MathDB

1

Part of 2014 District Olympiad

Problems(8)

Find all possible integers

Source: Romanian District Olympiad 2014, Grade 5, P1

6/15/2014
Find with proof all positive 33 digit integers abc‾\overline{abc} satisfying b⋅ac‾=c⋅ab‾+10 b\cdot \overline{ac}=c \cdot \overline{ab} +10
number theory proposednumber theory
An equality and an inequality

Source: Romanian District Olympiad 2014, Grade 6, P1

6/15/2014
Prove that: [*](12)3+(23)3+(56)3=1\displaystyle\left( \frac{1}{2}\right) ^{3}+\left( \frac{2}{3}\right)^{3}+\left( \frac{5}{6}\right) ^{3}=1 [*]333+433+533<6333^{33}+4^{33}+5^{33}<6^{33}
inequalitiesalgebra proposedalgebra
Inequality for real numbers

Source: Romanian District Olympiad 2014, Grade 7, P1

6/15/2014
[*]Prove that for any real numbers aa and bb the following inequality holds: (a2+1)(b2+1)+50≥2(2a+1)(3b+1) \left( a^{2}+1\right) \left( b^{2}+1\right) +50\geq2\left( 2a+1\right)\left( 3b+1\right) [*]Find all positive integers nn and pp such that: (n2+1)(p2+1)+45=2(2n+1)(3p+1) \left( n^{2}+1\right) \left( p^{2}+1\right) +45=2\left( 2n+1\right)\left( 3p+1\right)
inequalitiesinequalities proposed
Find the length

Source: Romanian District Olympiad 2014, Grade 8, P1

6/15/2014
In the right parallelopiped ABCDA′B′C′D′ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}, with AB=123AB=12\sqrt{3} cm and AA′=18AA^{\prime}=18 cm, we consider the points P∈AA′P\in AA^{\prime} and N∈A′B′N\in A^{\prime}B^{\prime} such that A′N=3B′NA^{\prime}N=3B^{\prime}N. Determine the length of the line segment APAP such that for any position of the point M∈BCM\in BC, the triangle MNPMNP is right angled at NN.
analytic geometrygeometry proposedgeometry
Irrational number

Source: Romanian District Olympiad 2014, Grade 9, P1

6/15/2014
Find the x∈R∖Qx\in \mathbb{R}\setminus \mathbb{Q} such that x2+x∈Z and x3+2x2∈Z x^2+x\in \mathbb{Z}\text{ and }x^3+2x^2\in\mathbb{Z}
algebra solvedalgebra
A complex equation

Source: Romanian District Olympiad 2014, Grade 10, P1

6/15/2014
Solve for z∈Cz\in \mathbb{C} the equation : ∣z−∣z+1∣∣=∣z+∣z−1∣∣ |z-|z+1||=|z+|z-1||
conicsellipsecomplex numbersalgebra proposedalgebra
A pair of matrices

Source: Romanian District Olympiad 2014, Grade 11, P1

6/15/2014
[*]Give an example of matrices AA and BB from M2(R)\mathcal{M}_{2}(\mathbb{R}), such that A^{2}+B^{2}=\left( \begin{array} [c]{cc} 2 & 3\\ 3 & 2 \end{array} \right) . [*]Let AA and BB be matrices from M2(R)\mathcal{M}_{2}(\mathbb{R}), such that \displaystyle A^{2}+B^{2}=\left( \begin{array} [c]{cc} 2 & 3\\ 3 & 2 \end{array} \right) . Prove that AB≠BAAB\neq BA.
linear algebra
Proving Riemann-integrability

Source: Romanian District Olympiad 2014, Grade 12, P1

6/15/2014
For each positive integer nn we consider the function fn:[0,n]→Rf_{n}:[0,n]\rightarrow{\mathbb{R}} defined by fn(x)=arctan⁡(⌊x⌋)f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} , where ⌊x⌋\left\lfloor x\right\rfloor denotes the floor of the real number xx. Prove that fnf_{n} is a Riemann Integrable function and find lim⁡n→∞1n∫0nfn(x)dx.\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.
functionfloor functionlimitintegrationreal analysisreal analysis unsolved