International Mathematics Competition
for University Students
2020

IMC2020: Problems on Day 2

Problem 5. Find all twice continuously differentiable functions $\displaystyle f:\mathbb{R}\to(0,+\infty)$ satisfying

$\displaystyle f''(x)f(x)\geq {2(f'(x))^2}$

for all $\displaystyle x\in\mathbb{R}$.

Karen Keryan, Yerevan State University & American University of Armenia, Yerevan

  Hint    Solution  

Problem 6. Find all prime numbers $\displaystyle p$ for which there exists a unique $\displaystyle a\in \{1,2,\ldots,p\}$ such that $\displaystyle a^3-3a+1$ is divisible by $\displaystyle p$.

Géza Kós, Loránd Eötvös University, Budapest

  Hint    Solution  

Problem 7. Let $\displaystyle G$ be a group and $\displaystyle n\ge2$ be an integer. Let $\displaystyle H_1$ and $\displaystyle H_2$ be two subgroups of $\displaystyle G$ that satisfy

$\displaystyle [G:H_1]=[G:H_2]=n \quad\text{and}\quad [G:(H_1\cap H_2)]=n(n-1).$

Prove that $\displaystyle H_1$ and $\displaystyle H_2$ are conjugate in $\displaystyle G$.

(Here $\displaystyle [G:H]$ denotes the index of the subgroup $\displaystyle H$, i.e. the number of distinct left cosets $\displaystyle xH$ of $\displaystyle H$ in $\displaystyle G$. The subgroups $\displaystyle H_1$ and $\displaystyle H_2$ are conjugate if there exists an element $\displaystyle g\in G$ such that $\displaystyle g^{-1}H_1g=H_2$.)

Ilya Bogdanov and Alexander Matushkin, Moscow Institute of Physics and Technology

  Hint    Solution  

Problem 8. Compute

$\displaystyle \lim_{n\to \infty} \frac1{\log \log n}\sum_{k=1}^n (-1)^k\binom{n}{k} \log k.$

(Here $\displaystyle \log$ denotes the natural logarithm.)

Fedor Petrov, St. Petersburg State University

  Hint    Solution