# International Mathematics Competition for University Students 2021

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IMC 2021
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## IMC2021: Day 1, Problem 1

Problem 1. Let $\displaystyle A$ be a real $\displaystyle n\times n$ matrix such that $\displaystyle A^3=0$.

(a) Prove that there is a unique real $\displaystyle n\times n$ matrix $\displaystyle X$ that satisfies the equation

$\displaystyle X+AX+XA^2=A.$

(b) Express $\displaystyle X$ in terms of $\displaystyle A$.

Bekhzod Kurbonboev, Institute of Mathematics, Tashkent

Hint: (a) Multiply the equation by some power of $\displaystyle A$ from left and another power of $\displaystyle A$ from right.

(b) Substitute repeatedly $\displaystyle X=A-AX-XA^2$.

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