Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Mexico National Olympiad
Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.
2014 Mexico National Olympiad
Problem 4



Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$. Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$, $F$ on $CD$ such that $E$ is midpoint of $DF$, and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$). Prove that the circumference through $D$, $F$ and $G$ is tangent to $BG$.
2014 Mexico National Olympiad
Let $a, b, c$ be positive reals such that $a + b + c = 3$. Prove:



$$ \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2} $$



And determine when equality holds.
2014 Mexico National Olympiad
Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.
2013 Mexico National Olympiad
All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$

Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.
2013 Mexico National Olympiad
Let $ABCD$ be a parallelogram with the angle at $A$ obtuse. Let $P$ be a point on segment $BD$. The circle with center $P$ passing through $A$ cuts line $AD$ at $A$ and $Y$ and cuts line $AB$ at $A$ and $X$. Line $AP$ intersects $BC$ at $Q$ and $CD$ at $R$. Prove $\angle XPY = \angle XQY + \angle XRY$.
2013 Mexico National Olympiad
What is the largest amount of elements that can be taken from the set $\{1, 2, ... , 2012, 2013\}$, such that within them there are no distinct three, say $a$, $b$,and $c$, such that $a$ is a divisor or multiple of $b-c$?
2013 Mexico National Olympiad
A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0).

Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$, $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.
2013 Mexico National Olympiad
A pair of integers is special if it is of the form $(n, n-1)$ or $(n-1, n)$ for some positive integer $n$. Let $n$ and $m$ be positive integers such that pair $(n, m)$ is not special. Show $(n, m)$ can be expressed as a sum of two or more different special pairs if and only if $n$ and $m$ satisfy the inequality $ n+m\geq (n-m)^2 $.

Note: The sum of two pairs is defined as $ (a, b)+(c, d) = (a+c, b+d) $.
2013 Mexico National Olympiad
Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$, define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$, where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$. Show some number $i$, amongst 1, 2, 3, and 4 satisfies

$$\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}$$
2012 Mexico National Olympiad
Let $\mathcal{C}_1$ be a circumference with center $O$, $P$ a point on it and $\ell$ the line tangent to $\mathcal{C}_1$ at $P$. Consider a point $Q$ on $\ell$ different from $P$, and let $\mathcal{C}_2$ be the circumference passing through $O$, $P$ and $Q$. Segment $OQ$ cuts $\mathcal{C}_1$ at $S$ and line $PS$ cuts $\mathcal{C}_2$ at a point $R$ diffferent from $P$. If $r_1$ and $r_2$ are the radii of $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, Prove

$$\frac{PS}{SR} = \frac{r_1}{r_2}.$$
2012 Mexico National Olympiad
Let $n \geq 4$ be an even integer. Consider an $n \times n$ grid. Two cells ($1 \times 1$ squares) are neighbors if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors.

An integer from 1 to 4 is written inside each square according to the following rules:



If a cell has a 2 written on it, then at least two of its neighbors contain a 1.

If a cell has a 3 written on it, then at least three of its neighbors contain a 1.

If a cell has a 4 written on it, then all of its neighbors contain a 1.





Among all arrangements satisfying these conditions, what is the maximum number that can be obtained by adding all of the numbers on the grid?
2012 Mexico National Olympiad
Prove among any $14$ consecutive positive integers there exist $6$ which are pairwise relatively prime.
2012 Mexico National Olympiad
The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied three times the result is $1$, and applying it four times the result is $0$. When the process is applied one or more times to an integer $n$, the result is eventually $0$. The number obtained before obtaining $0$ is called the house of $n$.



How many integers less than $26000$ share the same house as $2012$?
2012 Mexico National Olympiad
Some frogs, some red and some others green, are going to move in an $11 \times 11$ grid, according to the following rules. If a frog is located, say, on the square marked with # in the following diagram, then



If it is red, it can jump to any square marked with an x.

if it is green, it can jump to any square marked with an o.



$$\begin{tabular}{| p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | l}

\hline

&&&&&&\\ \hline

&&x&&o&&\\ \hline

&o&&&&x&\\ \hline

&&&\small{\#}&&&\\ \hline

&x&&&&o&\\ \hline

&&o&&x&&\\ \hline

&&&&&&\\ \hline

\end{tabular}

$$

We say 2 frogs (of any color) can meet at a square if both can get to the same square in one or more jumps, not neccesarily with the same amount of jumps.



