Navvye/Olympiad-Problems-CSV huggingface dataset:
2004 Mexico National Olympiad
Let $\omega_1$ and $\omega_2$ be two circles such that the center $O$ of $\omega_2$ lies in $\omega_1$. Let $C$ and $D$ be the two intersection points of the circles. Let $A$ be a point on $\omega_1$ and let $B$ be a point on $\omega_2$ such that $AC$ is tangent to $\omega_2$ in C and BC is tangent to $\omega_1$ in $C$. The line segment $AB$ meets $\omega_2$ again in $E$ and also meets $\omega_1$ again in F. The line $CE$ meets $\omega_1$ again in $G$ and the line $CF$ meets the line $GD$ in $H$. Prove that the intersection point of $GO$ and $EH$ is the center of the circumcircle of the triangle $DEF$.
2004 Mexico National Olympiad
What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of $2004 \times 2004$, if the path does not cross the same place twice?.
2003 Mexico National Olympiad
Find all positive integers with two or more digits such that if we insert a $0$ between the units and tens digits we get a multiple of the original number.
2003 Mexico National Olympiad
$A, B, C$ are collinear with $B$ betweeen $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$.
2003 Mexico National Olympiad
At a party there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of then women. For which $r$ and $s$ must there be a man and a woman who like each other?
2003 Mexico National Olympiad
The quadrilateral $ABCD$ has $AB$ parallel to $CD$. $P$ is on the side $AB$ and $Q$ on the side $CD$ such that $\frac{AP}{PB}= \frac{DQ}{CQ}$. M is the intersection of $AQ$ and $DP$, and $N$ is the intersection of $PC$ and $QB$. Find $MN$ in terms of $AB$ and $CD$.
2003 Mexico National Olympiad
Some cards each have a pair of numbers written on them. There is just one card for each pair $(a,b)$ with $1 \leq a < b \leq 2003$. Two players play the following game. Each removes a card in turn and writes the product $ab$ of its numbers on the blackboard. The first player who causes the greatest common divisor of the numbers on the blackboard to fall to $1$ loses. Which player has a winning strategy?
2003 Mexico National Olympiad
Given a positive integer $n$, an allowed move is to form $2n+1$ or $3n+2$. The set $S_{n}$ is the set of all numbers that can be obtained by a sequence of allowed moves starting with $n$. For example, we can form $5 \rightarrow 11 \rightarrow 35$ so $5, 11$ and $35$ belong to $S_{5}$. We call $m$ and $n$ compatible if $S_{m}$ and $S_{n}$ has a common element. Which members of $\{1, 2, 3, ... , 2002\}$ are compatible with $2003$?
2002 Mexico National Olympiad
The numbers $1$ to $1024$ are written one per square on a $32 \times  32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times  16$ boards and the position of these boards is moved round clockwise, so that



$AB$ goes to $DA$

$DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,  CB$



then each of the $16 \times  16 $ boards is divided into four equal $8 \times  8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times  8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times  2$ parts which are moved around, and finally the squares of each $2 \times  2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?
2002 Mexico National Olympiad
$ABCD$ is a parallelogram. $K$ is the circumcircle of $ABD$. The lines $BC$ and $CD$ meet $K$ again at $E$ and $F$. Show that the circumcenter of $CEF$ lies on $K$.
2002 Mexico National Olympiad
Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?
2002 Mexico National Olympiad
A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?
2002 Mexico National Olympiad
A trio  is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which trio  contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all trios  with the maximum sum.
2002 Mexico National Olympiad
Let $ABCD$ be a quadrilateral with $\measuredangle DAB=\measuredangle ABC=90^{\circ}$. Denote by $M$ the midpoint of the side $AB$, and assume that $\measuredangle CMD=90^{\circ}$. Let $K$ be the foot of the perpendicular from the point $M$ to the line $CD$. The line $AK$ meets $BD$ at $P$, and the line $BK$ meets $AC$ at $Q$. Show that $\angle{AKB}=90^{\circ}$ and $\frac{KP}{PA}+\frac{KQ}{QB}=1$.



