MathDB

3

Part of 2019 District Olympiad

Problems(6)

(x^3-x)/24 is integer 2019 Romania District VII p3

Source:

9/1/2024
Consider the sets M={0,1,2,,2019}M = \{0,1,2,, 2019\} and A={x∈M  ∣x3−x24∈N}A=\left\{ x\in M\,\, | \frac{x^3-x}{24} \in N\right\} a) How many elements does the set AA have? b) Determine the smallest natural number nn, n≥2n \ge 2, which has the property that any nn-element subset of the set AA contains two distinct elements whose difference is divisible by 4040.
number theoryInteger
dihedral angles wanted and given, parallelepiped (2019 Romania District VIII p3)

Source:

5/23/2020
Consider the rectangular parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' as such the measure of the dihedral angle formed by the planes (A′BD)(A'BD) and (C′BD)(C'BD) is 90o90^o and the measure of the dihedral angle formed by the planes (AB′C)(AB'C) and (D′B′C)(D'B'C) is 60o60^o. Determine and measure the dihedral angle formed by the planes (BC′D)(BC'D) and (A′C′D)(A'C'D).
3D geometrygeometryanglesparallelepiped
Romanian District Olympiad 2019 - Grade 9 - Problem 3

Source: Romanian District Olympiad 2019 - Grade 9 - Problem 3

3/18/2019
Let (an)n∈N(a_n)_{n \in \mathbb{N}} be a sequence of real numbers such that 2(a1+a2+…+an)=nan+1 ∀ n≥1.2(a_1+a_2+…+a_n)=na_{n+1}~\forall~n \ge 1. <spanclass=′latex−bold′>a)</span><span class='latex-bold'>a)</span> Prove that the given sequence is an arithmetic progression. <spanclass=′latex−bold′>b)</span><span class='latex-bold'>b)</span> If ⌊a1⌋+⌊a2⌋+…+⌊an⌋=⌊a1+a2+…+an⌋ ∀ n∈N,\lfloor a_1 \rfloor + \lfloor a_2 \rfloor +…+ \lfloor a_n \rfloor = \lfloor a_1+a_2+…+a_n \rfloor~\forall~ n \in \mathbb{N}, prove that every term of the sequence is an integer.
Sequencefloor functionArithmetic Progressionalgebra
Romanian District Olympiad 2019 - Grade 10 - Problem 3

Source: Romanian District Olympiad 2019 - Grade 10 - Problem 3

3/17/2019
Let a,b,ca,b,c be distinct complex numbers with ∣a∣=∣b∣=∣c∣=1.|a|=|b|=|c|=1. If ∣a+b−c∣2+∣b+c−a∣2+∣c+a−b∣2=12,|a+b-c|^2+|b+c-a|^2+|c+a-b|^2=12, prove that the points of affixes a,b,ca,b,c are the vertices of an equilateral triangle.
complex numberscomplex number geometryalgebra
Romanian District Olympiad 2019 - Grade 11 - Problem 3

Source: Romanian District Olympiad 2019 - Grade 11 - Problem 3

3/16/2019
Let nn be an odd natural number and A,B∈Mn(C)A,B \in \mathcal{M}_n(\mathbb{C}) be two matrices such that (A−B)2=On.(A-B)^2=O_n. Prove that det⁡(AB−BA)=0.\det(AB-BA)=0.
determinanMatriceslinear algebra
Romanian District Olympiad 2019 - Grade 12 - Problem 3

Source: Romanian District Olympiad 2019 - Grade 12 - Problem 3

3/16/2019
Let GG be a finite group and let x1,…,xnx_1,…,x_n be an enumeration of its elements. We consider the matrix (aij)1≤i,j≤n,(a_{ij})_{1 \le i,j \le n}, where aij=0a_{ij}=0 if xixj−1=xjxi−1,x_ix_j^{-1}=x_jx_i^{-1}, and aij=1a_{ij}=1 otherwise. Find the parity of the integer det⁡(aij).\det(a_{ij}).
group theoryabstract algebrasuperior algebra