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