MathDB

4

Part of 2010 Romania National Olympiad

Problems(6)

Find the angles of triangle ABC

Source: Romanian MO 2010 Grade 7

8/6/2012
In the isosceles triangle ABCABC, with AB=ACAB=AC, the angle bisector of ∠B\angle B meets the side ACAC at B′B'. Suppose that BB′+B′A=BCBB'+B'A=BC. Find the angles of the triangle ABCABC.
Dan Nedeianu
geometryangle bisectorgeometry proposed
Prime p does not divide ab-cd

Source: Romanian MO 2010 Grade 8

8/6/2012
Let a,b,c,da,b,c,d be positive integers, and let p=a+b+c+dp=a+b+c+d. Prove that if pp is a prime, then pp is not a divisor of ab−cdab-cd.
Marian Andronache
number theory proposednumber theory
Prove that F has exactly two elements

Source: Romanian MO 2010 Grade 9

8/6/2012
Consider the set F\mathcal{F} of functions f:N→Nf:\mathbb{N}\to\mathbb{N} (where N\mathbb{N} is the set of non-negative integers) having the property that f(a2−b2)=f(a)2−f(b)2, for all a,b∈N, a≥b.f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b. a) Determine the set {f(1)∣f∈F}\{f(1)\mid f\in\mathcal{F}\}. b) Prove that F\mathcal{F} has exactly two elements.
Nelu Chichirim
functioninductionalgebrafunctional equationnumber theory proposednumber theory
ABM, BCN and CAP are similar

Source: Romanian MO 2010 Grade 10

8/6/2012
On the exterior of a non-equilateral triangle ABCABC consider the similar triangles ABM,BCNABM,BCN and CAPCAP, such that the triangle MNPMNP is equilateral. Find the angles of the triangles ABM,BCNABM,BCN and CAPCAP.
Nicolae Bourbacut
searchgeometrysimilar trianglesgeometry proposed
Romania National Olympiad 2010 - Grade XI

Source:

4/10/2011
Let a∈R+a\in \mathbb{R}_+ and define the sequence of real numbers (xn)n(x_n)_n by x1=ax_1=a and xn+1=∣xn−1n∣, n≥1x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1. Prove that the sequence is convergent and find it's limit.
logarithmslimitreal analysisreal analysis unsolved
Sequence is convergent iff f(0)=0

Source: Romanian MO 2010 Grade 12

8/6/2012
Let f:[−1,1]→Rf:[-1,1]\to\mathbb{R} be a continuous function having finite derivative at 00, and I(h)=∫−hhf(x) dx, h∈[0,1].I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1]. Prove that a) there exists M>0M>0 such that ∣I(h)−2f(0)h∣≤Mh2|I(h)-2f(0)h|\le Mh^2, for any h∈[0,1]h\in [0,1]. b) the sequence (an)n≥1(a_n)_{n\ge 1}, defined by an=∑k=1nk∣I(1/k)∣a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|, is convergent if and only if f(0)=0f(0)=0.
Calin Popescu
functioncalculusderivativeintegrationlimitreal analysisreal analysis unsolved