Found problems: 35
2014 Contests, 1
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
2014 Tuymaada Olympiad, 3
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
2023 Tuymaada Olympiad, 4
Two players play a game. They have $n > 2$ piles containing $n^{10}+1$ stones each. A move consists of removing all the piles but one and dividing the remaining pile into $n$ nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?
2016 Tuymaada Olympiad, 5
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
2023 Tuymaada Olympiad, 2
Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence $(a_1, a_2, \ldots , a_n)$, where all $a_k$ equal $0$ or $1$. After that Tanya writes down a sequence $(b_1, b_2, \ldots , b_n)$, where all $b_k$ also equal $0$ or $1$. Then Masha either does nothing or says “Mutabor” and replaces both sequences: her own sequence by $(a_n, a_{n-1}, \ldots , a_1)$, and Tanya’s sequence by $(1 - b_n, 1 - b_{n-1}, \ldots , 1 - b_1)$. Masha’s sequence is covered by a napkin, and Serge is invited to the room. Serge should look at Tanya’s sequence and tell the sequence covered by the napkin. For what $n$ Serge and Tanya can prepare and show such a trick? Serge does not have to determine whether the word “Mutabor” has been pronounced.
2018 Tuymaada Olympiad, 1
Do there exist three different quadratic trinomials $f(x), g(x), h(x)$ such that the roots of the equation $f(x)=g(x)$ are $1$ and $4$, the roots of the equation $g(x)=h(x)$ are $2$ and $5$, and the roots of the equation $h(x)=f(x)$ are $3$ and $6$?
[i]Proposed by A. Golovanov[/i]
2018 Azerbaijan Senior NMO, 5
Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then
\[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\]
[i]A. Golovanov[/i]
2023 Tuymaada Olympiad, 7
$3n$ people forming $n$ families of a mother, a father and a child, stand in a circle. Every two neighbours can exchange places except the case when a parent exchanges places with his/her child (this is forbidden). For what $n$ is it possible to obtain every arrangement of those people by such
exchanges? The arrangements differing by a circular shift are considered distinct.
2014 Tuymaada Olympiad, 1
Given are three different primes. What maximum number of these primes can divide their sum?
[i](A. Golovanov)[/i]
2014 Tuymaada Olympiad, 3
Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality
\[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \]
[i](N. Alexandrov)[/i]