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Part of 2011 District Olympiad

Problems(6)

N has digits 1-7, not perfect square - 2011 Romania District VII p3

Source:

9/1/2024
A positive integer NN has the digits 1,2,3,4,5,61, 2, 3, 4, 5, 6 and 77, so that each digit ii, i∈{1,2,3,4,5,6,7}i \in \{1, 2, 3, 4, 5, 6, 7\} occurs 4i4i times in the decimal representation of NN. Prove that NN is not a perfect square.
number theoryPerfect SquaresPerfect Square
angle between planes wanted, right triangular prism, S_ {ABE}+S_ {ACF}=S_{AEF}

Source: 2011 Romania District VIII P3

5/19/2020
Let ABCA′B′C′ABCA'B'C' a right triangular prism with the bases equilateral triangles. A plane α\alpha containing point AA intersects the rays BB′BB' and CC′CC' at points E and FF, so that SABE+SACF=SAEFS_ {ABE} + S_{ACF} = S_{AEF}. Determine the measure of the angle formed by the plane (AEF)(AEF) with the plane (BCC′)(BCC').
geometry3D geometryprismareasangles
Functions for which fof=[id] are not short maps, nor contractions

Source: Romanian District Olympiad 2011, Grade IX, Problem 3

10/8/2018
Let f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} be a function with the property that (f∘f)(x)=[x], (f\circ f) (x) =[x], for any real number x. x. Show that there exist two distinct real numbers a,b a,b so that ∣f(a)−f(b)∣≥∣a−b∣. |f(a)-f(b)|\ge |a-b|.
[] [] denotes the integer part.
functionalgebraInteger PartFloor
Characterization of complex numbers that satisfy a³=b³ or a̅=b.

Source: Romanian District Olympiad 2011, Grade X, Problem 3

10/8/2018
Let be two complex numbers a,b. a,b. Show that the following affirmations are equivalent:
(i) \text{(i)} there are four numbers x1,x2,x3,x4∈C x_1,x_2,x_3,x_4\in\mathbb{C} such that ∣x1∣=∣x3∣,∣x2∣=∣x4∣, \big| x_1 \big| =\big| x_3 \big|, \big| x_2 \big| =\big| x_4 \big|, and x_{j_1}^2-ax_{j_1}+b=0=x_{j_2}^2-bx_{j_2}+a, \forall j_1\in\{ 1,2\} , \forall j_2\in\{ 3,4\} .
(ii)a3=b3 \text{(ii)} a^3=b^3 or b=a‾ b=\overline{a} (the conjugate of a).
complex numbersalgebra
Romania District Olympiad 2011 - Grade XI

Source:

3/12/2011
Let A,B∈M2(C)A,B\in \mathcal{M}_2(\mathbb{C}) two non-zero matrices such that AB+BA=O2AB+BA=O_2 and det⁡(A+B)=0\det(A+B)=0. Prove AA and BB have null traces.
linear algebralinear algebra unsolved
Sufficient condition for continuous and nondecreasing function to be constant

Source: Romanian District Olympiad 2011, Grade XII, Problem 3

10/8/2018
Let f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} be a continuous and nondecreasing function.
a) Show that the sequence (12n∑i=12nf(i2n))n≥1 \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} is nonincreasing.
b) Prove that, if there exists some natural index at which the sequence above is equal to ∫01f(x)dx, \int_0^1 f(x)dx, then f f is constant.
functionIntegralRiemann sumreal analysis