Olympiad problems

2018 Iran Team Selection Test, 1

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]

2017 Iran Team Selection Test, 6

Tags: combinatorics , Iran , Iranian TST , Hamiltonian path , catalan

In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom. Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$. ([i]e.g.[/i] We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape) [i]Proposed by Morteza Saghafian[/i]

2015 Iran Team Selection Test, 2

Tags: number theory , Iran , Iranian TST , least common multiple

Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence: $a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$ Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$. ($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)

2017 Iran Team Selection Test, 2

Tags: Iran , Iranian TST , number theory , combinatorics

Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two. [i]Proposed by Morteza Saghafian[/i]

2020 Iran Team Selection Test, 6

Tags: combinatorics , triangulation , Iranian TST , Morteza Saghafiyan

$n$ positive numbers are given. Is it always possible to find a convex polygon with $n+3$ edges and a triangulation of it so that the length of the diameters used in the triangulation are the given $n$ numbers? [i]Proposed by Morteza Saghafian[/i]

2017 Iran Team Selection Test, 6

Tags: Iran , Iranian TST , geometry

In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$ [i] Proposed by Iman Maghsoudi[/i]

2019 Iran Team Selection Test, 5

Tags: function , algebra , TST , Iran , Iranian TST , functional equation

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$: $$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$ [i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]

2019 Iran Team Selection Test, 1

Tags: Iranian TST , TST

$S$ is a subset of Natural numbers which has infinite members. $$S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\}$$ Prove the set of prime divisors of $S’$ has also infinite members [i]Proposed by Yahya Motevassel[/i]

2019 Iran Team Selection Test, 5

Tags: function , algebra , TST , Iran , Iranian TST , functional equation

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$: $$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$ [i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]

2017 Iran Team Selection Test, 1

Tags: algebra , inequalities , Iran , Iranian TST

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

2020 Iran Team Selection Test, 1

Tags: polynomial , number theory , Iranian TST

We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$. [i]Proposed by Masud Shafaie[/i]

2017 Iran Team Selection Test, 3

Tags: combinatorics , Iranian TST , Iran

There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a [i]match[/i] be cause they have distinct amount of shapes, distinct shapes but the same color of shapes. What is the maximum number of cards that we can choose such that non of the triples make a [i]match[/i]? [i]Proposed by Amin Bahjati[/i]

2017 Iran Team Selection Test, 1

Tags: number theory , Iran , Iranian TST

Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$ $$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$ [i]Proposed by Navid Safaei[/i]

2020 Iran Team Selection Test, 1

Tags: combinatorics , graph theory , Iranian TST

A weighted complete graph with distinct positive wights is given such that in every triangle is [i]degenerate [/i] that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints. [i]Proposed by Morteza Saghafian [/i]

2017 Iran Team Selection Test, 4

Tags: Iran , Iranian TST , geometry , combinatorics

There are $6$ points on the plane such that no three of them are collinear. It's known that between every $4$ points of them, there exists a point that it's power with respect to the circle passing through the other three points is a constant value $k$.(Power of a point in the interior of a circle has a negative value.) Prove that $k=0$ and all $6$ points lie on a circle. [i]Proposed by Morteza Saghafian[/I]

2017 Iran Team Selection Test, 5

Tags: Iran , Iranian TST , polynomial , algebra

Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as $$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$ Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that $$P_{2n}(x)=P_n(x^2+c).$$ [i]Proposed by Navid Safaei[/i]

2017 Iran Team Selection Test, 6

Tags: Iran , Iranian TST , geometry

In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$ [i] Proposed by Iman Maghsoudi[/i]

2017 Iran Team Selection Test, 4

Tags: Iran , Iranian TST , geometry , combinatorics

There are $6$ points on the plane such that no three of them are collinear. It's known that between every $4$ points of them, there exists a point that it's power with respect to the circle passing through the other three points is a constant value $k$.(Power of a point in the interior of a circle has a negative value.) Prove that $k=0$ and all $6$ points lie on a circle. [i]Proposed by Morteza Saghafian[/I]

2018 Iran Team Selection Test, 4

Tags: number theory , Iran , Iranian TST

We say distinct positive integers $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. [i]Proposed by Morteza Saghafian[/i]

2018 Iran Team Selection Test, 2

Tags: combinatorics , Iran , Iranian TST , game strategy , game , Game Theory

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

2018 Iran Team Selection Test, 6

Tags: combinatorics , Iran , Iranian TST , Sequence

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

2017 Iran Team Selection Test, 5

Tags: geometry , Iran , Iranian TST , circumcircle , Hi

In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively. Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other. [i]Proposed by Iman Maghsoudi[/i]

2018 Iran Team Selection Test, 2

Tags: Iran , Iranian TST , maximum value , algebra

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ [i]Proposed by Morteza Saghafian[/i]

2017 Iran Team Selection Test, 2

Tags: combinatorics , Iran , Iranian TST

In the country of [i]Sugarland[/i], there are $13$ students in the IMO team selection camp. $6$ team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these $6$ tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation. Is it possible that all $13$ students have a chance of being a team member? [i]Proposed by Morteza Saghafian[/i]

2015 Iran Team Selection Test, 2

Tags: number theory , Iran , Iranian TST , least common multiple

Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence: $a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$ Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$. ($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)