Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 Iran MO (3rd Round)
Let $a,b,c\in\mathbb R^{n}, a+b+c=0$ and $\lambda>0$. Prove that $$\prod_{cycle}\frac{|a|+|b|+(2\lambda+1)|c|}{|a|+|b|+|c|}\geq(2\lambda+3)^{3}$$
2002 Iran MO (3rd Round)
$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that $$f(a+\frac1{f(a)})<2f(a)$$
2002 Iran MO (3rd Round)
$a_{n}$ is a sequence that $a_{1}=1,a_{2}=2,a_{3}=3$, and $$a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}$$ Prove that for each natural $n$, $a_{n}$ is integer.
2002 Iran MO (3rd Round)
$a_{n}$ ($n$ is integer) is a sequence from positive reals that $$a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4$$ Prove $a_{n}$ is constant.
2002 Iran MO (3rd Round)
$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.
2002 Iran MO (3rd Round)
$M$ is midpoint of $BC$.$P$ is an arbitary point on $BC$.

$C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$

Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P.

$C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$.

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Prove : $$r_{1}+r_{2}+r_{3}=R$$ ($R$ radius of big circle)
2002 Iran MO (3rd Round)
In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?
2002 Iran MO (3rd Round)
Circles $C_{1}$ and $C_{2}$ are tangent to each other at $K$ and are tangent to circle $C$ at $M$ and $N$. External tangent of $C_{1}$ and $C_{2}$ intersect $C$ at $A$ and $B$. $AK$ and $BK$  intersect with circle $C$ at $E$ and $F$ respectively. If AB is diameter of $C$, prove that $EF$ and $MN$ and $OK$ are concurrent. ($O$ is center of circle $C$.)
2002 Iran MO (3rd Round)
Let $ M$ and $ N$ be points on the side $ BC$ of triangle $ ABC$, with the point $ M$ lying on the segment $ BN$, such that $ BM = CN$. Let $ P$ and $ Q$ be points on the segments $ AN$ and $ AM$, respectively, such that $ \measuredangle PMC =\measuredangle MAB$ and $ \measuredangle QNB =\measuredangle NAC$. Prove that $ \measuredangle QBC =\measuredangle PCB$.
2002 Iran MO (3rd Round)
$H,I,O,N$ are orthogonal center, incenter, circumcenter, and Nagelian point of triangle $ABC$. $I_{a},I_{b},I_{c}$ are excenters of $ABC$ corresponding vertices $A,B,C$. $S$ is point that $O$ is midpoint of $HS$. Prove that centroid of triangles $I_{a}I_{b}I_{c}$ and $SIN$ concide.
2002 Iran MO (3rd Round)
In an $m\times n$ table there is a policeman in cell $(1,1)$, and there is a thief in cell $(i,j)$. A move is going from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move, then the policeman moves and ... For which $(i,j)$ the policeman can catch the thief?
2002 Iran MO (3rd Round)
We have a bipartite graph $G$ (with parts $X$ and $Y$). We orient each edge arbitrarily. Hessam chooses a vertex at each turn and reverse the orientation of all edges that $v$ is one of their endpoint. Prove that with these steps we can reach to a graph that for each vertex $v$ in part $X$, $\deg^{+}(v)\geq \deg^{-}(v)$ and for each vertex in part $Y$, $\deg^{+}v\leq \deg^{-}v$
2002 Iran MO (3rd Round)
$f,g$ are two permutations of set $X=\{1,\dots,n\}$. We say $f,g$ have common points iff there is a $k\in X$ that $f(k)=g(k)$.

a) If $m>\frac{n}{2}$, prove that there are $m$ permutations $f_{1},f_{2},\dots,f_{m}$ from $X$ that for each permutation $f\in X$, there is an index $i$ that $f,f_{i}$ have common points.

b) Prove that if $m\leq\frac{n}{2}$, we can not find permutations $f_{1},f_{2},\dots,f_{m}$ satisfying the above condition.
2002 Iran MO (3rd Round)
A subset $S$ of $\mathbb N$ is eventually linear iff there are $k,N\in\mathbb N$ that for $n>N,n\in S\Longleftrightarrow k|n$. Let $S$ be a subset of $\mathbb N$ that is closed under addition. Prove that $S$ is eventually linear.
2002 Iran MO (3rd Round)
Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)
2002 Iran MO (3rd Round)
For positive $a,b,c$, $$a^{2}+b^{2}+c^{2}+abc=4$$ Prove $a+b+c \leq3$
2002 Iran MO (3rd Round)
Find the smallest natural number $n$ that the following statement holds :

