Navvye/Olympiad-Problems-CSV huggingface dataset:
1990 Brazil National Olympiad
$ABCD$ is a quadrilateral,

$E,F,G,H$ are midpoints of $AB,BC,CD,DA$.

Find the point P such that

$area (PHAE) = area (PEBF) = area (PFCG) = area (PGDH)$.
1990 Brazil National Olympiad
Let

$f(x)=\frac{ax+b}{cx+d}$

$F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$)



Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$,

show that $F_n(x)=x$ (for any valid x).
1989 Brazil National Olympiad
The sides of a triangle $T$, with vertices $(0,0)$,$(3,0)$ and $(0,3)$ are mirrors.



Show that one of the images of the triagle $T_1$ with vertices $(0,0)$,$(0,1)$ and $(2,0)$ is the triangle with vertices $(24,36)$,$(24,37)$ and $(26,36)$.
1989 Brazil National Olympiad
Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.
1989 Brazil National Olympiad
A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$.



Determine, justifying your answer, the set of all possible values for $f$.
1989 Brazil National Olympiad
A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second.



At the $n$-th turn, $n=1,2,\ldots,10$, A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A.



B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins.



Which player has a winning strategy?

Suppose now both players has $k$ chips numbered 1 to $k$.  Which player has a winning strategy?

Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy?
1989 Brazil National Olympiad
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.



Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
1988 Brazil National Olympiad
Find all primes which are sum of two primes and difference of two primes.
1988 Brazil National Olympiad
Show that, among all triangles whose vertices are at distances 3,5,7 respectively from a given point P, the ones with largest area have P as orthocenter.



(You can suppose, without demonstration, the existence of a triangle with maximal area in this question.)
1988 Brazil National Olympiad
Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that



$f(x \cdot y) = f(x) + f(y)$

$f(30) = 0$

$f(x)=0$ always when the units digit of $x$ is $7$
1988 Brazil National Olympiad
Two triangles are circumscribed to a circumference. Show that if a circumference containing five of their vertices exists, then it will contain the sixth vertex too.
1988 Brazil National Olympiad
A figure on a computer screen shows $n$ points on a sphere, no four coplanar. Some pairs of points are joined by segments. Each segment is colored red or blue. For each point there is a key that switches the colors of all segments with that point as endpoint. For every three points there is a sequence of key presses that makes the three segments between them red. Show that it is possible to make all the segments on the screen red. Find the smallest number of key presses that can turn all the segments red, starting from the worst case.
1987 Brazil National Olympiad
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i  \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot  k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.
1987 Brazil National Olympiad
Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$.
1987 Brazil National Olympiad
Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = min(x_n, y_n)$ and $y_{n+1} = max(x_n, y_n)- k\cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $x_1/y_1$ the first player has a winning strategy.
1987 Brazil National Olympiad
Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$  be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$.
1987 Brazil National Olympiad
$A$ and $B$ wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. $A$ chooses a point $P$ on the top face. $B$ then chooses a vertical plane through the point $P$ to divide the cake. $B$ chooses which piece to take. Which point $P$ should $A $ choose in order to secure as large a slice as possible?
1986 Brazil National Olympiad
A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times.
1986 Brazil National Olympiad
Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.
1986 Brazil National Olympiad
The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be

(1) the half-lines perpendicular to $R$, and

(2) the semicircles with center on $R$.

Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.
1986 Brazil National Olympiad
Find all $10$ digit numbers $a_0a_1...a_9$ such that for each $k, a_k$ is the number of times that the digit $k$ appears in the number.
1986 Brazil National Olympiad
A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.
1985 Brazil National Olympiad
$a, b, c, d$ are integers with $ad \ne bc$. Show that $1/((ax+b)(cx+d))$ can be written in the form $ r/(ax+b) + s/(cx+d)$. Find the sum $1/1\cdot  4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$.
1985 Brazil National Olympiad
Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$.
1985 Brazil National Olympiad
A convex quadrilateral is inscribed in a circle of radius $1$. Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$.
1985 Brazil National Olympiad
$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.
1985 Brazil National Olympiad
$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.
1984 Brazil National Olympiad
Find all solutions in positive integers to $(n+1)^k -1 = n!$
1984 Brazil National Olympiad
Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done?
1984 Brazil National Olympiad
Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.
1984 Brazil National Olympiad
$ABC$ is a triangle with $\angle A = 90^o$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and$ F$. For which point $D$ is $ EF$ a minimum?
1984 Brazil National Olympiad
$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.
1984 Brazil National Olympiad
There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$?
1983 Brazil National Olympiad
Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.
1983 Brazil National Olympiad
An equilateral triangle $ABC$ has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side $a$ is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done, the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.
1983 Brazil National Olympiad
Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$.
1983 Brazil National Olympiad
Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color.
1983 Brazil National Olympiad
Show that $1 \le  n^{1/n} \le 2$ for all positive integers $n$.

Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.
1983 Brazil National Olympiad
Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$.
1982 Brazil National Olympiad
The angles of the triangle $ABC$ satisfy $\angle A / \angle C = \angle B / \angle A = 2$. The incenter is $O. K, L$  are the excenters of the excircles opposite $B$ and $A$ respectively. Show that triangles $ABC$ and $OKL$ are similar.
1982 Brazil National Olympiad
Any positive integer $n$ can be written in the form $n = 2^b(2c+1)$. We call $2c+1$ the odd part of $n$. Given an odd integer $n > 0$, define the sequence $ a_0, a_1, a_2, ...$ as follows: $a_0 = 2^n-1, a_{k+1} $ is the odd part of $3a_k+1$. Find $a_n$.
1982 Brazil National Olympiad
$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?
1982 Brazil National Olympiad
Three numbered tiles are arranged in a tray as shown:



Show that we cannot interchange the $1$ and the $3$ by a sequence of moves where we slide a tile to the adjacent vacant space.
1982 Brazil National Olympiad
Show how to construct a line segment length $(a^4 + b^4)^{1/4}$ given segments lengths $a$ and $b$.
1982 Brazil National Olympiad
Five spheres of radius $r$ are inside a right circular cone. Four of the spheres lie on the base of the cone. Each touches two of the others and the sloping sides of the cone. The fifth sphere touches each of the other four and also the sloping sides of the cone. Find the volume of the cone.
1981 Brazil National Olympiad
For which $k$ does the system $x^2 - y^2 = 0, (x-k)^2 + y^2 = 1$ have exactly:

(i) two,

(ii) three real solutions?
1981 Brazil National Olympiad
Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?
1981 Brazil National Olympiad
Given a sheet of paper and the use of a rule, compass and pencil, show how to draw a straight line that passes through two given points, if the length of the ruler and the maximum opening of the compass are both less than half the distance between the two points. You may not fold the paper.
1981 Brazil National Olympiad
A graph has $100$ points. Given any four points, there is one joined to the other three. Show that one point must be joined to all $99$ other points. What is the smallest number possible of such points (that are joined to all the others)?
1981 Brazil National Olympiad
Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.
1981 Brazil National Olympiad
The centers of the faces of a cube form a regular octahedron of volume $V$. Through each vertex of the cube we may take the plane perpendicular to the long diagonal from the vertex. These planes also form a regular octahedron. Show that its volume is $27V$.
1980 Brazil National Olympiad
Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$. Then take the same number of balls from $B$ and place them in $A$. Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$?
1980 Brazil National Olympiad
Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$.
1980 Brazil National Olympiad
Given a triangle $ABC$ and a point $P_0$ on the side $AB$. Construct points $P_i, Q_i, R_i $ as follows. $Q_i$ is the foot of the perpendicular from $P_i$ to $BC, R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$. Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$.
1980 Brazil National Olympiad
Given $5$ points of a sphere radius $r$, show that two of the points are a distance $\le r \sqrt2$ apart.
1979 Brazil National Olympiad
Show that if $a < b$ are in the interval $\left[0, \frac{\pi}{2}\right]$ then $a - \sin a < b - \sin b$. Is this true for $a < b$ in the interval $\left[\pi,\frac{3\pi}{2}\right]$?
1979 Brazil National Olympiad
The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.
1979 Brazil National Olympiad
The vertex C of the triangle ABC is allowed to vary along a line parallel to AB. Find the locus of the orthocenter.
1979 Brazil National Olympiad
Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.
1979 Brazil National Olympiad




ABCD is a square with side 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN meet at I. Find the area of the triangle CIN.

