Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 Iran MO (3rd Round)
Prove that there is no polynomial $P \in \mathbb C[x]$ such that set $\left \{ P(z) \; | \; \left | z \right | =1 \right \}$ in complex plane forms a polygon. In other words, a complex polynomial can't map the unit circle to a polygon.

(30 points)
2013 Iran MO (3rd Round)
Let $p$ a prime number and $d$ a divisor of $p-1$. Find the product of elements in $\mathbb Z_p$ with order $d$. ($\mod p$).

(10 points)
2013 Iran MO (3rd Round)
Suppose that $a,b$ are two odd positive integers such that $2ab+1 \mid a^2 + b^2 + 1$. Prove that $a=b$.

(15 points)
2013 Iran MO (3rd Round)
Let $p>3$ a prime number. Prove that there exist $x,y \in \mathbb Z$ such that $p = 2x^2 + 3y^2$ if and only if $p \equiv 5, 11 \;  (\mod 24)$

(20 points)
2013 Iran MO (3rd Round)
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation:

$$x^2 - (n^2 +1)y^2 = n^2.$$

(25 points)
2013 Iran MO (3rd Round)
$p=3k+1$ is a prime number. For each $m \in \mathbb Z_p$, define function $L$ as follow:

$L(m) = \sum_{x \in \mathbb{Z}_p}^{ } \left ( \frac{x(x^3 + m)}{p} \right )$

a) For every $m \in \mathbb Z_p$ and $t \in {\mathbb Z_p}^{*}$ prove that $L(m) = L(mt^3)$. (5 points)



b) Prove that there is a partition of ${\mathbb Z_p}^{*} = A \cup B \cup C$ such that $|A| = |B| = |C| = \frac{p-1}{3}$ and $L$ on each set is constant.

Equivalently there are $a,b,c$ for which $L(x) = \left\{\begin{matrix}

a &  &  &x \in A \\ 

 b&  &  &x \in B \\ 

 c&  &  & x \in C

\end{matrix}\right.$ . (7 points)



c) Prove that $a+b+c = -3$. (4 points)



d) Prove that $a^2 + b^2 + c^2 = 6p+3$. (12 points)



e) Let $X= \frac{2a+b+3}{3},Y= \frac{b-a}{3}$, show that $X,Y \in \mathbb Z$ and also show that :$p= X^2 + XY +Y^2$. (2 points)



(${\mathbb Z_p}^{*} =  \mathbb Z_p \setminus  \{0\}$)
2013 Iran MO (3rd Round)
Let $ABCDE$ be a pentagon inscribe in a circle $(O)$. Let $ BE \cap AD = T$. Suppose the parallel line with $CD$ which passes through $T$ which cut $AB,CE$ at $X,Y$. If $\omega$ be the circumcircle of triangle $AXY$ then prove that $\omega$ is tangent to $(O)$.
2013 Iran MO (3rd Round)
Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.
2013 Iran MO (3rd Round)
Suppose line $\ell$ and four points $A,B,C,D$ lies on $\ell$. Suppose that circles $\omega_1 , \omega_2$ passes through $A,B$ and circles $\omega'_1 , \omega'_2$ passes through $C,D$. If $\omega_1 \perp \omega'_1$ and $\omega_2 \perp \omega'_2$ then prove that lines $O_1O'_2 , O_2O'_1 , \ell $ are concurrent where $O_1,O_2,O'_1,O'_2$ are center of $\omega_1 , \omega_2 , \omega'_1 , \omega'_2$.
2013 Iran MO (3rd Round)
In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.
2013 Iran MO (3rd Round)
Let $ABC$ be triangle with circumcircle $(O)$. Let $AO$ cut $(O)$ again at $A'$. Perpendicular bisector of $OA'$ cut $BC$ at $P_A$. $P_B,P_C$ define similarly. Prove that :



I) Point $P_A,P_B,P_C$ are collinear.



