Olympiad problems

2014 Iran Geometry Olympiad (senior), 3:

Tags: geometry , Iran

Let $ABC$ be an acute triangle.A circle with diameter $BC$ meets $AB$ and $AC$ at $E$ and $F$,respectively. $M$ is midpoint of $BC$ and $P$ is point of intersection $AM$ with $EF$. $X$ is an arbitary point on arc $EF$ and $Y$ is the second intersection of $XP$ with a circle with diameter $BC$.Prove that $ \measuredangle XAY=\measuredangle XYM $. Author:Ali zo'alam , Iran

2017 Iran MO (3rd round), 3

Tags: algebra , Inequality , AM-GM , Iran , inequalities

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$

2017 Iranian Geometry Olympiad, 3

Tags: IGO , Iran , geometry

On the plane, $n$ points are given ($n>2$). No three of them are collinear. Through each two of them the line is drawn, and among the other given points, the one nearest to this line is marked (in each case this point occurred to be unique). What is the maximal possible number of marked points for each given $n$? [i]Proposed by Boris Frenkin (Russia)[/i]

2019 Iranian Geometry Olympiad, 1

Tags: IGO , Iran , geometry , Angle Chasing

Circles $\omega_1$ and $\omega_2$ intersect each other at points $A$ and $B$. Point $C$ lies on the tangent line from $A$ to $\omega_1$ such that $\angle ABC = 90^\circ$. Arbitrary line $\ell$ passes through $C$ and cuts $\omega_2$ at points $P$ and $Q$. Lines $AP$ and $AQ$ cut $\omega_1$ for the second time at points $X$ and $Z$ respectively. Let $Y$ be the foot of altitude from $A$ to $\ell$. Prove that points $X, Y$ and $Z$ are collinear. [i]Proposed by Iman Maghsoudi[/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]

2018 Iran Team Selection Test, 3

Tags: polynomial , Integer Polynomial , Iranian TST , number theory , Iran

$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions? a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $ b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $ [i]Proposed by Mojtaba Zare[/i]

2000 Iran MO (3rd Round), 2

Tags: geometry , 3D geometry , sphere , circumcircle , Iran

Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.

2019 Iran Team Selection Test, 2

Tags: Iranian TST , TST , Iran

$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square. Prove $a$ is one of $a_1,\dots ,a_n$ [i]Proposed by Mohsen Jamali[/i]

2014 Iran MO (3rd Round), 3

Tags: geometry , circumcircle , Iran

Distinct points $B,B',C,C'$ lie on an arbitrary line $\ell$. $A$ is a point not lying on $\ell$. A line passing through $B$ and parallel to $AB'$ intersects with $AC$ in $E$ and a line passing through $C$ and parallel to $AC'$ intersects with $AB$ in $F$. Let $X$ be the intersection point of the circumcircles of $\triangle{ABC}$ and $\triangle{AB'C'}$($A \neq X$). Prove that $EF \parallel AX$.

2017 Iranian Geometry Olympiad, 5

Tags: IGO , Iran , geometry , 3D geometry , sphere , tetrahedron

Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$. [i]Proposed by Alexey Zaslavsky (Russia)[/i]

2023 Iran Team Selection Test, 2

Tags: Iran , TST , Iranian TST , combinatorics

Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$. [i]Proposed by Navid Safaei [/i]

2009 Iran MO (3rd Round), 5

Tags: Pythagorean Theorem , geometry , combinatorics , Iran

A ball is placed on a plane and a point on the ball is marked. Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling. Prove that it is possible. Time allowed for this problem was 90 minutes.

2018 Iran Team Selection Test, 4

Tags: number theory , Iran , Iranian TST , TST

Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$. Prove that there exist infinitely many positive integers which they are not "useful but not optimized". (e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number) [i]Proposed by Mohsen Jamali[/i]

2019 Iranian Geometry Olympiad, 2

Tags: IGO , Iran , geometry

Find all quadrilaterals $ABCD$ such that all four triangles $DAB$, $CDA$, $BCD$ and $ABC$ are similar to one-another. [i]Proposed by Morteza Saghafian[/i]

2019 Iranian Geometry Olympiad, 4

Tags: IGO , Iran , geometry

Let $ABCD$ be a parallelogram and let $K$ be a point on line $AD$ such that $BK=AB$. Suppose that $P$ is an arbitrary point on $AB$, and the perpendicular bisector of $PC$ intersects the circumcircle of triangle $APD$ at points $X$, $Y$. Prove that the circumcircle of triangle $ABK$ passes through the orthocenter of triangle $AXY$. [i]Proposed by Iman Maghsoudi[/i]

2016 Iran MO (3rd Round), 1

Tags: number theory , Iran , prime numbers , modular arithmetic

Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that: $$q |(x+1)^p-x^p$$ If and only if $$q \equiv 1 \pmod p$$

2005 Germany Team Selection Test, 2

Tags: algebra , polynomial , function , Iran , power of 2

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

2017 Iran MO (3rd round), 1

Tags: Iran , combinatorics

There's a tape with $n^2$ cells labeled by $1,2,\ldots,n^2$. Suppose that $x,y$ are two distinct positive integers less than or equal to $n$. We want to color the cells of the tape such that any two cells with label difference of $x$ or $y$ have different colors. Find the minimum number of colors needed to do so.

2019 Iran Team Selection Test, 2

Tags: TST , Iranian TST , geometry , Iran

In a triangle $ABC$, $\angle A$ is $60^\circ$. On sides $AB$ and $AC$ we make two equilateral triangles (outside the triangle $ABC$) $ABK$ and $ACL$. $CK$ and $AB$ intersect at $S$ , $AC$ and $BL$ intersect at $R$ , $BL$ and $CK$ intersect at $T$. Prove the radical centre of circumcircle of triangles $BSK, CLR$ and $BTC$ is on the median of vertex $A$ in triangle $ABC$. [i]Proposed by Ali Zamani[/i]

2015 Iran MO (2nd Round), 2

Tags: combinatorics , Iran

A circle is divided into $2n$ equal by $2n$ points. Ali draws $n+1$ arcs, of length $1,2,\ldots,n+1$. Prove that we can find two arcs, such that one of them is inside in the other one.

2022 Iran-Taiwan Friendly Math Competition, 2

Tags: algebra , functional equation , function , Taiwan , Iran

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that: $\bullet$ $f(x)<2$ for all $x\in (0,1)$; $\bullet$ for all real numbers $x,y$ we have: $$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$ Proposed by Navid Safaei

2017 Iran Team Selection Test, 6

Tags: algebra , number theory , Iran , Iranian TST

Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as: $a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers $n$ such that $a_n$ is a power of $k$. [i]Proposed by Amirhossein Pooya[/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]

2016 Iran MO (3rd Round), 3

Tags: Iran , combinatorics , graph theory , Directed graphs

There are $24$ robots on the plane. Each robot has a $70^{\circ}$ field of view. What is the maximum number of observing relations? (Observing is a one-sided relation)

2017 Iran MO (3rd round), 2

Tags: Iran , combinatorics

Two persons are playing the following game on a $n\times m$ table, with drawn lines: Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game. Who has a winning strategy?