[Moderator edit: The proposed solution can be found at  .]
2001 Mexico National Olympiad
Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$.
2001 Mexico National Olympiad
Given some colored balls (at least three different colors) and at least three boxes. The balls are put into the boxes so that no box is empty and we cannot find three balls of different colors which are in three different boxes. Show that there is a box such that all the balls in all the other boxes have the same color.
2001 Mexico National Olympiad
$ABCD$ is a cyclic quadrilateral. $M$ is the midpoint of $CD$. The diagonals meet at $P$. The circle through $P$ which touches $CD$ at $M$ meets $AC$ again at $R$ and $BD$ again at $Q$. The point $S$ on $BD$ is such that $BS = DQ$. The line through $S$ parallel to $AB$ meets $AC$ at $T$. Show that $AT = RC$.
2001 Mexico National Olympiad
For positive integers $n, m$ define $f(n,m)$ as follows. Write a list of $ 2001$ numbers $a_i$, where $a_1 = m$, and $a_{k+1}$ is the residue of $a_k^2$ $mod \, n$ (for $k = 1, 2,..., 2000$). Then put $f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}$. For which $n \ge 5$ can we find m such that $2 \le  m \le n/2$ and $f(m,n) > 0$?
2001 Mexico National Olympiad
$ABC$ is a triangle with $AB < AC$ and $\angle A = 2 \angle C$. $D$ is the point on $AC$ such that $CD = AB$. Let L be the line through $B$ parallel to $AC$. Let $L$ meet the external bisector of $\angle A$ at $M$ and the line through $C$ parallel to $AB$ at $N$. Show that $MD = ND$.
2001 Mexico National Olympiad
A collector of rare coins has coins of denominations $1, 2,..., n$ (several coins for each denomination).

He wishes to put the coins into $5$ boxes so that:

(1) in each box there is at most one coin of each denomination;

(2) each box has the same number of coins and the same denomination total;

(3) any two boxes contain all the denominations;

(4) no denomination is in all $5$ boxes.

For which $n$ is this possible?
2000 Mexico National Olympiad
Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$, and $D$ and $A$ at $S$. Circles $A$ and $C$ do not meet, and so do not $B$ and $D$.

(a) Prove that the points $P,Q,R,S$ lie on a circle.

(b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$, and the distance between the centers of $A$ and $C$ is $6$. Compute the area of the quadrilateral $PQRS$.
2000 Mexico National Olympiad
A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row?



1 2 3 4 5

3 5 7 9

8 12 16

20 28

4
2000 Mexico National Olympiad
Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way:

Write down some elements of $A$ one after another without repeating, write a sign $+ $ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$.

For example, if $A = \{2,8,13,20\}$, numbers $8$ and $14 = 20-2+8$ are elements of $A'$.

Set $A''$ is constructed from $A'$ in the same manner.

Find the smallest possible number of elements of $A$, if $A''$ contains all the integers from $1$ to $40$.
2000 Mexico National Olympiad
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number  of primes that can occur before encoutering the first composite term?
2000 Mexico National Olympiad
A board $n$×$n$ is coloured black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells)whose side lengths are both odd or both even, but not both equal to $1$, and invert the colours of all cells inside the rectangle. Determine the values of $n$ for which it is possible to make all the cells have the same colour in a finite number of such steps.
2000 Mexico National Olympiad
Let $ABC$ be a triangle with $\angle B > 90^o$ such that there is a point $H$ on side $AC$ with $AH = BH$ and BH perpendicular to $BC$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$ respectively. A line through $H$ parallel to $AB$ cuts $DE$ at $F$. Prove that $\angle BCF = \angle ACD$.
1999 Mexico National Olympiad
On a table there are $1999$ counters, red on one side and black on the other side, arranged arbitrarily. Two people alternately make moves, where each move is of one of the following two types:

(1) Remove several counters which all have the same color up;

(2) Reverse several counters which all have the same color up.

The player who takes the last counter wins. Decide which of the two players (the one playing first or the other one) has a wining strategy.
1999 Mexico National Olympiad
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.
1999 Mexico National Olympiad
A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively.

(a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$.

