MathDB

2

Part of 2012 District Olympiad

Problems(6)

a^2+ab+ac-bc = 0, irrational, NT 2012 Romania District VII p2

Source:

9/1/2024
Let a,ba, b and cc be positive real numbers such that a2+ab+ac−bc=0.a^2+ab+ac-bc = 0.
a) Show that if two of the numbers a,ba, b and cc are equal, then at least one of the numbers a,ba, b and cc is irrational.
b) Show that there exist infinitely many triples (m,n,p)(m, n, p) of positive integers such that m2+mn+mp−np=0.m^2 + mn + mp -np = 0.
algebranumber theoryirrational number
Graphical equation 2^x=x+1

Source: Romanian District Olympiad 2012, Grade X, Problem 2

10/9/2018
a) Solve in R \mathbb{R} the equation 2x=x+1. 2^x=x+1.
b) If a function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} has the property that (f\circ f)(x)=2^x-1, \forall x\in\mathbb{R} , then f(0)+f(1)=1. f(0)+f(1)=1.
functionequationsalgebra
plane forms equal angles with 2 planes, pyramid with base rectangle

Source: 2012 Romania District VIII P2

5/19/2020
The pyramid VABCDVABCD has base the rectangle ABCD, and the side edges are congruent. Prove that the plane (VCD)(VCD) forms congruent angles with the planes (VAC)(VAC) and (BAC)(BAC) if and only if ∠VAC=∠BAC\angle VAC = \angle BAC .
geometry3D geometrypyramidanglesplanes
Sum less than 3/4.

Source: Romanian District Olympiad 2012, Grade IX, Problem 2

10/9/2018
If a,b,c>0, a,b,c>0, then ∑cyca2a+b+c≤3/4. \sum_{\text{cyc}} \frac{a}{2a+b+c}\le 3/4.
inequalitiesalgebra
x->det(A²+B²+xBA) has degree <3 if AB=0

Source: Romanian District Olympiad 2012, Grade XI, Problem 2

10/9/2018
Let A,B∈M(R) A,B\in\mathcal{M} \left( \mathbb{R} \right) that satisfy AB=O3. AB=O_3. Prove that:
a) The function f:C⟶C f:\mathbb{C}\longrightarrow\mathbb{C} defined as f(x)=det⁡(A2+B2+xBA) f(x)=\det \left( A^2+B^2+xBA \right) is a polynomial one, of degree at most 2. 2. b) det⁡(A2+B2)≥0. \det\left( A^2+B^2 \right)\ge 0.
functionalgebrapolynomiallinear algebra
Romania District Olympiad 2012 - Grade XII

Source:

3/10/2012
Let (A,+,⋅)(A,+,\cdot) a 9 elements ring. Prove that the following assertions are equivalent:
(a) For any x∈A\{0}x\in A\backslash\{0\} there are two numbers a∈{−1,0,1}a\in \{-1,0,1\} and b∈{−1,1}b\in \{-1,1\} such that x2+ax+b=0x^2+ax+b=0.
(b) (A,+,⋅)(A,+,\cdot) is a field.
quadraticsalgebrapolynomialsuperior algebrasuperior algebra unsolved