MathDB

3

Part of 2015 District Olympiad

Problems(8)

Romanian District Olympiad 2015, Grade V, Problem 3

Source:

9/25/2018
Consider the following sequence of sets: {1,2},{3,4,5},{6,7,8,9},... \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,...
a) Find the samllest element of the 100-th 100\text{-th} term. b) Is 2015 2015 the largest element of one of these sets?
SequenceSetsromania
Romanian District Olympiad 2015, Grade VI, Problem 3

Source:

9/25/2018
Determine the perfect squares aabcd‾ \overline{aabcd} of five digits such that dcbaa‾ \overline{dcbaa} is a perfect square of five digits.
arithmeticbase 10 arithmeticromania
Romanian District Olympiad 2015, Grade VII, Problem 3

Source:

9/25/2018
On the segment AC AC of the triangle ABC, ABC, let M M be the midpoint of it, and let N N a point on AM, AM, distinct from A A and M. M. The parallel through N N with respect to AB AB intersects BM BM on P, P, the parallel through M M with respect to BC BC intersects BN BN on Q, Q, and the parallel through N N with respect to AQ AQ intersects BC BC on S. S. Prove that PS PS and AC AC are parallel.
geometry
number of solutions of a diophantine

Source: Romanian District Olympiad 2015, Grade VIII, Problem 3

9/25/2018
Find #{(x,y)∈N2∣1x−1y=12016}, \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , where #A \# A is the cardinal of A. A .
Diophantine equationalgebra
complex equation

Source: Romanian District Olympiad 2015, Grade X, Problem 3

9/25/2018
Solve in C \mathbb{C} the following equation: ∣z∣+∣z−5i∣=∣z−2i∣+∣z−3i∣. |z|+|z-5i|=|z-2i|+|z-3i|.
equationcomplex numbersalgebra
number theory: existence of some numbers

Source: Romanian District Olympiad 2015, Grade IX, Problem 3

9/25/2018
Let m,n m, n natural numbers with m≥2,n≥3. m\ge 2,n\ge 3. Prove that there exist m m distinct multiples of n−1, n-1, namely, a1,a2,a3,...,am, a_1,a_2,a_3,...,a_m, such that: 1n=∑i=1m(−1)i−1ai. \frac{1}{n} =\sum_{i=1}^m \frac{(-1)^{i-1}}{a_i} .
number theorycontests
matrices problem

Source: Romanian District Olympiad 2015, Grade XI, Problem 3

9/26/2018
Find all natural numbers k≥1 k\ge 1 and n≥2, n\ge 2, which have the property that there exist two matrices A,B∈Mn(Z) A,B\in M_n\left(\mathbb{Z}\right) such that A3=On A^3=O_n and AkB+BA=In. A^kB +BA=I_n.
Liniar algebraMatriceslinear algebra
determine all functions...

Source: Romanian District Olympiad 2015, Grade XII, Problem 3

9/26/2018
Find all continuous and nondecreasing functions f:[0,∞)⟶R f:[0,\infty)\longrightarrow\mathbb{R} that satisfy the inequality: \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt, \forall x,y\in [0,\infty) .
functioncontinuitymonotone functionsFind all functionsreal analysis