(b) Prove that $DL,EM$, and $FN$ are concurrent.
1999 Mexico National Olympiad
An $8 \times 8$ board is divided into unit squares. Ten of these squares have their centers marked. Prove that either there exist two marked points on the distance at most $\sqrt2$, or there is a point on the distance $1/2$ from the edge of the board.
1999 Mexico National Olympiad
In a quadrilateral $ABCD$ with $AB  // CD$, the external bisectors of the angles at $B$ and $C$ meet at $P$, while the external bisectors of the angles at $A$ and $D$ meet at $Q$. Prove that the length of $PQ$ equals the semiperimeter of $ABCD$.
1999 Mexico National Olympiad
A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping $2 x 1$ dominos, then at least one of its sides has even length.
1998 Mexico National Olympiad
A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.
1998 Mexico National Olympiad
Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$. For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$, let $T$ be the intersection point of the bisector of angle $QPR$ with $C$. Describe the locus of $T$ and justify your answer.
1998 Mexico National Olympiad
Every side and diagonal of a regular octagon is color with red or black. Show that there is at least seven triangles whose vertices are vertices of the octagon and its three sides are of the same color.
1998 Mexico National Olympiad
Find all integers that can be written in the form $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{9}{a_9}$ where $a_1,a_2, ...,a_9$ are nonzero digits, not necessarily different.
1998 Mexico National Olympiad
The tangents at points $B$ and $C$ on a given circle meet at point $A$. Let $Q$ be a point on segment $AC$ and let $BQ$ meet the circle again at $P$. The line through $Q $ parallel to $AB$ intersects $BC$ at $J$. Prove that $PJ$ is parallel to $AC$ if and only if $BC^2 = AC\cdot QC$.
1998 Mexico National Olympiad
A plane in space is equidistant from a set of points if its distances from the points in the set are equal. What is the largest possible number of equidistant planes from five points, no four of which are coplanar?
1997 Mexico National Olympiad
Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.
1997 Mexico National Olympiad
In a triangle $ABC, P$ and $P'$ are points on side $BC, Q$ on side $CA$, and $R $ on side $AB$, such that $\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}$ . Let $G$ be the centroid of triangle $ABC$ and $K$ be the intersection point of $AP'$ and $RQ$. Prove that points $P,G,K$ are collinear.
1997 Mexico National Olympiad
The numbers $1$ through $16$ are to be written in the cells of a $4\times 4$ board.

(a) Prove that this can be done in such a way that any two numbers in cells that share a side differ by at most $4$.

(b) Prove that this cannot be done in such a way that any two numbers in cells that share a side differ by at most $3$.
1997 Mexico National Olympiad
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?
1997 Mexico National Olympiad
Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.
1997 Mexico National Olympiad
Prove that number $1$ has infinitely many representations of the form  $$1 =\frac{1}{5}+\frac{1}{a_1}+\frac{1}{a_2}+ ...+\frac{1}{a_n}$$, where$ n$ and $a_i $ are positive integers with $5 < a_1 < a_2 < ... < a_n$.
1996 Mexico National Olympiad
Let $P$ and $Q$ be the points on the diagonal $BD$ of a quadrilateral $ABCD$ such that $BP = PQ = QD$. Let $AP$ and $BC$ meet at $E$, and let $AQ$ meet $DC$ at $F$.

(a) Prove that if $ABCD$ is a parallelogram, then $E$ and $F$ are the midpoints of the corresponding sides.

(b) Prove the converse of (a).
1996 Mexico National Olympiad
There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip $1$ moves one booth, chip $2$ moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip $1$ a bulb is lit. Where is chip $1$ on the first minute in which all bulbs are lit?
1996 Mexico National Olympiad
Prove that it is not possible to cover a $6\times  6$ square board with eighteen $2\times 1$ rectangles, in such a way that each of the lines going along the interior gridlines cuts at least one of the rectangles. Show also that it is possible to cover a $6\times 5$ rectangle with fifteen $2\times 1 $ rectangles so that the above condition is fulfilled.
1996 Mexico National Olympiad
For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$?
1996 Mexico National Olympiad
The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of  right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes.

(a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube.

(b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.
1996 Mexico National Olympiad
In a triangle $ABC$ with $AB < BC < AC$, points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp  CA$ and $BB'=CA$, and $CC' \perp AB$ and $CC'= AB$, as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C' $ are collinear.
1995 Mexico National Olympiad
$N$ students are seated at desks in an $m \times  n$ array, where $m, n \ge  3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$?
1995 Mexico National Olympiad
Consider 6 points on a plane such that 8 of the distances between them are equal to 1. Prove that there are at least 3 points that form an equilateral triangle.
1995 Mexico National Olympiad
$A, B, C, D$ are consecutive vertices of a regular $7$-gon. $AL$ and $AM$ are tangents to the circle center $C$ radius $CB$. $N$ is the point of intersection of $AC$ and $BD$. Show that $L, M, N$ are collinear.
1995 Mexico National Olympiad
Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square.