Let $A$ be a finite subset of $\mathbb R^{2}$. For each $n$ points in $A$ there are two lines including these $n$ points. All of the points lie on two lines.
2002 Iran MO (3rd Round)
Find all continious $f: \mathbb R\longrightarrow\mathbb R$ that for any $x,y$ $$f(x)+f(y)+f(xy)=f(x+y+xy)$$
2002 Iran MO (3rd Round)
$I$ is incenter of triangle $ABC$. Incircle of $ABC$ touches $AB,AC$ at $X,Y$. $XI$ intersects incircle at $M$. Let $CM\cap AB=X'$. $L$ is a point on the segment $X'C$ that $X'L=CM$. Prove that $A,L,I$ are collinear iff $AB=AC$.
2002 Iran MO (3rd Round)
$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$

$m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.
2002 Iran MO (3rd Round)
Excircle of triangle $ABC$ corresponding vertex $A$, is tangent to $BC$ at $P$. $AP$ intersects circumcircle of $ABC$ at $D$. Prove $$r(PCD)=r(PBD)$$ whcih $r(PCD)$ and $r(PBD)$ are inradii of triangles $PCD$ and $PBD$.
2002 Iran MO (3rd Round)
15000 years ago Tilif ministry in Persia decided to define a code for $n\geq2$ cities. Each code is a sequence of $0,1$ such that no code start with another code. We know that from $2^{m}$ calls from foreign countries to Persia $2^{m-a_{i}}$ of them where from the $i$-th city (So $\sum_{i=1}^{n}\frac1{2^{a_{i}}}=1$). Let $l_{i}$ be length of code assigned to $i$-th city. Prove that $\sum_{i=1}^{n}\frac{l_{i}}{2^{i}}$ is minimum iff $\forall i,\ l_{i}=a_{i}$
2002 Iran MO (3rd Round)
Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.
2002 Iran MO (3rd Round)
$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$)

Prove that $P$ and $I$ and $D$ are on a line.
2002 Iran MO (3rd Round)
An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edge the he moves on a straight line on cube's net. Also if he reaches to a vertex he will return his path.

a) Prove that for each beginning point ant can has infinitely many choices for his direction that its path becomes periodic.

b) Prove that if if the ant starts from point $A$ and its path is periodic, then for each point $B$ if ant starts with this direction, then his path becomes periodic.
2001 Iran MO (3rd Round)
Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that:

$ f(x)+f(\frac1x)=1$

$ 2f(f(x))=f(2x)$
2001 Iran MO (3rd Round)
Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: $$ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m$$
2000 Iran MO (3rd Round)
Does there exist a natural number $N$ which is a power of$2$, such that one

can permute its decimal digits to obtain a different power of $2$?
2000 Iran MO (3rd Round)
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.
2000 Iran MO (3rd Round)
In a deck of $n > 1$ cards, some digits from $1$ to$8$are written on each card.

A digit may occur more than once, but at most once on a certain card.

On each card at least one digit is written, and no two cards are denoted

by the same set of digits. Suppose that for every $k=1,2,\dots,7$ digits, the

number of cards that contain at least one of them is even. Find $n$.
2000 Iran MO (3rd Round)
A sequence of natural numbers $c_1, c_2,\dots$ is called perfect if every natural

number $m$ with $1\lem\le c_1 +\dots+ c_n$ can be represented as

$m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}$

Given $n$, find the maximum possible value of $c_n$ in a perfect sequence $(c_i)$.
2000 Iran MO (3rd Round)
Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$.  Prove that $ MN = AE + AF$.
2000 Iran MO (3rd Round)
Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 + a'^2}{3}}$.
2000 Iran MO (3rd Round)
Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.
2000 Iran MO (3rd Round)
Let $A$ and $B$ be arbitrary finite sets and let $f: A\longrightarrow B$ and $g: B\longrightarrow A$

be functions such that $g$ is not onto. Prove that there is a subset $S$ of $A$ such that

$\frac{A}{S}=g(\frac{B}{f(S)})$.
2000 Iran MO (3rd Round)
Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and

$f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$



$(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$.

