing functions f:{1,2,…,10}→{1,2,…,100}f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \} such that x+yx+y divides xf(x)+yf(y)x f(x) + y f(y) for all x,y∈{1,2,…,10}x,y \in \{ 1,2,\ldots,10 \}. Cristinel Mortici
functioninequalitiesalgebra proposedalgebra
Arithmetic progression

Source: RMO 2004, 10th grade, problem 1

3/6/2005
Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a function such that ∣f(x)−f(y)∣≤∣x−y∣|f(x)-f(y)| \leq |x-y|, for all x,y∈Rx,y \in \mathbb{R}. Prove that if for any real xx, the sequence x,f(x),f(f(x)),…x,f(x),f(f(x)),\ldots is an arithmetic progression, then there is a∈Ra \in \mathbb{R} such that f(x)=x+af(x)=x+a, for all x∈Rx \in \mathbb R.
functionarithmetic sequencealgebra unsolvedalgebra
A parabola and a regular polygon

Source: Romanian MO 2004, Final Round, 11th Grade, Problem 1

2/26/2006
Let n≥3n \geq 3 be an integer and FF be the focus of the parabola y2=2pxy^2=2px. A regular polygon A1A2…AnA_1 A_2 \ldots A_n has the center in FF and none of its vertices lie on OxOx. (FA1,(FA2,…,(FAn\left( FA_1 \right., \left( FA_2 \right., \ldots, \left( FA_n \right. intersect the parabola at B1,B2,…,BnB_1,B_2,\ldots,B_n. Prove that FB1+FB2+…+FBn>np. FB_1 + FB_2 + \ldots + FB_n > np . Calin Popescu
conicsparabolageometry unsolvedgeometry
Determine continuous functions

Source: RMO 2004, Grade 12, Problem 1

2/26/2006
Find all continuous functions f:R→Rf : \mathbb R \to \mathbb R such that for all x∈Rx \in \mathbb R and for all n∈N∗n \in \mathbb N^{\ast} we have n2∫xx+1nf(t) dt=nf(x)+12. n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . Mihai Piticari
functionintegrationcalculusderivativelimitalgebrafunctional equation