Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Iran MO (2nd round)
Let $A_1,A_2,\cdots,A_n$ be $n$ distinct points on the plane ($n>1$). We consider all the segments $A_iA_j$ where $i<j\leq{n}$ and color the midpoints of them. What's the minimum number of colored points? (In fact, if $k$ colored points coincide, we count them $1$.)
1998 Iran MO (2nd round)
If $a_1<a_2<\cdots<a_n$ be real numbers, prove that:

$$ a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4. $$
1998 Iran MO (2nd round)
Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.
1998 Iran MO (2nd round)
Let $n$ be a positive integer. We call $(a_1,a_2,\cdots,a_n)$ a good $n-$tuple if $\sum_{i=1}^{n}{a_i}=2n$ and there doesn't exist a set of $a_i$s such that the sum of them is equal to $n$. Find all good $n-$tuple.

(For instance, $(1,1,4)$ is a good $3-$tuple, but $(1,2,1,2,4)$ is not a good $5-$tuple.)
1998 Iran MO (2nd round)
Let the positive integer $n$ have at least for positive divisors and $0<d_1<d_2<d_3<d_4$ be its least positive divisors. Find all positive integers $n$ such that:

$$ n=d_1^2+d_2^2+d_3^2+d_4^2. $$
1998 Iran MO (2nd round)
Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that:

$$ AQ+CQ=BP. $$
1998 Iran MO (2nd round)
If $A=(a_1,\cdots,a_n)$ , $B=(b_1,\cdots,b_n)$ be $2$ $n-$tuple that $a_i,b_i=0 \ or \ 1$ for $i=1,2,\cdots,n$, we define $f(A,B)$ the number of $1\leq i\leq n$ that $a_i\ne b_i$.

For instance, if $A=(0,1,1)$ , $B=(1,1,0)$, then $f(A,B)=2$.

Now, let $A=(a_1,\cdots,a_n)$ , $B=(b_1,\cdots,b_n)$ , $C=(c_1,\cdots,c_n)$ be 3 $n-$tuple, such that for $i=1,2,\cdots,n$, $a_i,b_i,c_i=0 \ or \ 1$ and $f(A,B)=f(A,C)=f(B,C)=d$.

$a)$ Prove that $d$ is even.

$b)$ Prove that there exists a $n-$tuple $D=(d_1,\cdots,d_n)$ that $d_i=0 \ or \ 1$ for $i=1,2,\cdots,n$, such that $f(A,D)=f(B,D)=f(C,D)=\frac{d}{2}$.
1997 Iran MO (2nd round)
Let $x,y$ be positive integers such that $3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.
1997 Iran MO (2nd round)
Let segments $KN,KL$ be tangent to circle $C$ at points $N,L$, respectively. $M$ is a point on the extension of the segment $KN$ and $P$ is the other meet point of the circle $C$ and the circumcircle of $\triangle KLM$. $Q$ is on $ML$ such that $NQ$ is perpendicular to $ML$. Prove that

$$ \angle MPQ=2\angle KML. $$
1997 Iran MO (2nd round)
We have a $n\times n$ table and we’ve written numbers $0,+1 \ or \ -1$ in each $1\times1$ square such that in every row or column, there is only one $+1$ and one $-1$. Prove that by swapping the rows with each other and the columns with each other finitely, we can swap $+1$s with $-1$s.
1997 Iran MO (2nd round)
Let $x_1,x_2,x_3,x_4$ be positive reals such that $x_1x_2x_3x_4=1$. Prove that:

$$ \sum_{i=1}^{4}{x_i^3}\geq\max\{ \sum_{i=1}^{4}{x_i},\sum_{i=1}^{4}{\frac{1}{x_i}} \}. $$
1997 Iran MO (2nd round)
In triangle $ABC$, angles $B,C$ are acute. Point $D$ is on the side $BC$ such that $AD\perp{BC}$. Let the interior bisectors of $\angle B,\angle C$ meet $AD$ at $E,F$, respectively. If $BE=CF$, prove that $ABC$ is isosceles.
1997 Iran MO (2nd round)
Let $a,b$ be positive integers and $p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$ be a prime number. Find the maximum value of $p$ and justify your answer.
1996 Iran MO (2nd round)
Let $a, b, c$ be real numbers. Prove that there exists a triangle with side lengths $a, b, c$ if and only if