$(b)$ Determine$f$.
2000 Iran MO (3rd Round)
Let us denote $\prod = \{(x, y) | y > 0\}$. We call a semicircle in $\prod$ with

center on the $x-\text{axis}$ a semi-line. Two intersecting semi-lines determine

four semi-angles. A bisector of a semi-angle is a semi-line that bisects

the semi-angle. Prove that in every semi-triangle (determined by three

semi-lines) the bisectors are concurrent.
2000 Iran MO (3rd Round)
Find all f:N $\longrightarrow$ N that:





a) $f(m)=1 \Longleftrightarrow m=1 $

b) $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $

c) $ f^{2000}(m)=f(m) $
2000 Iran MO (3rd Round)
Let $n$ points be given on a circle, and let $nk + 1$ chords between these points be drawn, where $2k+1 < n$. Show that it is possible to select $k+1$ of the chords so that no two of them intersect.
2000 Iran MO (3rd Round)
In a tennis tournament where $ n$ players $ A_1,A_2,\dots,A_n$ take part, any two

players play at most one match, and $ k \leq \frac {n(n - 1)}{2}$

$ 2$ matches are played. The winner of a match gets $ 1$ point while the loser gets $ 0$. Prove that a sequence

$ d_1,d_2,\dots,d_n$ of nonnegative integers can be the sequence of scores of the

players ($ d_i$ being the score of$ A_i$) if and only if

$ (i)\ \ d_1 + d_2 + \dots + d_n = k$, and

$ (ii)\ \text{for any} X\subset\{A_1,\dots,A_n\}$, the number of matches between the players in $ X$ is at most $ \sum_{A_j\in X}d_j$
2000 Iran MO (3rd Round)
Isosceles triangles $A_3A_1O_2$ and $A_1A_2O_3$ are constructed on the sides of

a triangle $A_1A_2A_3$ as the bases, outside the triangle. Let $O_1$ be a point

outside $\Delta A_1A_2A_3$ such that

$\angle O_1A_3A_2 =\frac 12\angle A_1O_3A_2$ and $\angle O_1A_2A_3 =\frac 12\angle A_1O_2A_3$.

Prove that $A_1O_1\perp O_2O_3$, and if $T$ is the projection of $O_1$ onto $A_2A_3$,

then $\frac{A_1O_1}{O_2O_3} = 2\frac{O_1T}{A_2A_3}$.
2000 Iran MO (3rd Round)
A circle$\Gamma$ with radius $R$ and center $\omega$, and a line $d$ are drawn on a plane,

such that the distance of $\omega$ from $d$ is greater than $R$. Two points $M$ and

$N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\Gamma$. Prove

that there is a point $P$ in the plane from which all the segments $MN$ are

visible at a constant angle.
2000 Iran MO (3rd Round)
Let $n$ be a positive integer. Suppose $S$ is a set of ordered $n-\mbox{tuples}$ of

nonnegative integers such that, whenever $(a_1,\dots,an)\in S$ and $b_i$ are nonnegative integers with$b_i\le a_i$, the $n-\text{tuple}$ $(b_1,\dots,b_n)$ is also in $S$. If $h_m$

is the number of elements of $S$ with the sum of components equal to$m$,

prove that $h_m$ is a polynomial in $m$ for all sufficiently large$m$.
2000 Iran MO (3rd Round)
Suppose that $a, b, c$ are real numbers such that for all positive numbers

$x_1,x_2,\dots,x_n$ we have



$(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$



Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors

$(-2,1,0)$ and $(-1,2,-1)$.
2000 Iran MO (3rd Round)
Prove that for every natural number $ n$  there exists a polynomial $ p(x)$ with

integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .
1998 Iran MO (3rd Round)
Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$

$$x_n=\{\begin{array}{cc}x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2) .\end{array}$$

where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.
1998 Iran MO (3rd Round)
Let $ABCD$ be a convex pentagon such that

$$\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.$$

Let $F$ be a point on AB such that $AF:BF=AE:BC$. Show that

$$\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.$$
1998 Iran MO (3rd Round)
Let $A,B$ be two matrices with positive integer entries such that sum of entries of a row in $A$ is equal to  sum of entries of the same row in $B$ and  sum of entries of a column in $A$ is equal to  sum of entries of the same column in $B$. Show that there exists a sequence of matrices $A_1,A_2,A_3,\cdots , A_n$ such that all entries of the matrix $A_i$ are positive integers and in the sequence

