MathDB

2019 Teodor Topan

Part of Teodor Topan

Subcontests

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a³,b³,a+b rational, but a,b not rational

a) Give example of two irrational numbers a,b a,b having the property that a3,b3,a+b a^3,b^3,a+b are all rational.
b) Prove that if x,y x,y are two nonnegative real numbers having the property that x3,y3,x+y x^3,y^3,x+y are rational, then x x and y y are both rational.
Mihai Piticari and Vladimir Cerbu

Logarithmic Diophantine

Solve in the natural numbers the equation log⁡6n−19(n!+1)=2. \log_{6n-19} (n!+1) =2.
Dragoș Crișan

Linear algebra impossibility

Do exist pairwise distinct matrices A,B,C∈M2(R) A,B,C\in \mathcal{M}_2(\mathbb{R}) verifying the following properties?
(i)det⁡A=det⁡C \text{(i)} \det A=\det C (ii)AB=C,BC=A,CA=B \text{(ii)} AB=C,BC=A,CA=B (iii)trA,trB≠0 \text{(iii)} \text{tr} A,\text{tr} B\neq 0
Robert Pop