MathDB

1

Part of 2013 IMC

Problems(2)

IMC 2013/1/1

Source: Imc 2013 Problem 1

8/8/2013
Let A\displaystyle{A} and B\displaystyle{B} be real symmetric matrixes with all eigenvalues strictly greater than 1\displaystyle{1}. Let λ\displaystyle{\lambda } be a real eigenvalue of matrix AB\displaystyle{{\rm A}{\rm B}}. Prove that ∣λ∣>1\displaystyle{\left| \lambda \right| > 1}.
Proposed by Pavel Kozhevnikov, MIPT, Moscow.
linear algebramatrixvectorinequalitiesIMCcollege contests
IMC 2013/2/1

Source: Imc 2013 Problem 6

8/9/2013
Let z\displaystyle{z} be a complex number with ∣z+1∣>2\displaystyle{\left| {z + 1} \right| > 2}. Prove that ∣z3+1∣>1\displaystyle{\left| {{z^3} + 1} \right| > 1}.
Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.
inequalitiestrigonometryquadraticsfunctioncomplex analysisIMCcollege contests