II ) Prove that the distance of $O$ from this line is equal to $\frac {R}{2}$ where $R$ is the radius of the circumcircle.
2013 Iran MO (3rd Round)
Assume that the following generating function equation is correct, prove the following statement:

$\Pi_{i=1}^{\infty} (1+x^{3i})\Pi_{j=1}^{\infty} (1-x^{6j+3})=1$

Statement: The number of partitions of $n$ to numbers not of the form $6k+1$ or $6k-1$ is equal to the number of partitions of $n$ in which each summand appears at least twice.

(10 points)

Proposed by Morteza Saghafian
2013 Iran MO (3rd Round)
How many rooks can be placed in an $n\times n$ chessboard such that each rook is threatened by at most $2k$ rooks?

(15 points)

Proposed by Mostafa Einollah zadeh
2013 Iran MO (3rd Round)
$n$ cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order.

A set of overtaking actions is called "small" if any car overtakes at most once.

A set of overtaking actions is called "complete" if any car overtakes exactly once.

If $F$ is the set of all possible orders of the cars after a small set of overtaking actions and $G$ is the set of all possible orders of the cars after a complete set of overtaking actions, prove that

$$\mid F\mid=2\mid G\mid$$

(20 points)

Proposed by Morteza Saghafian
2013 Iran MO (3rd Round)
We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles.

We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments.



(25 points)

Proposed by Amirali Moinfar
2013 Iran MO (3rd Round)
Consider a graph with $n$ vertices and $\frac{7n}{4}$ edges.

(a) Prove that there are two cycles of equal length.

(25 points)

(b) Can you give a smaller function than $\frac{7n}{4}$ that still fits in part (a)? Prove your claim.

We say function $a(n)$ is smaller than $b(n)$ if there exists an $N$ such that for each $n>N$ ,$a(n)<b(n)$

(At most 5 points)

Proposed by Afrooz Jabal'ameli
2013 Iran MO (3rd Round)
An $n$-stick is a connected figure consisting of $n$ matches of length $1$ which are placed horizontally or vertically and no two touch each other at points other than their ends. Two shapes that can be transformed into each other by moving, rotating or flipping are considered the same.

An $n$-mino is a shape which is built by connecting $n$ squares of side length 1 on their sides such that there's a path on the squares between each two squares of the $n$-mino.

Let $S_n$ be the number of $n$-sticks and $M_n$ the number of $n$-minos, e.g. $S_3=5$ And $M_3=2$.

(a) Prove that for any natural $n$, $S_n \geq M_{n+1}$.

(b) Prove that for large enough $n$ we have $(2.4)^n \leq S_n \leq (16)^n$.

A grid segment is a segment on the plane of length 1 which it's both ends are integer points. A polystick is called wise if using it and it's rotations or flips we can cover all grid segments without overlapping, otherwise it's called unwise.

(c) Prove that there are at least $2^{n-6}$ different unwise $n$-sticks.

(d) Prove that any polystick which is in form of a path only going up and right is wise.

(e) Extra points: Prove that for large enough $n$ we have $3^n \leq S_n \leq 12^n$



Time allowed for this exam was 2 hours.
2013 Iran MO (3rd Round)
We define the distance between two circles $\omega ,\omega '$by the length of the common external tangent of the circles and show it by $d(\omega , \omega ')$. If two circles doesn't have a common external tangent then the distance between them is undefined. A point is also a circle with radius $0$ and the distance between two cirlces can be zero.

(a) Centroid. $n$ circles $\omega_1,\dots, \omega_n$ are fixed on the plane. Prove that there exists a unique circle $\overline \omega$ such that for each circle $\omega$ on the plane the square of distance between $\omega$ and $\overline \omega$ minus the sum of squares of distances of $\omega$ from each of the $\omega_i$s $1\leq i \leq n$ is constant, in other words:$$d(\omega,\overline \omega)^2-\frac{1}{n}{\sum_{i=1}}^n d(\omega_i,\omega)^2= constant$$

(b) Perpendicular Bisector. Suppose that the circle $\omega$ has the same distance from $\omega_1,\omega_2$. Consider $\omega_3$ a circle tangent to both of the common external tangents of $\omega_1,\omega_2$. Prove that the distance of $\omega$ from centroid of $\omega_1 , \omega_2$ is not more than the distance of $\omega$ and $\omega_3$. (If the distances are all defined)

(c) Circumcentre. Let $C$ be the set of all circles that each of them has the same distance from fixed circles $\omega_1,\omega_2,\omega_3$. Prove that there exists a point on the plane which is the external homothety center of each two elements of $C$.