The midpoints of the sides AB, BC, CD, DA of the parallelogram ABCD are M, N, P, Q respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is $\frac{2}{5}$ the area of the parallelogram.

ABC is a triangle with CA = CB and centroid G. Show that the area of AGB is $\frac{1}{3}$ of the area of ABC.

Is (ii) true for all convex quadrilaterals ABCD?
Brazil NMO L2 (OBM) - geometry
A sheet of rectangular $ABCD$ paper, of area $1$, is folded along its diagonal $AC$ and then unfolded, then it is bent so that vertex $A$ coincides with vertex $C$ and then unfolded, leaving the crease $MN$, as shown below.

a) Show that the quadrilateral $AMCN$ is a rhombus.

b) If the diagonal $AC$ is twice the width $AD$, what is the area of the rhombus $AMCN$?

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Brazil NMO L2 (OBM) - geometry
Let $XYZ$ be a  right triangle of  area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric  of $Z$  wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$.
Brazil NMO L2 (OBM) - geometry
Let $ABC$ be a triangle inscribed in a circle of center $O$ and $P$  be a point on the arc $AB$, that does not contain $C$. The perpendicular drawn fom $P$ on line $BO$ intersects $AB$ at $S$ and $BC$ at $T$. The perpendicular drawn from $P$ on line $AO$ intersects $AB$ at $Q$ and $AC$ at $R$. Prove that:

a) $PQS$ is an isosceles triangle

b) $PQ^2=QR= ST$
Brazil NMO L2 (OBM) - geometry
The triangle $ABC$ is inscribed in the circle $S$ and $AB <AC$. The line containing $A$ and is perpendicular to $BC$ meets $S$ in $P$ ($P \ne A$). Point $X$ is on the segment $AC$ and the line $BX$ intersects $S$ in $Q$ ($Q \ne B$). Show that $BX = CX$ if, and only if, $PQ$ is a diameter of $S$.
Brazil NMO L2 (OBM) - geometry
In the figure, $ABC$ and $DAE$ are isosceles triangles ($AB = AC = AD = DE$) and the angles $BAC$ and $ADE$ have measures $36^o$.

a) Using geometric properties, calculate the measure of angle $\angle EDC$.

b) Knowing that $BC = 2$, calculate the length of segment $DC$.

c) Calculate the length of segment $AC$ .

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Brazil NMO L2 (OBM) - geometry
Let $D$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. Let $O_1$ and $O_2$ be the circumcenters of the $ADC$ and $DBC$ triangles, respectively.

a) Prove that $\angle O_1DO_2$ is right.

b) Prove that $AB$ is tangent to the circle of diameter $O_1O_2$ .
Brazil NMO L2 (OBM) - geometry
In the right triangle $ABC$, the perpendicular sides $AB$ and $BC$ have lengths $3$ cm and $4$ cm, respectively. Let $M$ be the midpoint of the side $AC$ and let $D$ be a point, distinct from $A$, such that $BM = MD$ and $AB = BD$.

a) Prove that $BM$ is perpendicular to $AD$.

b) Calculate the area of the quadrilateral $ABDC$.
Brazil NMO L2 (OBM) - geometry
The angle $B$ of a  triangle $ABC$ is $120^o$. Let $M$ be a point on the side $AC$ and $K$ a point on the extension of the side $AB$, such that $BM$ is the internal bisector of the angle $\angle ABC$ and $CK$ is the external bisector corresponding to the angle $\angle ACB$ . The segment $MK$ intersects $BC$ at point $P$. Prove that $\angle APM = 30^o$.
Brazil NMO L2 (OBM) - geometry
Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$, $N$ and $R$ be the midpoints of $AB$, $BC$ an $AH$, respectively. If $A\hat{B}C=70^\large\circ$, compute $M\hat{N}R$.
Brazil NMO L2 (OBM) - geometry
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle  $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$.

