Navvye/Olympiad-Problems-CSV huggingface dataset:
2001 India IMO Training Camp
If on $ \triangle ABC$, trinagles $ AEB$ and $ AFC$ are constructed externally such that $ \angle AEB=2 \alpha$, $ \angle AFB= 2 \beta$.

$ AE=EB$, $ AF=FC$.

COnstructed externally on $ BC$ is triangle $ BDC$ with $ \angle DBC= \beta$ , $ \angle BCD= \alpha$.

Prove that 1. $ DA$ is perpendicular to $ EF$.

2. If $ T$ is the projection of $ D$ on $ BC$, then prove that $ \frac{DA}{EF}= 2 \frac{DT}{BC}$.
2001 India IMO Training Camp
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : $$f ( f (x)-x) = 2x$$ for all $x > 0$.
2001 India IMO Training Camp
Points $B = B_1 , B_2, \cdots  , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that :

$$(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)$$
2001 India IMO Training Camp
Complex numbers $\alpha$ , $\beta$  , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial $$(x-\alpha)(x-\beta )(x-\gamma )$$ has integer coefficients.
2001 India IMO Training Camp
Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv  1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that :

$$\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}$$
2001 India IMO Training Camp
Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if:

$(i)$ all the three colors occur at the vertices of the square, and

$(ii)$ one side of the square has the endpoints of the same color.

Show that the number of properly colored squares is even.
2001 India IMO Training Camp
Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$.

Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.

Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.

Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$.

(a) Prove that $r_{1}+r_{2}=2r$.

(b) Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.
2001 India IMO Training Camp
A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.
2001 India IMO Training Camp
Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq  3$. Prove that

$$\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1$$
2010 Indonesia TST
Let $ a$, $ b$, and $ c$ be non-negative real numbers and let $ x$, $ y$, and $ z$ be positive real numbers such that $ a+b+c=x+y+z$. Prove that

$$ \dfrac{a^3}{x^2}+\dfrac{b^3}{y^2}+\dfrac{c^3}{z^2} \ge a+b+c.$$

Hery Susanto, Malang
2010 Indonesia TST
Let $ A=\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S=\{n: n \in A,\gcd \left(n,2009^{2009}\right)=1\}$. Let $ P$ be the product of all elements of $ S$. Prove that $$ P \equiv 1 \pmod{2009^{2009}}.$$

Nanang Susyanto, Jogjakarta
2010 Indonesia TST
In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party?

Yudi Satria, Jakarta
2010 Indonesia TST
Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that $$ \angle ABP=\angle PBQ=\angle QBC,$$

(a) prove that $ ABC$ is a right-angled triangle, and

(b) calculate $ \dfrac{BP}{CH}$.

Soewono, Bandung
2010 Indonesia TST
Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE = \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$,  the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$.

Utari Wijayanti, Bandung
2010 Indonesia TST
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$ f(x^3+y^3)=xf(x^2)+yf(y^2)$$ for all real numbers $ x$ and $ y$.

Hery Susanto, Malang
2010 Indonesia TST
Let $ x$, $ y$, and $ z$ be integers satisfying the equation $$ \dfrac{2008}{41y^2}=\dfrac{2z}{2009}+\dfrac{2007}{2x^2}.$$ Determine the greatest value that $ z$ can take.

Budi Surodjo, Jogjakarta
2010 Indonesia TST
For each positive integer $ n$, define $ f(n)$ as the number of digits $ 0$ in its decimal representation. For example, $ f(2)=0$, $ f(2009)=2$, etc. Please, calculate $$ S=\sum_{k=1}^{n}2^{f(k)},$$ for $ n=9,999,999,999$.

Yudi Satria, Jakarta
2010 Indonesia TST
Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)=41$ and $ f(b)=49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$.

Nanang Susyanto, Jogjakarta
2010 Indonesia TST
Given an equilateral triangle, all points on its sides are colored in one of two given colors. Prove that the is a right-angled triangle such that its three vertices are in the same color and on the sides of the equilateral triangle.

Alhaji Akbar, Jakarta
2010 Indonesia TST
Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1=1$, $ a_2=\dfrac{4}{3}$, and $$ a_{n+1}=\sqrt{1+a_na_{n-1}}, \quad \forall n \ge 2.$$ Prove that for all $ n \ge 2$, $$ a_n^2>a_{n-1}^2+\dfrac{1}{2}$$ and $$ 1+\dfrac{1}{a_1}+\dfrac{1}{a_2}+\dots+\dfrac{1}{a_n}>2a_n.$$

Fajar Yuliawan, Bandung
2010 Indonesia TST
Let $ ABC$ be an acute-angled triangle such that there exist points $ D,E,F$ on side $ BC,CA,AB$, respectively such that the inradii of triangle $ AEF,BDF,CDE$ are all equal to $ r_0$. If the inradii of triangle $ DEF$ and $ ABC$ are $ r$ and $ R$, respectively, prove that $$ r+r_0=R.$$

Soewono, Bandung
2010 Indonesia TST
The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: choose two integers $ a$ and $ b$ such that $ a-b \ge 2$ and replace them with $ a-1$ and $ b+1$. Please, determine the maximum number of steps that can be done.

