Found problems: 14
2019 VJIMC, 4
Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} $. Let $n \geq 1 $ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers.
Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}$$
Prove that $f'$ has at least one root in $D$.
[i]Proposed by Géza Kós (Lorand Eotvos University, Budapest)[/i]
2025 VJIMC, 3
Evaluate the integral $\int_0^{\infty} \frac{\log(x+2)}{x^2+3x+2}\mathrm{d}x$.
2019 VJIMC, 2
Find all twice differentiable functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x)) $$ for every $x\in \mathbb{R}$.
[i]Proposed by Orif Ibrogimov (Czech Technical University of Prague), Karim Rakhimov (University of Pisa)[/i]
2025 VJIMC, 3
Let us call a sequence $(b_1, b_2, \ldots)$ of positive integers fast-growing if $b_{n+1} \geq b_n + 2$ for all $n \geq 1$. Also, for a sequence $a = (a(1), a(2), \ldots)$ of real numbers and a sequence $b = (b_1, b_2, \ldots)$ of positive integers, let us denote
\[
S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|.
\]
a) Do there exist two fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]
are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?
b) Do there exist three fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$, $d = (d_1, d_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]
are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?
2022 VJIMC, 4
Let $g$ be the multiplicative function given by $$g(p^{\alpha}) = \alpha p^{\alpha-1},$$ for all $\alpha\in\mathbb Z^+$ and primes $p$. Prove that there exist infinitely many integers $n$ such that $$g(n+1) = g(n) + g(1).$$
2019 VJIMC, 1
Let $\{a_n \}_{n=0}^{\infty}$ be a sequence given recrusively such that $a_0=1$ and $$a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2}$$ for $n\geq 0$
Show that :
a) $a_n$ is a positive integer.
b) $a_n a_{n+1}-1$ is a square of an integer.
[i]Proposed by Stefan Gyurki (Matej Bel University, Banska Bystrica).[/i]
2019 VJIMC, 1
a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the conditions $$x*x*y=y \;\;\; \text{and}\; \;\; y*x*x=y$$ for every $x,y\in A$ imply commutativity of $*$?
b)a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the condition$$x*x*y=y $$ for every $x,y\in A$ implies commutativity of $*$?
[i]Proposed by Paulius Drungilas, Arturas Dubickas (Vilnius University).
[/i]
2019 VJIMC, 2
A triplet of polynomials $u,v,w \in \mathbb{R}[x,y,z]$ is called [i]smart[/i] if there exists polynomials $P,Q,R\in \mathbb{R}[x,y,z]$ such that the following polynomial identity holds :$$u^{2019}P +v^{2019 }Q+w^{2019} R=2019$$
a) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z$$ [i]smart[/i]?
b) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z$$ [i]smart[/i]?
[i]Proposed by Arturas Dubickas (Vilnius University).
[/i]
2025 VJIMC, 4
Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.
2025 VJIMC, 2
Let $A,B$ be two $n\times n$ complex matrices of the same rank, and let $k$ be a positive integer. Prove that $A^{k+1}B^k = A$ if and only if $B^{k+1}A^k = B$.
2019 VJIMC, 3
Let $p$ be an even non-negative continous function with $\int _{\mathbb{R}} p(x) dx =1$ and let $n$ be a positive integer. Let $\xi_1,\xi_2,\xi_3 \dots ,\xi_n$ be independent identically distributed random variables with density function $p$ .
Define
\begin{align*}
X_{0} & = 0 \\
X_{1} & = X_0+ \xi_1 \\
& \vdotswithin{ = }\notag \\
X_{n} & = X_{n-1} + \xi_n
\end{align*}
Prove that the probability that all random variables $X_1,X_2 \dots X_{n-1}$ lie between $X_0$ and $X_n$ is $\frac{1}{n}$.
[i]Proposed by Fedor Petrov (St.Petersburg State University).[/i]
2019 VJIMC, 3
For an invertible $n\times n$ matrix $M$ with integer entries we define a sequence $\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}$ by the recurrence $M_0=M$ ,$M_{i+1}=(M_i^T)^{-1}M_i$ for $i\geq 0$.
Find the smallest integer $n\geq 2 $ for wich there exists a normal $n\times n$ matrix with integer entries such that its sequence $\mathcal{S}_M$ is not constant and has period $P=7$ i.e $M_{i+7}=M_i$.
($M^T$ means the transpose of a matrix $M$ . A square matrix is called normal if $M^T M=M M^T$ holds).
[i]Proposed by Martin Niepel (Comenius University, Bratislava)..[/i]
2025 VJIMC, 1
Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.
2019 VJIMC, 4
Determine the largest constant $K\geq 0$ such that $$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$.
[i]Proposed by Orif Ibrogimov (Czech Technical University of Prague).[/i]