Olympiad problems

2013 Serbia National Math Olympiad, 2

For a natural number $n$, set $S_n$ is defined as: \[S_n = \left \{ {n\choose n}, {2n \choose n}, {3n\choose n},..., {n^2 \choose n} \right \}.\] a) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is not complete residue system mod $n$; b) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is complete residue system mod $n$.

2023 Serbia Team Selection Test, P5

Tags: nt , Factorials , Weird , Serbia , TST , modular arithmetic , number theory

For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\] Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.

2015 Serbia National Math Olympiad, 6

Tags: number theory , number theory proposed , exponential equations , congruence , Serbia

In nonnegative set of integers solve the equation: $$(2^{2015}+1)^x + 2^{2015}=2^y+1$$

2021 Serbia Team Selection Test, P3

Tags: system of equations , number theory , Serbia , prime numbers , Diophantine Equations

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

2023 Serbia Team Selection Test, P4

Tags: algebra , polynomial , Serbia , TST

Let $p$ be a prime and $P\in \mathbb{R}[x]$ be a polynomial of degree less than $p-1$ such that $\lvert P(1)\rvert=\lvert P(2)\rvert=\ldots=\lvert P(p)\rvert$. Prove that $P$ is constant.

2023 Serbia Team Selection Test, P3

Tags: Serbia , TST , algebra , broken

The positive integers are partitioned into 2 sequences $a_1<a_2<\dots$ and $b_1<b_2<\dots$ such that $b_n=a_n+n$ for every positive integer $n$. Show that $a_n+b_n=a_{b_n}$.

2009 Serbia Team Selection Test, 2

Tags: inequalities , inequalities unsolved , Serbia

Let $ x,y,z$ be positive real numbers such that $ xy \plus{} yz \plus{} zx \equal{} x \plus{} y \plus{} z$. Prove the inequality $ \frac1{x^2 \plus{} y \plus{} 1} \plus{} \frac1{y^2 \plus{} z \plus{} 1} \plus{} \frac1{z^2 \plus{} x \plus{} 1}\le1$ When does the equality hold?

2016 Serbia National Math Olympiad, 1

Tags: number theory , Serbia , Divisibility

Let $n>1$ be an integer. Prove that there exist $m>n^n $ such that $\frac {n^m-m^n}{m+n} $ is a positive integer.

2009 Serbia National Math Olympiad, 3

Tags: Sets , combinatorics , Set systems , Serbia , set theory , graph theory , Inclusion-exclusion

Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , ..., \mathbb{S}_n$ with the following properties: $1$) $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$ for any two indices $1 \leq i, j\leq n$, and $2$) $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$ for any $1 \leq i < j < k \leq n$ [i]Proposed by Ivan Matic[/i]

2021 Serbia National Math Olympiad, 4

Tags: geometry , Serbia

A convex quadrilateral $ABCD$ will be called [i]rude[/i] if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of quadrilateral $ABCD$ such that the sum of diagonals of $PQRS$ is larger than the sum of diagonals of $ABCD$. Let $r>0$ be a real number. Let us assume that a convex quadrilateral $ABCD$ is not rude, but every quadrilateral $A'BCD$ such that $A'\neq A$ and $A'A\leq r$ is rude. Find all possible values of the largest angle of $ABCD$.

2004 Serbia Team Selection Test, 2

Tags: TST , inequalities , Serbia , three variable inequality

Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that the most two of numbers $$2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}$$ are greater than $1$.

2018 Serbia National Math Olympiad, 1

Tags: geometry , incenter , moving points , Serbia , Triangle

Let $\triangle ABC$ be a triangle with incenter $I$. Points $P$ and $Q$ are chosen on segmets $BI$ and $CI$ such that $2\angle PAQ=\angle BAC$. If $D$ is the touch point of incircle and side $BC$ prove that $\angle PDQ=90$.

2009 Serbia National Math Olympiad, 1

Tags: geometry , Serbia , Triangle

In a scalene triangle $ABC$, $\alpha$ and $\beta$ respectively denote the interior angles at vertixes $A$ and $B$. The bisectors of these two angles meet the opposite sides of the triangle at points $D$ and $E$, respectively. Prove that the acute angle between the lines $DE$ and $AB$ does not exceed $ \frac{ | \alpha - \beta |}{3}$ . [i]Proposed by Dusan Djukic[/i]

2023 Serbia Team Selection Test, P2

Tags: geometry , Serbia , TST , circumcircle

A circle centered at $A$ intersects sides $AC$ and $AB$ of $\triangle ABC$ at $E$ and $F$, and the circumcircle of $\triangle ABC$ at $X$ and $Y$. Let $D$ be the point on $BC$ such that $AD$, $BE$, $CF$ concur. Let $P=XE\cap YF$ and $Q=XF\cap YE$. Prove that the foot of the perpendicular from $D$ to $EF$ lies on $PQ$.

2014 Serbia National Math Olympiad, 1

Tags: Olympiad , function , functional equation , Serbia

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$ hold: $$f(xf(y)-yf(x))=f(xy)-xy$$ [i]Proposed by Dusan Djukic[/i]

2018 Serbia National Math Olympiad, 3

Tags: combinatorics , inequalities , combinatorial geometry , discrete geometry , Serbia

Let $n$ be a positive integer. There are given $n$ lines such that no two are parallel and no three meet at a single point. a) Prove that there exists a line such that the number of intersection points of these $n$ lines on both of its sides is at least $$\left \lfloor \frac{(n-1)(n-2)}{10} \right \rfloor.$$ Notice that the points on the line are not counted. b) Find all $n$ for which there exists a configurations where the equality is achieved.

2021 Serbia National Math Olympiad, 2

Tags: graph theory , combinatorics , Serbia

In the country of Graphia there are $100$ towns, each numbered from $1$ to $100$. Some pairs of towns may be connected by a (direct) road and we call such pairs of towns [i]adjacent[/i]. No two roads connect the same pair of towns. Peter, a foreign tourist, plans to visit Graphia $100$ times. For each $i$, $i=1,2,\dots, 100$, Peter starts his $i$-th trip by arriving in the town numbered $i$ and then each following day Peter travels from the town he is currently in to an adjacent town with the lowest assigned number, assuming such that a town exists and that he hasn't visited it already on the $i$-th trip. Otherwise, Peter deems his $i$-th trip to be complete and returns home. It turns out that after all $100$ trips, Peter has visited each town in Graphia the same number of times. Find the largest possible number of roads in Graphia.