Yudi Satria, Jakarta
2010 Indonesia TST
Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear.

Rudi Adha Prihandoko, Bandung
2010 Indonesia TST
Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$ x+f(y)=af(y+f(x))$$ for all real numbers $ x$ and $ y$.

Hery Susanto, Malang
2010 Indonesia TST
Prove that for all integers $ m$ and $ n$, the inequality

$$ \dfrac{\phi(\gcd(2^m + 1,2^n + 1))}{\gcd(\phi(2^m + 1),\phi(2^n + 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}$$

holds.

Nanang Susyanto, Jogjakarta
2010 Indonesia TST
Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number?

Nanang Susyanto, Jogjakarta
2010 Indonesia TST
Consider a polynomial with coefficients of real numbers $ \phi(x)=ax^3+bx^2+cx+d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that $$ 2b^3+9a^2d-7abc \le 0.$$

Hery Susanto, Malang
2010 Indonesia TST
Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows:

(1). $ \dfrac{1}{2} \in \mathbb{H}$,

(2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1+x} \in \mathbb{H}$ and also $ \dfrac{x}{1+x} \in \mathbb{H}$.

Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.
2010 Indonesia TST
Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation).

Yudi Satria, Jakarta
2009 Indonesia TST
Prove that for all odd $ n > 1$, we have $ 8n + 4|C^{4n}_{2n}$.
2009 Indonesia TST
Let $ f(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\cdots+a_1x+a_0$, with $ a_i=a_{2n-1}$ for all $ i=1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x+\frac1x\right)x^n=f(x)$.
2009 Indonesia TST
In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.
2009 Indonesia TST
Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly.

a) Prove that $ AA_0,BB_0,CC_0$ are concurrent.

b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2=b^2+c^2$.
2009 Indonesia TST
Given an $ n\times n$ chessboard.

a) Find the number of rectangles on the chessboard.

b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?
2009 Indonesia TST
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that

$$ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.

$$
2009 Indonesia TST
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2+y^2+z^2$ if $ x^3+y^3+z^3-3xyz=1$.
2009 Indonesia TST
Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2+1$.
2009 Indonesia TST
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime?

b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?
2009 Indonesia TST
Consider the following array:

$$ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots

$$ Find the 5-th number on the $ n$-th row with $ n>5$.
2009 Indonesia TST
Let $ ABC$ be an acute triangle with $ \angle BAC=60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB=\angle BPC=\angle CPA=120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$.

a) Prove that $ \angle YXZ=60^{\circ}$

b) Prove that $ X,P,M$ are collinear.
2009 Indonesia TST
Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x+y-z)+f(2\sqrt{xz})+f(2\sqrt{yz})=f(x+y+z)$ for all nonnegative $ x,y,z$ with $ x+y\ge z$.
2009 Indonesia TST
Let $ [a]$ be the integer such that $ [a]\le a<[a]+1$. Find all real numbers $ (a,b,c)$ such that $$ \{a\}+[b]+\{c\}=2.9\\\{b\}+[c]+\{a\}=5.3\\\{c\}+[a]+\{b\}=4.0.$$
2009 Indonesia TST
Let $ ABC$ be a triangle with $ \angle BAC=60^{\circ}$. The incircle of $ ABC$ is tangent to $ AB$ at $ D$. Construct a circle with radius $ DA$ and cut the incircle of $ ABC$ at $ E$. If $ AF$ is an altitude, prove that $ AE\ge AF$.
2009 Indonesia TST
Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n - 1$ divides $ 2^{k(n - 1)! + k^n} - 1$ for all integers $ k > n$.
2009 Indonesia TST
2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.
2009 Indonesia TST
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.
2009 Indonesia TST
Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that $$ \frac{x_1x_2}{x_3}+\frac{x_2x_3}{x_4}+\cdots+\frac{x_nx_1}{x_2}\ge4n$$ and determine when the equality holds.
2009 Indonesia TST
Let $ S=\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x+y\in A$ or $ x+y-n\in A$. Show that the number of elements of $ A$ divides $ n$.
2009 Indonesia TST
Let $ ABCD$ be a convex quadrilateral. Let $ M,N$ be the midpoints of $ AB,AD$ respectively. The foot of perpendicular from $ M$ to $ CD$ is $ K$, the foot of perpendicular from $ N$ to $ BC$ is $ L$. Show that if $ AC,BD,MK,NL$ are concurrent, then $ KLMN$ is a cyclic quadrilateral.
2021 Iran Team Selection Test
In acute scalene triangle $ABC$  the external angle bisector of $\angle BAC$ meet $BC$ at point $X$.Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$.The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$. Show that $ND$ is angle bisector of $\angle YNZ$.