$$A=A_0,A_1,A_2,A_3,\cdots , A_n=B,$$

for each index $i$, there exist indexes $k,j,m,n$ such that

$$\begin{array}{*{20}{c}}

   \\ 

  {{A_{i + 1}} - {A_{i}} = } 

\end{array}\begin{array}{*{20}{c}}

  {\begin{array}{*{20}{c}}

 \quad \quad \ \ j& \ \ \ {k} 

\end{array}} \\ 

  {\begin{array}{*{20}{c}}

  m \\ 

  n 

\end{array}\left( {\begin{array}{*{20}{c}}

  { + 1}&{ - 1} \\ 

  { - 1}&{ + 1} 

\end{array}} \right)} 

\end{array} \ \text{or} \ \begin{array}{*{20}{c}}

  {\begin{array}{*{20}{c}}

 \quad \quad \ \ j& \ \ \ {k} 

\end{array}} \\ 

  {\begin{array}{*{20}{c}}

  m \\ 

  n 

\end{array}\left( {\begin{array}{*{20}{c}}

  { - 1}&{ + 1} \\ 

  { + 1}&{ - 1} 

\end{array}} \right)} 

\end{array}.$$

That is, all indices of ${A_{i + 1}} - {A_{i}}$ are zero, except the indices $(m,j), (m,k), (n,j)$, and $(n,k)$.
1998 Iran MO (3rd Round)
Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$,



(i) $mf(f(m))=\left( f(m) \right)^2$,

(ii) If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$,

(iii) $f(m)=m$ if and only if $m=1$.
1998 Iran MO (3rd Round)
Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.
1998 Iran MO (3rd Round)
Let $n(r)$ be the maximum possible number of points with integer coordinates on a circle with radius $r$ in Cartesian plane. Prove that $n(r) < 6\sqrt[3]{3 \pi r^2}.$
1998 Iran MO (3rd Round)
A one-player game is played on a $m \times n$ table with $m \times n$ nuts. One of the nuts' sides is black, and the other side of them is white. In the beginning of the game, there is one nut in each cell of the table and all nuts have their white side upwards except one cell in one corner of the table which has the black side upwards. In each move, we should remove a nut which has its black side upwards from the table and reverse all nuts in adjacent cells (i.e. the cells which share a common side with the removed nut's cell). Find all pairs $(m,n)$ for which we can remove all nuts from the table.
1998 Iran MO (3rd Round)
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that

$$ \angle MAB = \angle NAC\quad \mbox{and}\quad \angle MBA = \angle NBC.

$$

Prove that

$$ \frac {AM \cdot AN}{AB \cdot AC} + \frac {BM \cdot BN}{BA \cdot BC} + \frac {CM \cdot CN}{CA \cdot CB} = 1.

$$
1998 Iran MO (3rd Round)
Find all functions $f : \mathbb R \to \mathbb R$ such that for all $x, y,$

$$f(f(x) + y) = f(x^2 - y) + 4f(x)y.$$
1998 Iran MO (3rd Round)
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
1998 Iran MO (3rd Round)
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE$ and $EF = FA$. Prove that

$$\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.$$
1998 Iran MO (3rd Round)
Let $ABC$ be a given triangle. Consider any painting of points of the plane in red and green. Show that there exist either two red points on the distance $1$, or three green points forming a triangle congruent to triangle $ABC$.
1998 Iran MO (3rd Round)
Let be given $r_1,r_2,\ldots,r_n \in \mathbb R$. Show that there exists a subset $I$ of $\{1,2,\ldots,n \}$ which which has one or two elements in common with the sets $\{i,i + 1,i + 2\} , (1 \leq i \leq n- 2)$ such that

$$\left| {\mathop \sum \limits_{i \in I} {r_i}} \right| \geqslant \frac{1}{6}\mathop \sum \limits_{i = 1}^n \left| {{r_i}} \right|.$$
1998 Iran MO (3rd Round)
In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$
1998 Iran MO (3rd Round)
For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows:



- $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$;



- $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$.



Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.
1997 Iran MO (3rd Round)
Find all strictly ascending functions $f$ such that for all $x\in \mathbb R$,

$$f(1-x)=1-f(f(x)).$$
1997 Iran MO (3rd Round)
Show that for any arbitrary triangle $ABC$, we have

$$\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.$$
1997 Iran MO (3rd Round)
There are $30$ bags and there are $100$ similar coins in each bag (coins in each bag are similar, coins of different bags can be different). The weight of each coin is an one digit number in grams. We have a digital scale which can weigh at most $999$ grams in each weighing. Using this scale, we want to find the weight of coins of each bag.



