MathDB

2

Part of 2015 District Olympiad

Problems(8)

Romanian District Olympiad 2015, Grade VII, Problem 2

Source:

9/25/2018
a) Show that if two non-negative integers p,q p,q satisfy the property that both 2p−q \sqrt{2p-q} and 2p+q \sqrt{2p+q} are non-negative integers, then q q is even.
b) Determine how many natural numbers m m are there such that 2m−4030 \sqrt{2m-4030} and 2m+4030 \sqrt{2m+4030} are both natural.
number theory
Romanian District Olympiad 2015, Grade V, Problem 2

Source:

9/25/2018
At a math contest there were 50 50 participants, where they were given 3 3 problems each to solve. The results have shown that every candidate has solved correctly at least one problem, and that a total of 100 100 problems have been evaluated by the jury as correct. Show that there were, at most, 25 25 winners who got the maximum score.
discrete mathsreal life problemromania
Romanian District Olympiad 2015, Grade VI, Problem 2

Source:

9/25/2018
Let ABC ABC be an obtuse triangle with AB=AC,M AB=AC, M the symmetric point of A A with respect to C, C, and P P the intersection of the line AB AB with the perpendicular bisector of the segment AB‾. \overline{AB} . Knowing that PM PM is perpendicular to BC, BC, show that APM APM is equilateral.
geometryperpendicular bisectorromania
integer part equation

Source: Romanian District Olympiad 2015, Grade IX, Problem 2

9/25/2018
Determine the real numbers a,b, a,b, such that [ax+by]+[bx+ay]=(a+b)\cdot [x+y], \forall x,y\in\mathbb{R} , where [t] [t] is the greatest integer smaller than t. t.
Diophantine equationInteger Partnumber theory
fancy formulation, but easy concept

Source: Romanian District Olympiad 2015, Grade VIII, Problem 2

9/25/2018
For every real number a, a, define the set Aa={n∈{0}∪N∣n2+an∈{0}∪N}. A_a=\left\{ n\in\{ 0\}\cup\mathbb{N}\bigg|\sqrt{n^2+an}\in\{ 0\}\cup\mathbb{N}\right\} .
a) Show the equivalence: #Aa∈N  ⟺  a≠0, \# A_a\in\mathbb{N}\iff a\neq 0, where #B \# B is the cardinal of B. B. b) Determine max⁡A40. \max A_{40} .
algebra
easy system of equations

Source: Romanian District Olympiad 2015, Grade X, Problem 2

9/25/2018
Solve in Z \mathbb{Z} the following system of equations: {5x−log⁡2(y+3)=3y5y−log⁡2(x+3)=3x. \left\{\begin{matrix} 5^x-\log_2 (y+3) = 3^y\\ 5^y -\log_2 (x+3)=3^x\end{matrix}\right. .
algebrasystem of equationsexponential equations
2x2 matrices

Source: Romanian District Olympiad 2015, Grade XI, Problem 2

9/26/2018
Let be two matrices A,B∈M2(R) A,B\in M_2\left(\mathbb{R}\right) that satisfy the equality (A−B)2=O2. \left( A-B\right)^2 =O_2.
a) Show that det⁡(A2−B2)=(det⁡A−det⁡B)2. \det\left( A^2-B^2\right) =\left( \det A -\det B\right)^2. b) Demonstrate that det⁡(AB−BA)=0  ⟺  det⁡A=det⁡B. \det\left( AB-BA\right) =0\iff \det A=\det B.
linear algebraMatrices
limit of sum of integrals

Source: Romanian District Olympiad 2015, Grade XII, Problem 2

9/26/2018
a) Calculate ∫01xsin⁡(πx2)dx. \int_{0}^1 x\sin\left( \pi x^2\right) dx.
b) Calculate lim⁡n→∞1n∑k=0n−1k∫knk+1nsin⁡(πx2)dx. \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx.
Florin Stănescu
integralsreal analysiscalculuscontestsintegration