(d) Regular Tetrahedron. Does there exist 4 circles on the plane which the distance between each two of them equals to $1$?



Time allowed for this problem was 150 minutes.
2013 Iran MO (3rd Round)
Real function $f$ generates real function $g$ if there exists a natural $k$ such that $f^k=g$ and we show this by $f \rightarrow g$. In this question we are trying to find some properties for relation $\rightarrow$, for example it's trivial that if $f \rightarrow g$ and $g \rightarrow h$ then $f \rightarrow h$.(transitivity)



(a) Give an example of two real functions $f,g$ such that $f\not = g$ ,$f\rightarrow g$ and $g\rightarrow f$.

(b) Prove that for each real function $f$ there exists a finite number of real functions $g$ such that $f \rightarrow g$ and $g \rightarrow f$.

(c) Does there exist a real function $g$ such that no function generates it, except for $g$ itself?

(d) Does there exist a real function which generates both $x^3$ and $x^5$?

(e) Prove that if a function generates two polynomials of degree 1 $P,Q$ then there exists a polynomial $R$ of degree 1 which generates $P$ and $Q$.



Time allowed for this problem was 75 minutes.
2013 Iran MO (3rd Round)
A polygon $A$ that doesn't intersect itself and has perimeter $p$ is called Rotund if for each two points $x,y$ on the sides of this polygon which their distance on the plane is less than $1$ their distance on the polygon is at most $\frac{p}{4}$. (Distance on the polygon is the length of smaller path between two points on the polygon)

Now we shall prove that we can fit a circle with radius $\frac{1}{4}$ in any rotund polygon.

The mathematicians of two planets earth and Tarator have two different approaches to prove the statement. In both approaches by "inner chord" we mean a segment with both endpoints on the polygon, and "diagonal" is an inner chord with vertices of the polygon as the endpoints.

Earth approach: Maximal Chord

We know the fact that for every polygon, there exists an inner chord $xy$ with a length of at most 1 such that for any inner chord $x'y'$ with length of at most 1 the distance on the polygon of $x,y$ is more than the distance on the polygon of $x',y'$. This chord is called the maximal chord.

On the rotund polygon $A_0$ there's two different situations for maximal chord:

(a) Prove that if the length of the maximal chord is exactly $1$, then a semicircle with diameter maximal chord fits completely inside $A_0$, so we can fit a circle with radius $\frac{1}{4}$ in $A_0$.

(b) Prove that if the length of the maximal chord is less than one we still can fit a circle with radius $\frac{1}{4}$ in $A_0$.

Tarator approach: Triangulation

Statement 1: For any polygon that the length of all sides is less than one and no circle with radius $\frac{1}{4}$ fits completely inside it, there exists a triangulation of it using diagonals such that no diagonal with length more than $1$ appears in the triangulation.

Statement 2: For any polygon that no circle with radius $\frac{1}{4}$ fits completely inside it, can be divided into triangles that their sides are inner chords with length of at most 1.

The mathematicians of planet Tarator proved that if the second statement is true, for each rotund polygon there exists a circle with radius $\frac{1}{4}$ that fits completely inside it.

(c) Prove that if the second statement is true, then for each rotund polygon there exists a circle with radius $\frac{1}{4}$ that fits completely inside it.

They found out that if the first statement is true then the second statement is also true, so they put a bounty of a  doogh on proving the first statement. A young earth mathematician named J.N., found a counterexample for statement 1, thus receiving the bounty.