(a) Show that this operation is possible by $10$ times of weighing, and



(b) It's not possible by $9$ times of weighing.
1997 Iran MO (3rd Round)
Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.
1997 Iran MO (3rd Round)
Let $ABC$ and $XYZ$ be two triangles. Define

$$A_1=BC\cap ZX, A_2=BC\cap XY,$$$$B_1=CA\cap XY, B_2=CA\cap YZ,$$$$C_1=AB\cap YZ, C_2=AB\cap ZX.$$

Hereby, the abbreviation $g\cap h$ means the point of intersection of two lines $g$ and $h$.



Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.
1997 Iran MO (3rd Round)
Let $d$ be a real number such that $d^2=r^2+s^2$, where $r$ and $s$ are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance $d$ have different colors.
1997 Iran MO (3rd Round)
Suppose that $a, b, x$ are positive integers such that

$$x^{a+b}=a^bb$$

Prove that $a=x$ and $b=x^x$.
1997 Iran MO (3rd Round)
In an acute triangle $ABC$, points $D,E,F$ are the feet of the altitudes from $A,B,C$, respectively. A line through $D$ parallel to $EF$ meets $AC$ at $Q$ and $AB$ at $R$. Lines $BC$ and $EF$ intersect at $P$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of $BC$.
1997 Iran MO (3rd Round)
Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$. We consider pairwise distances between numbers in $S$. If every distance that appears, except the distance $1$, occurs at least twice, prove that all the $x_i$ are rational.
1997 Iran MO (3rd Round)
Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that

$$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.$$
1997 Iran MO (3rd Round)
In an acute triangle $ABC$ let $AD$ and $BE$ be altitudes, and $AP$ and $BQ$ be bisectors. Let $I$ and $O$ be centers of incircle and circumcircle, respectively. Prove that the points $D, E$, and $I$ are collinear if and only if the points $P, Q$, and $O$ are collinear.
1997 Iran MO (3rd Round)
Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$, one can move from $A$ to $B$ if the distance $AB$ is $1$. Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$.
1996 Iran MO (3rd Round)
Let $a,b,c,d$ be positive real numbers. Prove that

$$\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.$$
1996 Iran MO (3rd Round)
Let $ABCD$ be a parallelogram. Construct the equilateral triangle $DCE$ on the side $DC$ and outside of parallelogram. Let $P$ be an arbitrary point in plane of $ABCD$. Show that

$$PA+PB+AD \geq PE.$$
1996 Iran MO (3rd Round)
Find all sets of real numbers $\{a_1,a_2,\ldots, _{1375}\}$ such that

$$2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \quad \forall n \in \{1,2,\ldots,1374\},$$

and

$$2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.$$
1996 Iran MO (3rd Round)
Show that there doesn't exist two infinite and separate sets $A,B$ of points such that



(i) There are no three collinear points in $A \cup B$,



(ii) The distance between every two points in $A \cup B$ is at least $1$, and



(iii) There exists at least one point belonging to set $B$ in interior of each triangle which all of its vertices are chosen from the set $A$, and there exists at least one point belonging to set $A$ in interior of each triangle which all of its vertices are chosen from the set $B$.
1996 Iran MO (3rd Round)
Find all non-negative integer solutions of the equation

$$2^x + 3^y = z^2 .$$
1996 Iran MO (3rd Round)
Let $ABCD$ be a convex quadrilateral. Construct the points $P,Q,R,$ and $S$ on continue of $AB,BC,CD,$ and $DA$, respectively, such that

$$BP=CQ=DR=AS.$$

Show that if $PQRS$ is a square, then $ABCD$ is also a square.
1996 Iran MO (3rd Round)
Let $a_1 \geq a_2 \geq \cdots \geq a_n$ be $n$ real numbers such that $a_1^k +a_2^k + \cdots + a_n^k \geq 0$ for all positive integers $k$. Suppose that $p=\max\{|a_1|,|a_2|, \ldots,|a_n|\}$. Prove that $p=a_1$, and

$$(x-a_1)(x-a_2)\cdots(x-a_n)\leq x^n-a_1^n \qquad \forall x>a_1.$$