Found problems: 7
2022 Serbia National Math Olympiad, P4
Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$, $n\in\mathbb{N}$, such that $gcd(x, n)$ is either $1$ or prime. Prove
$$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$
For which $n$ does equality hold?
2022 Serbia National Math Olympiad, P2
Let $a$, $b$ and $c$ be positive real numbers and $a^3+b^3+c^3=3$. Prove
$$\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3$$
2018 Serbia JBMO TST, 2
Show that for $a,b,c > 0$ the following inequality holds:
$\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}$.
2022 Serbia National Math Olympiad, P6
Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence
$$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$
exists number less than $2pq$, divisible by $p$ or $q$.
2022 Serbia National Math Olympiad, P1
Let $k$ be incircle of acute triangle $ABC$, $AC\neq BC$, and $l$ be excircle that touches $AB$. Line $p$ through the $C$ is orthogonal to $AB$, $p\cap k = \{X, Y\}$ , $p\cap l = \{Z, T\}$ and the point arrangement is $X-Y-Z-T$. Circle $m$ through $X$ and $Z$ intersects $AB$ at $D$ and $E$. Prove that points $D,Y,E,T$ are concyclic.
2022 Serbia National Math Olympiad, P5
On the board are written $n$ natural numbers, $n\in \mathbb{N}$. In one move it is possible to choose two
equal written numbers and increase one by $1$ and decrease the other by $1$. Prove that in this
the game cannot be played more than $\frac{n^3}{6}$ moves.
2022 Serbia National Math Olympiad, P3
The table of dimensions $n\times n$, $n\in\mathbb{N}$, is filled with numbers from $1$ to $n^2$, but the difference
any two numbers on adjacent fields is at most $n$, and that for every $k = 1, 2,\dots , n^2$ set of fields
whose numbers are $1, 2,\dots , k$ is connected, as well as the set of fields whose numbers are $k, k + 1,\dots , n^2$. Neighboring fields are fields with a common side, while a set of fields is considered connected if from each field to every other field of that set can be reached going only to the neighboring fields within that set.
We call a pair of adjacent numbers, ie. numbers on adjacent fields, good, if their absolute difference is exactly $n$
(one number can be found in several good pairs). Prove that the table has at least $2 (n - 1)$ good pairs.