(d) Find a 1392-gon that is counterexample for statement 1.

But the Tarators are not disappointed and they are still trying to prove the second statement.

(e) (Extra points) Prove or disprove the second statement.



Time allowed for this problem was 150 minutes.
2013 Iran MO (3rd Round)
A subsum of $n$ real numbers $a_1,\dots,a_n$ is a sum of elements of a subset of the set $\{a_1,\dots,a_n\}$. In other words a subsum is $\epsilon_1a_1+\dots+\epsilon_na_n$ in which for each $1\leq i \leq n$ ,$\epsilon_i$ is either $0$ or $1$.

Years ago, there was a valuable list containing $n$ real not necessarily distinct numbers and their $2^n-1$ subsums. Some mysterious creatures from planet Tarator has stolen the list, but we still have the subsums.

(a) Prove that we can recover the numbers uniquely if all of the subsums are positive.

(b) Prove that we can recover the numbers uniquely if all of the subsums are non-zero.

(c) Prove that there's an example of the subsums for $n=1392$ such that we can not recover the numbers uniquely.



Note: If a subsum is sum of element of two different subsets, it appears twice.

Time allowed for this question was 75 minutes.
2013 Iran MO (3rd Round)
Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator.

We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money.

On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money?



Note 1: Tourists spend no money during travel!

Note 2: The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables.

Note 3: The maximum must be among all possible polyhedras.



Time allowed for this problem was 90 minutes.
2013 Iran MO (3rd Round)
An equation $P(x)=Q(y)$ is called Interesting if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$.

An interesting equation $P(x)=Q(y)$ yields in interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that  $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$.

(a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$.$S$ is an answer of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation.

(b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$. An interesting equation is called primary if there's no other interesting equation with lower degree that yields in it.

Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$.



Time allowed for this question was 2 hours.
2013 Iran MO (3rd Round)
Let $A_1A_2A_3A_4A_5$ be a convex 5-gon in which the coordinates of all of it's vertices are rational. For each $1\leq i \leq 5$ define $B_i$ the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i+3}A_{i+4}$.

($A_i=A_{i+5}$) Prove that at most 3 lines from the lines $A_iB_i$ ($1\leq i \leq 5$) are concurrent.



Time allowed for this problem was 75 minutes.
2012 Iran MO (3rd Round)
Prove that the number of incidences of $n$ distinct points on $n$ distinct lines in plane is $\mathcal O (n^{\frac{4}{3}})$. Find a configuration for which $\Omega (n^{\frac{4}{3}})$ incidences happens.
2012 Iran MO (3rd Round)
Consider a set of $n$ points in plane. Prove that the number of isosceles triangles having their vertices among these $n$ points is $\mathcal O (n^{\frac{7}{3}})$. Find a configuration of $n$ points in plane such that the number of equilateral triangles with vertices among these $n$ points is $\Omega (n^2)$.
2012 Iran MO (3rd Round)
Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?
2012 Iran MO (3rd Round)
Prove that from an $n\times n$ grid, one can find $\Omega (n^{\frac{5}{3}})$ points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of $\frac{5}{3}$!
2012 Iran MO (3rd Round)
Prove that for each coloring of the points inside or on the boundary of a square with $1391$ colors, there exists a monochromatic regular hexagon.
2012 Iran MO (3rd Round)
Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that





$W(k,2)=\Omega (2^{\frac{k}{2}})$.
2012 Iran MO (3rd Round)
Prove that if $n$ is large enough, then for each coloring of the subsets of the set $\{1,2,...,n\}$ with $1391$ colors, two non-empty disjoint subsets $A$ and $B$ exist such that $A$, $B$ and $A\cup B$ are of the same color.
2012 Iran MO (3rd Round)
Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.
2012 Iran MO (3rd Round)
$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$.



