MathDB

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Part of 2012 Centers of Excellency of Suceava

Problems(4)

Prove that (function)

Source:

3/31/2013
Function f ⁣:[0,+∞)→[0,+∞){{f\colon \mathbb[0, +\infty)}\to\mathbb[0, +\infty)} satisfies the condition f(x)+f(y)≥2f(x+y)f(x)+f(y){\ge}2f(x+y) for all x,y≥0x,y{\ge}0. Prove that f(x)+f(y)+f(z)≥3f(x+y+z)f(x)+f(y)+f(z){\ge}3f(x+y+z) for all x,y,z≥0x,y,z{\ge}0. Mathematical induction? __________________________________ Azerbaijan Land of the Fire :lol:
functioninductionalgebra unsolvedalgebra
Easy algebraic identity

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10/31/2019
Let be three nonzero rational numbers a,b,c a,b,c under the relation (a+b)(b+c)(c+a)=a2b2c2. (a+b)(b+c)(c+a)=a^2b^2c^2. Show that the expression 3+1/a3+1/b3+1/c33 \sqrt[3]{3+1/a^3+1/b^3+1/c^3} is rational.
Ion Bursuc
algebraformula
Determinant of sum of matrix with its adjugate

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10/31/2019
Let be a natural number n n and a n×n n\times n nilpotent real matrix A. A. Prove that 0=det⁡(A+adjA). 0=\det\left( A+\text{adj} A \right) .
Neculai Moraru
linear algebramatrix
Group theory operations

Source:

10/31/2019
Let be a natural number n≥2, n\ge 2, a group G G and two elements of it e1,e2 e_1,e_2 such that e2e1x=xe2e1, e_2e_1x=xe_2e_1, for any element x x of G. G. Prove that (e1xe2)n=e1xne2, \left( e_1xe_2 \right)^n =e_1x^ne_2, for any element x x of G, G, if and only if e2e1=(e2e1)n. e_2e_1=\left( e_2e_1\right)^n.
Ion Bursuc
group theoryabstract algebra