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Brazil NMO L2 (OBM) - geometry
$ \triangle ABC$ is a right isosceles triangle. Choose points $ K$ and $ M$ from the hypotenuse $ AB$, such that $ K \in AM$ and $ \angle KCM = 45$º. Prove that $ (AK)^2 + (MB)^2 = (KM)^2$



Thanks for any help.
Brazil NMO L2 (OBM) - geometry
Let $ABC$ be an acutangle triangle and $O, H$ its circumcenter, orthocenter, respectively. If  $\frac{AB}{\sqrt2}=BH=OB$, calculate the angles of the triangle $ABC$ .
Brazil NMO L2 (OBM) - geometry
Let $ A$ be one of the two points of intersection of two circles with centers $ X, Y$ respectively.The tangents at $ A$ to the two circles meet the circles again at $ B, C$. Let a point $ P$ be located so that $ PXAY$ is a parallelogram. Show that $ P$ is also the circumcenter of triangle $ ABC$.
Brazil NMO L2 (OBM) - geometry
Let $ABCD$ be a parallelogram and $\omega$ be  the circumcircle of the triangle $ABD$. Let $E ,F$ be the intersections of $\omega$  with lines $BC ,CD$ respectively . Prove that the circumcenter of the triangle $CEF$ lies on  $\omega$.
Brazil NMO L2 (OBM) - geometry
The diagonals of an cyclic quadrilateral $ABCD$ intersect at $O$. The circumcircles of triangle $AOB$ and $COD$ intersect lines $BC$ and $AD$, for the second time, at points $M, N, P$and $Q$. Prove that the $MNPQ$ quadrilateral is inscribed in a circle of center $O$.
Brazil NMO L2 (OBM) - geometry
The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?
Brazil NMO L2 (OBM) - geometry
Let $ ABCD $ be a convex quadrilateral such that $ AD = DC, AC = AB $ and $ \angle ADC = \angle CAB $. If $ M $ and $ N $ are midpoints of the $ AD $ and $ AB $ sides, prove that the $ MNC $ triangle is isosceles.
Brazil NMO L2 (OBM) - geometry
Let be a triangle $ ABC $, the midpoint of the $ AC $ and $ N $ side, and the midpoint of the $ AB $ side. Let $ r $ and $ s $ reflect the straight lines $ BM $ and $ CN $ on the straight $ BC $, respectively. Also define $ D $ and $ E $ as the intersection of the lines $ r $ and $ s $ and the line $ MN $, respectively. Let $ X $ and $ Y $ be the intersection points between the circumcircles of the triangles $ BDM $ and $ CEN $, $ Z $ the intersection of the lines $ BE $ and $ CD $ and $ W $ the intersection between the lines $ r $ and $ s $. Prove that $ XY, WZ $, and $ BC $ are concurrents.
Brazil NMO L2 (OBM) - geometry
The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$.
Brazil NMO L2 (OBM) - geometry
Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of  the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.
Brazil NMO L2 (OBM) - geometry
Let ABC be a scalene triangle and AM is the median relative to side BC. The diameter circumference AM intersects for the second time the side AB and AC at points P and Q, respectively, both different from A. Assuming that PQ is parallel to BC, determine the angle measurement <BAC.



Any solution without trigonometry?
Brazil NMO L2 (OBM) - geometry
Let $AB$ be a diameter of the circunference $\omega$, let $C$ and $D$ be point in this circunference, such that $CD$ is perpedicular to $AB$. Let $E$ be the point of intersection of the segment $CD$ and the segment $AB$, and a point $P$ that is in the segment $CD, P$ is different of $E$. The lines $AP$ and $BP$ intersects $\omega$, in $F$ and $G$ respectively. If $O$ is the circumcenter of triangle $EFG$, show that the area of triangle $OCD$ is invariant, independent of the position of the point $P$.
Brazil NMO L2 (OBM) - geometry
Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$. The line $OA$ intersects the segment $EH$ in the point $X$, $OB$ intersects $EF$ in the point $Y$, $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W$. If the $(EFGH) = 1$. Find:

$(ABCD) \times (XYZW)$

Note $(P)$ denote the area of the polygon $P$.