Found problems: 112
2020 JBMO Shortlist, 2
Viktor and Natalia bought $2020$ buckets of ice-cream and want to organize a degustation schedule with $2020$ rounds such that:
- In every round, both of them try $1$ ice-cream, and those $2$ ice-creams tried in a single round
are different from each other.
- At the end of the $2020$ rounds, both of them have tried each ice-cream exactly once.
We will call a degustation schedule fair if the number of ice-creams that were tried by Viktor before Natalia is equal to the number of ice creams tried by Natalia before Viktor.
Prove that the number of fair schedules is strictly larger than $2020!(2^{1010} + (1010!)^2)$.
[i]Proposed by Viktor Simjanoski, Macedonia
[/i]
2022 JBMO Shortlist, G6
Let $ABC$ be a right triangle with hypotenuse $BC$. The tangent to the circumcircle of triangle $ABC$ at $A$ intersects the line $BC$ at $T$. The points $D$ and $E$ are chosen so that $AD = BD, AE = CE,$ and $\angle CBD = \angle BCE < 90^{\circ}$. Prove that $D, E,$ and $T$ are collinear.
Proposed by [i]Nikola Velov, Macedonia[/i]
2021 Balkan MO Shortlist, N3
Let $n$ be a positive integer. Determine, in terms of $n$, the greatest integer which divides
every number of the form $p + 1$, where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$.
2021 JBMO Shortlist, G3
Let $ABC$ be an acute triangle with circumcircle $\omega$ and circumcenter $O$. The perpendicular from $A$ to $BC$ intersects $BC$ and $\omega$ at $D$ and $E$, respectively. Let $F$ be a point on the segment $AE$, such that $2 \cdot FD = AE$. Let $l$ be the perpendicular to $OF$ through $F$. Prove that $l$, the tangent to $\omega$ at $E$, and the line $BC$ are concurrent.
Proposed by [i] Stefan Lozanovski, Macedonia[/i]
2011 Balkan MO Shortlist, A4
Let $x,y,z \in \mathbb{R}^+$ satisfying $xyz=3(x+y+z)$. Prove, that
\begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}
2022 Greece Team Selection Test, 4
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]
2012 Balkan MO, 2
Prove that
\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]
for all positive real numbers $x,y$ and $z$.
2021 Balkan MO Shortlist, N7
A [i]super-integer[/i] triangle is defined to be a triangle whose lengths of all sides and at least
one height are positive integers. We will deem certain positive integer numbers to be [i]good[/i] with
the condition that if the lengths of two sides of a super-integer triangle are two (not necessarily
different) good numbers, then the length of the remaining side is also a good number. Let $5$ be
a good number. Prove that all integers larger than $2$ are good numbers.
2019 Azerbaijan BMO TST, 3
Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that:
$$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$
2024 Junior Balkan MO, 1
Let $a, b, c$ be positive real numbers such that
$$a^2 + b^2 + c^2 = \frac{1}{4}.$$
Prove that
$$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$
[i]Proposed by Petar Filipovski, Macedonia[/i]
2021 Balkan MO Shortlist, N5
A natural number $n$ is given. Determine all $(n - 1)$-tuples of nonnegative integers $a_1, a_2, ..., a_{n - 1}$ such that
$$\lfloor \frac{m}{2^n - 1}\rfloor + \lfloor \frac{2m + a_1}{2^n - 1}\rfloor + \lfloor \frac{2^2m + a_2}{2^n - 1}\rfloor + \lfloor \frac{2^3m + a_3}{2^n - 1}\rfloor + ... + \lfloor \frac{2^{n - 1}m + a_{n - 1}}{2^n - 1}\rfloor = m$$
holds for all $m \in \mathbb{Z}$.
2021 Balkan MO Shortlist, C4
A sequence of $2n + 1$ non-negative integers $a_1, a_2, ..., a_{2n + 1}$ is given. There's also a sequence of $2n + 1$ consecutive cells enumerated from $1$ to $2n + 1$ from left to right, such that initially the number $a_i$ is written on the $i$-th cell, for $i = 1, 2, ..., 2n + 1$. Starting from this initial position, we repeat the following sequence of steps, as long as it's possible:
[i]Step 1[/i]: Add up the numbers written on all the cells, denote the sum as $s$.
[i]Step 2[/i]: If $s$ is equal to $0$ or if it is larger than the current number of cells, the process terminates. Otherwise, remove the $s$-th cell, and shift shift all cells that are to the right of it one position to the
left. Then go to Step 1.
Example: $(1, 0, 1, \underline{2}, 0) \rightarrow (1, \underline{0}, 1, 0) \rightarrow (1, \underline{1}, 0) \rightarrow (\underline{1}, 0) \rightarrow (0)$.
A sequence $a_1, a_2,. . . , a_{2n+1}$ of non-negative integers is called balanced, if at the end of this
process there’s exactly one cell left, and it’s the cell that was initially enumerated by $(n + 1)$,
i.e. the cell that was initially in the middle.
Find the total number of balanced sequences as a function of $n$.
[i]Proposed by Viktor Simjanoski, North Macedonia[/i]