$$2(a^4 + b^4 + c^4) < (a^2 + b^2 + c^2)^2.$$
1996 Iran MO (2nd round)
Let $a,b,c,d$ be positive integers such that $ab=cd$. Prove that $a+b+c+d$ is a composite number.
1996 Iran MO (2nd round)
Let $N$ be the midpoint of side $BC$ of triangle $ABC$. Right isosceles triangles $ABM$ and $ACP$ are constructed outside the triangle, with bases $AB$ and $AC$. Prove that $\triangle MNP$ is also a right isosceles triangle.
1996 Iran MO (2nd round)
Let $n$ blue points $A_i$ and $n$ red points $B_i  \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that

$$\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.$$
1995 Iran MO (2nd round)
Prove that for every positive integer $n \geq 3$ there exist two sets $A =\{ x_1, x_2,\ldots, x_n\}$ and $B =\{ y_1, y_2,\ldots, y_n\}$ for which



i) $A \cap B = \varnothing.$



ii) $x_1+ x_2+\cdots+ x_n= y_1+ y_2+\cdots+ y_n.$



ii) $x_1^2+ x_2^2+\cdots+ x_n^2= y_1^2+ y_2^2+\cdots+ y_n^2.$
1995 Iran MO (2nd round)
Let $ABC$ be an acute triangle and let $\ell$ be a line in the plane of triangle $ABC.$ We've drawn the reflection of the line $\ell$ over the sides $AB, BC$ and $AC$ and they intersect in the points $A', B'$ and $C'.$ Prove that the incenter of the triangle $A'B'C'$ lies on the circumcircle of the triangle $ABC.$
1995 Iran MO (2nd round)
Let $k$ be a positive integer. $12k$ persons have participated in a party and everyone shake hands with $3k+6$ other persons. We know that the number of persons who shake hands with every two persons is a fixed number. Find $k.$
1995 Iran MO (2nd round)
Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another.

(For example, $23=9+8+6.)$
1995 Iran MO (2nd round)
Let $n \geq 0$ be an integer. Prove that

$$ \lceil \sqrt n +\sqrt{n+1}+\sqrt{n+2} \rceil = \lceil  \sqrt{9n+8} \rceil$$

Where $\lceil  x \rceil $ is the smallest integer which is greater or equal to $x.$
1995 Iran MO (2nd round)
In a quadrilateral $ABCD$ let $A', B', C'$ and $D'$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Denote by $S(X, YZ)$ the plane which passes through the point $X$ and is perpendicular to the line $YZ.$ Prove that if $A', B', C'$ and $D'$ don't lie in a plane, then four planes $S(A, C'D'), S(B, A'D'), S(C, A'B')$ and $S(D, B'C')$ pass through a common point.
1994 Iran MO (2nd round)
In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$

[asy]

unitsize(40);

import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0);

draw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){

return (0.6*cos(t)+0.64,0.6*sin(t)+2.24);

}

draw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){

return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);

}

draw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){

return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);

}

draw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq);

dot((0,0),ds); label("$O$", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label("$A$", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label("$B$", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label("$C$", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label("$E$", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label("$D$", (-0.92,0.7), NE*lsf); label("$\alpha$", (0.48,1.38),NE*lsf); label("$\beta$", (-0.02,0.94),NE*lsf); label("$\gamma$", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle);

[/asy]
1994 Iran MO (2nd round)
Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$:

$$ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}$$
1994 Iran MO (2nd round)
Let $\overline{a_1a_2a_3\ldots a_n}$ be the representation of a $n-$digits number in base $10.$ Prove that there exists a one-to-one function like $f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}$ such that $f(a_1) \neq 0$ and the number $\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }$ is divisible by $3.$
1994 Iran MO (2nd round)
The incircle of triangle $ABC$ meet the sides $AB, AC$ and $BC$ in $M,N$ and $P$, respectively. Prove that the orthocenter of triangle $MNP,$ the incenter and the circumcenter of triangle $ABC$ are collinear.



 [asy]

import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffwwww = rgb(1,0.4,0.4); pen xdxdff = rgb(0.49,0.49,1);

draw((8,17.58)--(2.84,9.26)--(20.44,9.21)--cycle); draw((8,17.58)--(2.84,9.26),ttttff+linewidth(2pt)); draw((2.84,9.26)--(20.44,9.21),ttttff+linewidth(2pt)); draw((20.44,9.21)--(8,17.58),ttttff+linewidth(2pt)); draw(circle((9.04,12.66),3.43),blue+linewidth(1.2pt)+linetype("8pt 8pt")); draw((6.04,14.42)--(8.94,9.24),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(11.12,15.48),ffwwww+linewidth(1.2pt)); draw((11.12,15.48)--(6.04,14.42),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(7.81,14.79)); draw((11.12,15.48)--(6.95,12.79)); draw((6.04,14.42)--(10.12,12.6));