Show that one cannot find $27$ such elements.
1995 Mexico National Olympiad
$ABCDE$ is a convex pentagon such that the triangles $ABC, BCD, CDE, DEA$ and $EAB$ have equal areas. Show that $(1/4)$ area $(ABCDE) <$ area $(ABC) < (1/3)$ area $(ABCDE)$.
1995 Mexico National Olympiad
A $1$ or $0$ is placed on each square of a $4 \times  4$ board. One is allowed to change each symbol in a row, or change each symbol in a column, or change each symbol in a diagonal (there are $14$ diagonals of lengths $1$ to $4$). For which arrangements can one make changes which end up with all $0$s?
1994 Mexico National Olympiad
The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$.
1994 Mexico National Olympiad
The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.
1994 Mexico National Olympiad
$ABCD$ is a parallelogram. Take $E$ on the line $AB$ so that $BE = BC$ and $B$ lies between $A$ and $E$. Let the line through $C$ perpendicular to $BD$ and the line through $E$ perpendicular to $AB$ meet at $F$. Show that $\angle DAF = \angle BAF$.
1994 Mexico National Olympiad
A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be read in the following order. Take the last unread page ($400$), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$. What is the last page to be read?
1994 Mexico National Olympiad
$ABCD$ is a convex quadrilateral. Take the $12$ points which are the feet of the altitudes in the triangles $ABC, BCD, CDA, DAB$. Show that at least one of these points must lie on the sides of $ABCD$.
1994 Mexico National Olympiad
Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$, or with $25$ pieces of type $B$, or with $25$ pieces of type $C$.
1993 Mexico National Olympiad
$ABC$ is a triangle with $\angle A = 90^o$. Take $E$ such that the triangle $AEC$ is outside $ABC$ and $AE = CE$ and $\angle AEC = 90^o$. Similarly, take $D$ so that $ADB$ is outside $ABC$ and similar to $AEC$. $O$ is the midpoint of $BC$. Let the lines $OD$ and $EC$ meet at $D'$, and the lines $OE$ and $BD$ meet at $E'$. Find area $DED'E'$ in terms of the sides of $ABC$.
1993 Mexico National Olympiad
Find all numbers between $100$ and $999$ which equal the sum of the cubes of their digits.
1993 Mexico National Olympiad
Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.
1993 Mexico National Olympiad
$f(n,k)$ is defined by

(1) $f(n,0) = f(n,n) = 1$ and

(2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$.

How many times do we need to use (2) to find $f(3991,1993)$?
1993 Mexico National Olympiad
$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$, the circles with diameters $OB, OC$ meet again at $X$, and the circles with diameters $OC, OA$ meet again at $Y$. Show that $X, Y, Z$ are collinear.
1993 Mexico National Olympiad
$p$ is an odd prime. Show that $p$ divides $n(n+1)(n+2)(n+3) + 1$ for some integer $n$ iff $p$ divides $m^2 - 5$ for some integer $m$.
1992 Mexico National Olympiad
The tetrahedron $OPQR$ has the $\angle POQ = \angle POR = \angle QOR = 90^o$. $X, Y, Z$ are the midpoints of $PQ, QR$ and $RP.$ Show that the four faces of the tetrahedron $OXYZ$ have equal area.
1992 Mexico National Olympiad
Given a prime number $p$, how many $4$-tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$?
1992 Mexico National Olympiad
Given $7$ points inside or on a regular hexagon, show that three of them form a triangle with area $\le 1/6$ the area of the hexagon.
1992 Mexico National Olympiad
Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.
1992 Mexico National Olympiad
$x, y, z$ are positive reals with sum $3$. Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$
1992 Mexico National Olympiad
$ABCD$ is a rectangle. $I$ is the midpoint of $CD$. $BI$ meets $AC$ at $M$. Show that the line $DM$ passes through the midpoint of $BC$. $E$ is a point outside the rectangle such that $AE = BE$ and $\angle AEB = 90^o$. If $BE = BC = x$, show that $EM$ bisects $\angle AMB$. Find the area of $AEBM$ in terms of $x$.
1991 Mexico National Olympiad
Evaluate the sum of all positive irreducible fractions less than $1$ and having the denominator $1991$.
1991 Mexico National Olympiad
A company of $n$ soldiers is such that

(i) $n$ is a palindrome number (read equally in both directions);

(ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively.

Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.
1991 Mexico National Olympiad
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?
1991 Mexico National Olympiad
The diagonals $AC$ and $BD$ of a convex quarilateral $ABCD$ are orthogonal. Let $M,N,R,S$ be the midpoints of the sides $AB,BC,CD$ and $DA$ respectively, and let $W,X,Y,Z$ be the projections of the points $M,N,R$ and $S$ on the lines $CD,DA,AB$ and $BC$, respectively. Prove that the points $M,N,R,S,W,X,Y$ and $Z$ lie on a circle.