MathDB

2

Part of 2007 Romania National Olympiad

Problems(6)

there exists an interval such that f(I)=[a,b]

Source: romania nmo 2007, grade 11, problem 2

4/15/2007
Let f:R→Rf: \mathbb{R}\to\mathbb{R} be a continuous function, and a<ba<b be two points in the image of ff (that is, there exists x,yx,y such that f(x)=af(x)=a and f(y)=bf(y)=b). Show that there is an interval II such that f(I)=[a,b]f(I)=[a,b].
functionalgebrapolynomialreal analysisreal analysis unsolved
unique division and limit

Source: romanian nmo 2007, grade 12, problem 2

4/15/2007
Let f:[0,1]→(0,+∞)f: [0,1]\rightarrow(0,+\infty) be a continuous function. a) Show that for any integer n≥1n\geq 1, there is a unique division 0=a0<a1<…<an=10=a_{0}<a_{1}<\ldots<a_{n}=1 such that ∫akak+1f(x) dx=1n∫01f(x) dx\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx holds for all k=0,1,…,n−1k=0,1,\ldots,n-1. b) For each nn, consider the aia_{i} above (that depend on nn) and define bn=a1+a2+…+annb_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}. Show that the sequence (bn)(b_{n}) is convergent and compute it's limit.
integrationcalculusreal analysisreal analysis unsolved
Grade IX - Problem II

Source: Romanian National Mathematical Olympiad 2007

4/14/2007
Let ABCABC be an acute angled triangle and point MM chosen differently from A,B,CA,B,C. Prove that MM is the orthocenter of triangle ABCABC if and only if BCMAMA⃗+CAMBMB⃗+ABMCMC⃗=0⃗\frac{BC}{MA}\vec{MA}+\frac{CA}{MB}\vec{MB}+\frac{AB}{MC}\vec{MC}= \vec{0}
geometrygeometry unsolved
Right-angled triangle

Source: Romanian NMO 2007, 7th grade, problem nr. 2

7/11/2008
Consider the triangle ABC ABC with m(\angle BAC \equal{} 90^\circ) and AC \equal{} 2AB. Let P P and Q Q be the midpoints of AB AB and AC AC,respectively. Let M M and N N be two points found on the side BC BC such that CM \equal{} BN \equal{} x. It is also known that 2S[MNPQ] \equal{} S[ABC]. Determine x x in function of AB AB.
functiongeometrytrapezoid
Grade X - Problem 2

Source: Romanian National Mathematical Olympiad 2007

4/13/2007
Solve the equation 2x2+x+log⁡2x=2x+12^{x^{2}+x}+\log_{2}x = 2^{x+1}
logarithmsalgebra unsolvedalgebra
2007 rooms in a building

Source: RMO, 8th grade, 2

2/22/2008
In a building there are 6018 desks in 2007 rooms, and in every room there is at least one desk. Every room can be cleared dividing the desks in the oher rooms such that in every room is the same number of desks. Find out what methods can be used for dividing the desks initially.