dot((8,17.58),ds); label("$A$", (8.11,18.05),NE*lsf); dot((2.84,9.26),ds); label("$B$", (2.11,8.85), NE*lsf); dot((20.44,9.21),ds); label("$C$", (20.56,8.52), NE*lsf); dot((9.04,12.66),ds); label("$O$", (8.94,12.13), NE*lsf); dot((6.04,14.42),ds); label("$M$", (5.32,14.52), NE*lsf); dot((11.12,15.48),ds); label("$N$", (11.4,15.9), NE*lsf); dot((8.94,9.24),ds); label("$P$", (8.91,8.58), NE*lsf); dot((7.81,14.79),ds); label("$D$", (7.81,15.14),NE*lsf); dot((6.95,12.79),ds); label("$F$", (6.64,12.07),NE*lsf); dot((10.12,12.6),ds); label("$G$", (10.41,12.35),NE*lsf); dot((8.07,13.52),ds); label("$H$", (8.11,13.88),NE*lsf); clip((-0.68,-0.96)--(-0.68,25.47)--(30.71,25.47)--(30.71,-0.96)--cycle);

[/asy]
1994 Iran MO (2nd round)
Let $n >3$ be an odd positive integer and $n=\prod_{i=1}^k p_i^{\alpha_i}$ where $p_i$ are primes and $\alpha_i$ are positive integers. We know that

$$m=n(1-\frac{1}{p_1})(1-\frac{1}{p_2})(1-\frac{1}{p_3}) \cdots (1-\frac{1}{p_n}).$$

Prove that there exists a prime $P$ such that $P|2^m -1$ but $P \nmid n.$
1993 Iran MO (2nd round)
Suppose that $p$ is a prime number and is greater than $3$. Prove that $7^{p}-6^{p}-1$ is divisible by $43$.
1993 Iran MO (2nd round)
Let $ABC$ be an acute triangle with sides and area equal to $a, b, c$ and $S$ respectively. Prove or disprove that a necessary and sufficient condition for existence of a point $P$ inside the triangle $ABC$ such that the distance between $P$ and the vertices of $ABC$ be equal to $x, y$ and $z$ respectively is that there be a triangle with sides $a, y, z$ and area $S_1$, a triangle with sides $b, z, x$ and area $S_2$ and a triangle with sides $c, x, y$ and area $S_3$ where $S_1 + S_2 + S_3 = S.$
1993 Iran MO (2nd round)
Let $n, r$ be positive integers. Find the smallest positive integer $m$ satisfying the following condition. For each partition of the set $\{1, 2, \ldots ,m \}$ into $r$ subsets $A_1,A_2, \ldots ,A_r$, there exist two numbers $a$ and $b$ in some $A_i, 1 \leq i \leq r$, such that

$$ 1 < \frac ab < 1 +\frac 1n.$$
1993 Iran MO (2nd round)
$G$ is a graph with $n$ vertices $A_1,A_2,\ldots,A_n,$ such that for each pair of non adjacent vertices $A_i$ and $A_j$ , there exist another vertex $A_k$ that is adjacent

to both $A_i$ and $A_j .$



(a) Find the minimum number of edges in such a graph.



(b) If $n = 6$ and $A_1,A_2,A_3,A_4,A_5,$ and $A_6$ form a cycle of length $6,$ find the number of edges that must be added to this cycle such that the above condition holds.
1993 Iran MO (2nd round)
Show that if $D_1$ and $D_2$ are two skew lines, then there are infinitely many straight lines such that their points have equal distance from $D_1$ and $D_2.$
1993 Iran MO (2nd round)
Let $f(x)$ and $g(x)$ be two polynomials with real coefficients such that for infinitely many rational values $x, \frac{f(x)}{g(x)}$ is rational. Prove that $\frac{f(x)}{g(x)}$ can be written as the ratio of two polynomials with rational coefficients.
1992 Iran MO (2nd round)
Let $ABC$ be a right triangle with $\angle A=90^\circ.$ The bisectors of the angles $B$ and $C$ meet each other in $I$ and meet the sides $AC$ and $AB$ in $D$ and $E$, respectively. Prove that $S_{BCDE}=2S_{BIC},$ where $S$ is the area function.