Proposed by Alireza Haghi
2021 Iran Team Selection Test
In the simple and connected graph $G$ let $x_i$ be the number of vertices with degree $i$. Let $d>3$ be the biggest degree in the graph $G$. Prove that if :

$$x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1$$Then there exists a vertex with degree $d$ such that after removing that vertex the graph $G$ is still connected.



Proposed by Ali Mirzaie
2021 Iran Team Selection Test
There exist $4$ positive integers $a,b,c,d$ such that $abcd \neq 1$ and each pair of them have a GCD of $1$. Two functions $f,g : \mathbb{N} \rightarrow \{0,1\}$ are multiplicative functions such that for each positive integer $n$ we have :

$$f(an+b)=g(cn+d)$$Prove that at least one of the followings hold.

$i)$ for each positive integer $n$ we have $f(an+b)=g(cn+d)=0$

$ii)$ There exists a positive integer $k$ such that for all $n$ where $(n,k)=1$ we have $g(n)=f(n)=1$



(Function $f$ is multiplicative if for any natural numbers $a,b$ we have $f(ab)=f(a)f(b)$)



Proposed by Navid Safaii
2021 Iran Team Selection Test
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have :

$$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$Who has the winning strategy.



Proposed by Alireza Haghi
2021 Iran Team Selection Test
Call a triple of numbers Nice if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$  Nice triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios



Proposed by Morteza Saghafian
2021 Iran Team Selection Test
Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$, also $A,X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$



Proposed by Alireza Danaie
2021 Iran Team Selection Test
Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.



(Two cells of the grid are adjacent if they have a common vertex)
2021 Iran Team Selection Test
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have :

$$f(n)+1400m^2|n^2+f(f(m))$$
2021 Iran Team Selection Test
Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have:

$$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$$$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$Then we have :

$$bP(\frac{a}{c})=dQ(\frac{a}{c})$$

(Two polynomials are relatively prime if they don't have a common root)



Proposed by Navid Safaii and Alireza Haghi
2021 Iran Team Selection Test
Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds :



$$f(ac)+f(bc)-f(c)f(ab) \ge 1$$

Proposed by Mojtaba Zare
2021 Iran Team Selection Test
Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$.

Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$.



Proposed by Alireza Dadgarnia
2021 Iran Team Selection Test
Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no  $3$ subsets $A,B,C$ with the same color where :

$$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$

Proposed by Morteza Saghafian and Amir Jafari
2020 Iran Team Selection Test
A weighted complete graph with distinct positive wights is given such that in every triangle is degenerate   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.



Proposed by Morteza Saghafian
2020 Iran Team Selection Test
Let $O$ be the circumcenter of the triangle $ABC$. Points $D,E$ are on sides $AC,AB$ and points $P,Q,R,S$ are given in plane such that $P,C$ and $R,C$ are on different sides of $AB$ and pints $Q,B$ and $S,B$ are on different sides of $AC$ such that $R,S$ lie on circumcircle of $DAP,EAQ$ and $\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP$(In that order), $\angle ARE=\angle ASD=\angle BAC$, If $RS\| PQ$ prove that $RE ,DS$ are concurrent on $AO$.



Proposed by Alireza Dadgarnia
2020 Iran Team Selection Test
We call a number $n$ interesting if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that:

$i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$

$ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$

$iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$

Find all interesting $n$.



Proposed by Mojtaba Zare Bidaki
2020 Iran Team Selection Test
Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$.



Proposed by Ali Zamani
2020 Iran Team Selection Test
Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that

\begin{align*}

n+s(2n)=m+s(2m) \\
kn+s(n^2)=km+s(m^2).

\end{align*}

Proposed by Mohammadamin Sharifi
2020 Iran Team Selection Test
$n$ 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?



Proposed by Morteza Saghafian
2020 Iran Team Selection Test
We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ square-free mod n if there dose not 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 square-free mod p $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$.



Proposed by Masud Shafaie
2020 Iran Team Selection Test
Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees?



Proposed by Seyed Reza Hosseini
2020 Iran Team Selection Test
Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$  and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic.



Proposed by Alireza Dadgarnia and Amir Parsa Hosseini
2020 Iran Team Selection Test
Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent.



Proposed by Alireza Dadgarnia
2020 Iran Team Selection Test
For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$:

$$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$

Proposed by Mohammad Amin Sharifi
2020 Iran Team Selection Test
$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that

$$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$

Proposed by Ali Partofard
2019 Iran Team Selection Test
A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.

On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one?



Proposed by Morteza Saghafian
2019 Iran Team Selection Test
$a, a_1,a_2,\dots ,a_n$ 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 $a_1,\dots ,a_n$

Proposed by Mohsen Jamali
2019 Iran Team Selection Test
Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$.



Proposed by Siamak Ahmadpour