MathDB

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Part of 2006 District Olympiad

Problems(6)

Square root of 11....44....4

Source: Romanian District Olympiad 2006, Grade 7, Problem 1

3/11/2006
Prove that for all positive integers nn, n>1n>1 the number 11…44…4‾\sqrt{ \overline{ 11\ldots 44 \ldots 4 }}, where 1 appears nn times, and 4 appears 2n2n times, is irrational.
modular arithmetic
Another solid geometry

Source: Romanian District Olympiad 2006, Grade 8, Problem 1

3/11/2006
On the plane of triangle ABCABC with ∠BAC=90∘\angle BAC = 90^\circ we raise perpendicular lines in AA and BB, on the same side of the plane. On these two perpendicular lines we consider the points MM and NN respectively such that BN<AMBN < AM. Knowing that AC=2aAC = 2a, AB=a3AB = a\sqrt 3, AM=aAM=a and that the plane MNCMNC makes an angle of 30∘30^\circ with the plane ABCABC find a) the area of the triangle MNCMNC; b) the distance from BB to the plane MNCMNC.
geometrytrigonometryvector
Yet another classical inequality

Source: Romanian District Olympiad 2006, Grade 9, Problem 1

3/11/2006
Let x,y,zx,y,z be positive real numbers. Prove the following inequality: 1x2+yz+1y2+zx+1z2+xy≤12(1xy+1yz+1zx). \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right).
inequalitiesfunctionalgebra
Exponential inequality

Source: Romanian District Olympiad 2006, Grade 10, Problem 1

3/11/2006
Let a,b,c∈(0,1) a,b,c\in (0,1) and x,y,z\in (0, \plus{} \infty) be six real numbers such that a^x \equal{} bc ,   b^y \equal{} ca ,   c^z \equal{} ab . Prove that \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 . Cezar Lupu
inequalitieslogarithmsinequalities proposed
det (A^2+A+xI_2) = x

Source: Romanian District Olympiad 2006, Grade 11, Problem 1

3/11/2006
Let x>0x>0 be a real number and AA a square 2×22\times 2 matrix with real entries such that det⁡(A2+xI2)=0\det {(A^2+xI_2 )} = 0. Prove that det⁡(A2+A+xI2)=x\det{ (A^2+A+xI_2) } = x.
linear algebramatrixalgebrapolynomiallinear algebra unsolved
Inequality between product of finite integrals

Source: Romanian District Olympiad 2006, Grade 12, Problem 1

3/11/2006
Let f1,f2,…,fn:[0,1]→(0,∞)f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty) be nn continuous functions, n≥1n\geq 1, and let σ\sigma be a permutation of the set {1,2,…,n}\{1,2,\ldots, n\}. Prove that ∏i=1n∫01fi2(x)fσ(i)(x)dx≥∏i=1n∫01fi(x)dx. \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx.
inequalitiescalculusintegrationfunctionreal analysisreal analysis unsolved