 [asy]

import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttqqcc = rgb(0.2,0,0.8); pen qqwuqq = rgb(0,0.39,0); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2); pen ccccff = rgb(0.8,0.8,1);

draw((1.89,4.08)--(1.89,4.55)--(1.42,4.55)--(1.42,4.08)--cycle,qqwuqq); draw((1.42,4.08)--(7.42,4.1),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(1.42,4.08),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(7.42,4.1),ttqqcc+linewidth(1.6pt)); draw((1.4,10.08)--(4,4.09),fftttt+linewidth(1.2pt)); draw((7.42,4.1)--(1.41,6.24),fftttt+linewidth(1.2pt)); draw((1.41,6.24)--(4,4.09),ccccff+linetype("5pt 5pt"));

dot((1.42,4.08),ds); label("$A$", (1.1,3.66),NE*lsf); dot((7.42,4.1),ds); label("$B$", (7.15,3.75),NE*lsf); dot((1.4,10.08),ds); label("$C$", (1.49,10.22),NE*lsf); dot((4,4.09),ds); label("$E$", (3.96,3.46),NE*lsf); dot((1.41,6.24),ds); label("$D$", (0.9,6.17),NE*lsf); dot((3.37,5.54),ds); label("$I$", (3.45,5.69),NE*lsf); clip((-6.47,-7.49)--(-6.47,11.47)--(16.06,11.47)--(16.06,-7.49)--cycle); [/asy]
1992 Iran MO (2nd round)
In the sequence $\{a_n\}_{n=0}^{\infty}$ we have $a_0=1$, $a_1=2$ and

$$a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1$$

Prove that

$$52 < a_{1371} < 65$$
1992 Iran MO (2nd round)
There are some cities in both sides of a river and there are some sailing channels between the cities. Each sailing channel connects exactly one city from a side of the river to a city on the other side. Each city has exactly $k$ sailing channels. For every two cities, there's a way which connects them together. Prove that if we remove any (just one) sailing channel, then again for every two cities, there's a way that connect them together. $( k \geq 2)$
1992 Iran MO (2nd round)
Prove that for any positive integer $t,$

$$1+2^t+3^t+\cdots+9^t - 3(1 + 6^t +8^t )$$

is divisible by $18.$
1992 Iran MO (2nd round)
In triangle $ABC,$ we have $\angle A \leq 90^\circ$ and $\angle B = 2 \angle C.$ The interior bisector of the angle $C$ meets the median $AM$ in $D.$ Prove that $\angle MDC \leq 45^\circ.$ When does equality hold?



[asy]

import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqqqzz = rgb(0,0,0.6); pen ffqqtt = rgb(1,0,0.2); pen qqwuqq = rgb(0,0.39,0);

draw((3.14,5.81)--(3.23,0.74),ffqqtt+linewidth(2pt)); draw((3.23,0.74)--(9.73,1.32),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.14,5.81),ffqqtt+linewidth(2pt)); draw((9.73,1.32)--(3.19,2.88)); draw((3.14,5.81)--(6.71,1.05)); pair parametricplot0_cus(real t){

return (0.7*cos(t)+5.8,0.7*sin(t)+2.26);

}

draw(graph(parametricplot0_cus,-0.9270442638657642,-0.23350086562377703)--(5.8,2.26)--cycle,qqwuqq);

dot((3.14,5.81),ds); label("$A$", (2.93,6.09),NE*lsf); dot((3.23,0.74),ds); label("$B$", (2.88,0.34),NE*lsf); dot((9.73,1.32),ds); label("$C$", (10.11,1.04),NE*lsf); dot((3.19,2.88),ds); label("$X$", (2.77,2.94),NE*lsf); dot((6.71,1.05),ds); label("$M$", (6.75,0.5),NE*lsf); dot((5.8,2.26),ds); label("$D$", (5.89,2.4),NE*lsf); label("$\alpha$", (6.52,1.65),NE*lsf); clip((-3.26,-11.86)--(-3.26,8.36)--(20.76,8.36)--(20.76,-11.86)--cycle);

[/asy]
1992 Iran MO (2nd round)
Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$



Note. ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$
1991 Iran MO (2nd round)
Prove that the equation $x+x^2=y+y^2+y^3$ do not have any solutions in positive integers.
1991 Iran MO (2nd round)
Let $ABCD$ be a tetragonal.



(a) If the plane $(P)$ cuts $ABCD,$ find the necessary and sufficient condition such that the area formed from the intersection of the plane $(P)$ and the tetragonal be a parallelogram. Prove that the problem has three solutions in this case.



(b) Consider one of the solutions of (a). Find the situation of the plane $(P)$ for which the parallelogram has maximum area.