Proposed by Mohammad Gharakhani
2012 Iran MO (3rd Round)
Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition:

$$|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.$$



Proposed by Mohammad Gharakhani
2012 Iran MO (3rd Round)
$p$ is an odd prime number. Prove that there exists a natural number $x$ such that $x$ and $4x$ are both primitive roots modulo $p$.



Proposed by Mohammad Gharakhani
2012 Iran MO (3rd Round)
$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$.



a) Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$.



b) If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$.



Proposed by Mohammad Gharakhani
2012 Iran MO (3rd Round)
Let $p$ be a prime number. We know that each natural number can be written in the form

$$\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)$$

Uniquely.



Now let $T$ be the set of all the sums of the form

$$\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).$$



(This means to allow numbers with an infinite base $p$ representation). So numbers that for some $N\in \mathbb N$ all the coefficients $a_i, i\ge N$ are zero are natural numbers. (In fact we can consider members of $T$ as sequences $(a_0,a_1,a_2,...)$ for which $\forall_{i\in \mathbb N}: 0\le a_i \le p-1$.) Now we generalize addition and multiplication of natural numbers to this set so that it becomes a ring (it's not necessary to prove this fact). For example:



$1+(\sum_{i=0}^{\infty} (p-1)p^i)=1+(p-1)+(p-1)p+(p-1)p^2+...$

$=p+(p-1)p+(p-1)p^2+...=p^2+(p-1)p^2+(p-1)p^3+...$

$=p^3+(p-1)p^3+...=...$



So in this sum, coefficients of all the numbers $p^k, k\in \mathbb N$ are zero, so this sum is zero and thus we can conclude that $\sum_{i=0}^{\infty}(p-1)p^i$ is playing the role of $-1$ (the additive inverse of $1$) in this ring. As an example of multiplication consider

$$(1+p)(1+p+p^2+p^3+...)=1+2p+2p^2+\cdots$$



Suppose $p$ is $1$ modulo $4$. Prove that there exists $x\in T$ such that $x^2+1=0$.



Proposed by Masoud Shafaei
2012 Iran MO (3rd Round)
Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant.



Proposed by Mehdi E'tesami Fard
2012 Iran MO (3rd Round)
Let the Nagel point of triangle $ABC$ be $N$. We draw lines from $B$ and $C$ to $N$ so that these lines intersect sides $AC$ and $AB$ in $D$ and $E$ respectively. $M$ and $T$ are midpoints of segments $BE$ and $CD$ respectively. $P$ is the second intersection point of circumcircles of triangles $BEN$ and $CDN$. $l_1$ and $l_2$ are perpendicular lines to $PM$ and $PT$ in points $M$ and $T$ respectively. Prove that lines $l_1$ and $l_2$ intersect on the circumcircle of triangle $ABC$.



Proposed by Nima Hamidi
2012 Iran MO (3rd Round)
Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively.  From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that

$$\frac{TP}{TQ}\ge \frac{b}{a}.$$



Proposed by Morteza Saghafian
2012 Iran MO (3rd Round)
The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$.



Proposed By Pedram Safaei
2012 Iran MO (3rd Round)
Two fixed lines $l_1$ and $l_2$ are perpendicular to each other at a point $Y$. Points $X$ and $O$ are on $l_2$ and both are on one side of line $l_1$. We draw the circle $\omega$ with center $O$ and radius $OY$. A variable point $Z$ is on line $l_1$. Line $OZ$ cuts circle $\omega$ in $P$. Parallel to $XP$ from $O$ intersects $XZ$ in $S$. Find the locus of the point $S$.



Proposed by Nima Hamidi
2012 Iran MO (3rd Round)
We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist?



Proposed by Morteza Saghafian
2012 Iran MO (3rd Round)
Suppose $s,k,t\in \mathbb N$. We've colored each natural number with one of the $k$ colors, such that each color is used infinitely many times. We want to choose a subset $\mathcal A$ of $\mathbb N$ such that it has $t$ disjoint monochromatic $s$-element subsets. What is the minimum number of elements of $A$?



Proposed by Navid Adham