MathDB

4

Part of 2001 District Olympiad

Problems(6)

Romania District Olympiad 2001 - VII Grade

Source:

3/12/2011
Consider a convex qudrilateral ABCDABCD and M∈(AB), N∈(CD)M\in (AB),\ N\in (CD) such that AMBM=DNCN=k\frac{AM}{BM}=\frac{DN}{CN}=k. Prove that BC∥ADBC\parallel AD if and only if
MN=1k+1AD+kk+1BCMN=\frac{1}{k+1} AD+\frac{k}{k+1} BC
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ratiogeometry proposedgeometry
Romania District Olympiad 2001 - Grade VIII

Source:

3/12/2011
Consider a rectangular parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' in which we denote AB=a, BC=b, AA′=cAB=a,\ BC=b,\ AA'=c. Let DE⊥AC, DF⊥A′C, E∈AC, F∈A′CDE\perp AC,\ DF\perp A'C,\ E\in AC,\ F \in A'C and C′P⊥B′D′, C′Q⊥BD′, P∈B′D′, Q∈BD′C'P\perp B'D',\ C'Q\perp BD',\ P\in B'D',\ Q\in BD'. Prove that the planes (DEF)(DEF) and (C′PQ)(C'PQ) are perpendicular if and only if a2+c2=b2a^2+c^2=b^2.
Sorin Peligrad
geometry proposedgeometry
Romania District Olympiad 2001 - Grade X

Source:

3/16/2011
Solve the equation:
2lg⁡x+8=(x−8)1lg⁡22^{\lg x}+8=(x-8)^{\frac{1}{\lg 2}}
Note: lg⁡x=log⁡10x\lg x=\log_{10}x.
Daniel Jinga
logarithmsalgebra proposedalgebra
Romania District Olympiad 2001 - Grade IX

Source:

3/12/2011
Consider a function f:Z→Zf:\mathbb{Z}\to \mathbb{Z} such that:
f(m2+f(n))=f2(m)+n, ∀m,n∈Zf(m^2+f(n))=f^2(m)+n,\ \forall m,n\in \mathbb{Z}
Prove that:
a)f(0)=0f(0)=0; b)f(1)=1f(1)=1; c)f(n)=n, ∀n∈Zf(n)=n,\ \forall n\in \mathbb{Z}
Lucian Dragomir
functioninductionalgebra proposedalgebra
Romania District Olympiad 2001 - Grade XI

Source:

3/16/2011
Prove that:
a) the sequence an=1n+1+1n+2+…+1n+n, n≥1a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1 is monotonic.
b) there is a sequence (an)n≥1∈{0,1}(a_n)_{n\ge 1}\in \{0,1\} such that:
lim⁡n→∞(a1n+1+a2n+2+…+ann+n)=12\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}
Radu Gologan
limitinductioninequalitiesreal analysisreal analysis unsolved
Romania District Olympiad 2001 - Grade XII

Source:

3/16/2011
a)Prove that ln⁡(1+x)≤x, (∀)x≥0\ln(1+x)\le x,\ (\forall)x\ge 0.
b)Let a>0a>0. Prove that:
lim⁡n→∞n∫01xna+xndx=ln⁡a+1a\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}
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limitintegrationlogarithmscalculusreal analysisreal analysis unsolved