(c) Find a plane $(P)$ for which the parallelogram be a lozenge and then find the length side of his lozenge in terms of the length of the edges of $ABCD.$
1991 Iran MO (2nd round)
Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and

$$f(x+y)=f(x)+f(y)$$And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$
1991 Iran MO (2nd round)
Prove that there exist at least six points with rational coordinates on the curve of the equation

$$y^3=x^3+x+1370^{1370}$$
1991 Iran MO (2nd round)
Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that

$$\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3$$$$, IA'+IB'+IC' \geq IA+IB+IC$$
1991 Iran MO (2nd round)
Three groups $A, B$ and $C$ of mathematicians from different countries have invited to a ceremony. We have formed meetings such that three mathematicians participate in every meeting and there is exactly one mathematician from each group in every meeting. Also every two mathematicians have participated in exactly one meeting with each other.



(a) Prove that if this is possible, then number of mathematicians of the groups is equal.



(b) Prove that if there exist $3$ mathematicians in each group, then that work is possible.



(c) Prove that if number mathematicians of the groups be equal, then that work is possible.
1990 Iran MO (2nd round)
Let $ABCD$ be a parallelogram. The line $\Delta$ meets the lines $AB, BC, CD$ and $DA$ at $M, N, P$ and $Q,$ respectively. Let $R$ be the intersection point of the lines $AB,DN$ and let $S$ be intersection point of the lines $AD, BP.$ Prove that $RS \parallel \Delta.$





 [asy]

import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); pen qqzzcc = rgb(0,0.6,0.8); pen wwwwff = rgb(0.4,0.4,1);

draw((2,2)--(6,2),qqzzcc+linewidth(1.6pt)); draw((6,2)--(4,0),qqzzcc+linewidth(1.6pt)); draw((-1.95,(+12-2*-1.95)/2)--(12.24,(+12-2*12.24)/2),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0+3*-1.95)/3)--(12.24,(-0+3*12.24)/3),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0-0*-1.95)/6)--(12.24,(-0-0*12.24)/6),qqzzcc+linewidth(1.6pt)); draw((4,0)--(4,4),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((2,2)--(8.14,0),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((-1.95,(+32.56-4*-1.95)/4.14)--(12.24,(+32.56-4*12.24)/4.14),qqzzcc+linewidth(1.6pt));

dot((0,0),ds); label("$A$", (0,-0.3),NE*lsf); dot((4,0),ds); label("$B$", (4.02,-0.33),NE*lsf); dot((2,2),ds); label("$D$", (1.81,2.07),NE*lsf); dot((6,2),ds); label("$C$", (6.16,2.08),NE*lsf); dot((3,3),ds); label("$Q$", (2.97,3.22),NE*lsf); dot((5,1),ds); label("$N$", (4.99,1.19),NE*lsf); label("$\Delta$", (1.7,3.76),NE*lsf); dot((6,0),ds); label("$M$", (5.9,-0.33),NE*lsf); dot((4,2),ds); label("$P$", (4.02,2.08),NE*lsf); dot((4,4),ds); label("$S$", (3.94,4.12),NE*lsf); dot((8.14,0),ds); label("$E$", (8.2,0.09),NE*lsf); clip((-1.95,-6.96)--(-1.95,4.99)--(12.24,4.99)--(12.24,-6.96)--cycle);

[/asy]
1990 Iran MO (2nd round)
Find all integer solutions to the equation

$$(x^2-x)(x^2-2x+2)=y^2-1$$
1990 Iran MO (2nd round)
(a) For every positive integer $n$ prove that

$$1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2$$



(b) Let $X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1)$ and let $A_k$ be non-empty subsets of $X \ (k=1,2,3, \ldots , 2^n -1).$ If $a_k$ be the product of all elements of the set $A_k,$ prove that

$$\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1$$
1990 Iran MO (2nd round)
(a) Consider the set of all triangles $ABC$ which are inscribed in a circle with radius $R.$ When is $AB^2+BC^2+CA^2$ maximum? Find this maximum.



(b) Consider the set of all tetragonals $ABCD$ which are inscribed in a sphere with radius $R.$ When is the sum of squares of the six edges of $ABCD$ maximum? Find this maximum, and in this case prove that all of the edges are equal.
1990 Iran MO (2nd round)
Let $\alpha$ be a root of the equation $x^3-5x+3=0$ and let $f(x)$ be a polynomial with rational coefficients. Prove that if $f(\alpha)$ be the root of equation above, then $f(f(\alpha))$ is a root, too.
1990 Iran MO (2nd round)
We want to cover a rectangular $5 \times 137$ with the following figures, prove that this is impossible.



$$\text{Squars are the same and all are } \Huge{1 \times 1}$$

[asy]

import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1);

draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt));

label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle);

[/asy]