# International Mathematics Competition for University Students 2024

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IMC 2024
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## IMC2024: Problems on Day 2

Problem 6. Prove that for any function $\displaystyle f\colon\mathbb Q\to\mathbb Z$, there exist $\displaystyle a,b,c\in\mathbb Q$ such that $\displaystyle a < b < c$, $\displaystyle f(b) \geq f(a)$, and $\displaystyle f(b) \geq f(c)$.

Mehdi Golafshan & Markus A. Whiteland, University of Liège

Problem 7. Let $\displaystyle n$ be a positive integer. Suppose that $\displaystyle A$ and $\displaystyle B$ are invertible $\displaystyle n\times n$ matrices with complex entries such that $\displaystyle A+B=I$ (where $\displaystyle I$ is the identity matrix) and

$\displaystyle (A^2 + B^2)(A^4 + B^4) = A^5 + B^5.$

Find all possible values of $\displaystyle \det(AB)$ for the given $\displaystyle n$.

Sergey Bondarev, Sergey Chernov, Belarusian State University, Minsk

Problem 8. Define the sequence $\displaystyle x_1,x_2,\ldots$ by the initial terms $\displaystyle x_1=2$, $\displaystyle x_2=4$, and the recurrence relation

$\displaystyle x_{n+2}=3x_{n+1}-2x_n+\frac{2^ n}{x_n}\quad\text{for }n\geq 1.$

Prove that $\displaystyle \displaystyle\lim\limits_{n \to \infty}\frac{x_n}{2^ n}$ exists and satisfies

$\displaystyle \frac{1+\sqrt{3}}{2}\leq\lim\limits_{n \to \infty}\dfrac{x_n}{2^ n}\leq\frac{3}{2}.$

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

Problem 9. A matrix $\displaystyle A=(a_{ij})$ is called nice, if it has the following properties:

(i) the set of all entries of $\displaystyle A$ is $\displaystyle \{1,2,\ldots,2t\}$ for some integer $\displaystyle t$;

(ii) the entries are non-decreasing in every row and in every column: $\displaystyle a_{i,j}\leq a_{i,j+1}$ and $\displaystyle a_{i,j}\leq a_{i+1,j}$;

(iii) equal entries can appear only in the same row or the same column: if $\displaystyle a_{i,j}=a_{k,\ell}$, then either $\displaystyle i=k$ or $\displaystyle j=\ell$;

(iv) for each $\displaystyle s=1,2,\ldots,2t-1$, there exist $\displaystyle i\ne k$ and $\displaystyle j\ne \ell$ such that $\displaystyle a_{i,j}=s$ and $\displaystyle a_{k,\ell}=s+1$.

Prove that for any positive integers $\displaystyle m$ and $\displaystyle n$, the number of nice $\displaystyle m\times n$ matrices is even.

For example, the only two nice $\displaystyle 2\times 3$ matrices are $\displaystyle \begin{pmatrix} 1&1&1\\2&2&2 \end{pmatrix}$ and $\displaystyle \begin{pmatrix} 1&1&3\\2&4&4 \end{pmatrix}$.

Fedor Petrov, St Petersburg State University

Problem 10. We say that a square-free positive integer $\displaystyle n$ is almost prime if

$\displaystyle n \mid x^{d_1}+x^{d_2}+...+x^{d_k}-kx$

for all integers $\displaystyle x$, where $\displaystyle 1=d_1<d_2<...<d_k=n$ are all the positive divisors of $\displaystyle n$. Suppose that $\displaystyle r$ is a Fermat prime (i.e., it is a prime of the form $\displaystyle 2^{2^m}+1$ for an integer $\displaystyle m\geq 0$), $\displaystyle p$ is a prime divisor of an almost prime integer $\displaystyle n$, and $\displaystyle p\equiv 1\ (\mathrm{mod}\ r)$. Show that, with the above notation, $\displaystyle d_i\equiv 1\ (\mathrm{mod}\ r)$ for all $\displaystyle 1\leq i\leq k$.

(An integer $\displaystyle n$ is called square-free if it is not divisible by $\displaystyle d^2$ for any integer $\displaystyle d>1$.)

Tigran Hakobyan, Yerevan State University, Vanadzor, Armenia

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