Prove if 6 frogs are placed, then there exist at least 2 that can meet at a square.

For which values of $k$ is it possible to place one green and one red frog such that they can meet at exactly $k$ squares?
2012 Mexico National Olympiad
Consider an acute triangle $ABC$ with circumcircle $\mathcal{C}$. Let $H$ be the orthocenter of $ABC$ and $M$ the midpoint of $BC$. Lines $AH$, $BH$ and $CH$ cut $\mathcal{C}$ again at points $D$, $E$, and $F$ respectively; line $MH$ cuts $\mathcal{C}$ at $J$ such that $H$ lies between $J$ and $M$. Let $K$ and $L$ be the incenters of triangles $DEJ$ and $DFJ$ respectively. Prove $KL$ is parallel to $BC$.
2011 Mexico National Olympiad
$25$ lightbulbs are distributed in the following way: the first $24$ are placed on a circumference, placing a bulb at each vertex of a regular $24$-gon, and the remaining bulb is placed on the center of said circumference.

At any time, the following operations may be applied:



Take two vertices on the circumference with an odd amount of vertices between them, and change the state of the bulbs on those vertices and the center bulb.

Take three vertices on the circumference that form an equilateral triangle, change the state of the bulbs on those vertices and the center bulb.





Prove from any starting configuration of on and off lightbulbs, it is always possible to reach a configuration where all the bulbs are on.
2011 Mexico National Olympiad
Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.
2011 Mexico National Olympiad
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:



$$a_1^2 + a_1 - 1 = a_2$$$$ a_2^2 + a_2 - 1 = a_3$$$$\hspace*{3.3em} \vdots $$$$a_{n}^2 + a_n - 1 = a_1$$
2011 Mexico National Olympiad
Find the smallest positive integer that uses exactly two different digits when written in decimal notation and is divisible by all the numbers from $1$ to $9$.
2011 Mexico National Olympiad
A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.
2011 Mexico National Olympiad
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circumferences intersecting at points $A$ and $B$. Let $C$ be a point on line $AB$ such that $B$ lies between $A$ and $C$. Let $P$ and $Q$ be points on $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively such that $CP$ and $CQ$ are tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, $P$ is not inside $\mathcal{C}_2$ and $Q$ is not inside $\mathcal{C}_1$. Line $PQ$ cuts $\mathcal{C}_1$ at $R$ and $\mathcal{C}_2$ at $S$, both points different from $P$, $Q$ and $B$. Suppose $CR$ cuts $\mathcal{C}_1$ again at $X$ and $CS$ cuts $\mathcal{C}_2$ again at $Y$. Let $Z$ be a point on line $XY$. Prove $SZ$ is parallel to $QX$ if and only if $PZ$ is parallel to $RX$.
2010 Mexico National Olympiad
Find all triplets of natural numbers $(a,b,c)$ that satisfy the equation $abc=a+b+c+1$.
2010 Mexico National Olympiad
In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off).



Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.
2010 Mexico National Olympiad
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be externally tangent at a point $A$. A line tangent to $\mathcal{C}_1$ at $B$ intersects $\mathcal{C}_2$ at $C$ and $D$; then the segment $AB$ is extended to intersect $\mathcal{C}_2$ at a point $E$. Let $F$ be the midpoint of $\overarc{CD}$ that does not contain $E$, and let $H$ be the intersection of $BF$ with $\mathcal{C}_2$. Show that $CD$, $AF$, and $EH$ are concurrent.
2010 Mexico National Olympiad
Let $n$ be a positive integer. In an $n\times4$ table, each row is equal to



$$\begin{tabular}{| c | c | c | c |}

\hline

2 & 0 & 1 & 0 \\
\hline

\end{tabular}$$



A change is taking three consecutive boxes in the same row with different digits in them and changing the digits in these boxes as follows:



$$0\to1\text{, }1\to2\text{, }2\to0\text{.}$$



For example, a row $\begin{tabular}{| c | c | c | c |}\hline 2 & 0 & 1 & 0 \\ \hline\end{tabular}$ can be changed to the row $\begin{tabular}{| c | c | c | c |}\hline 0 & 1 & 2 & 0 \\ \hline\end{tabular}$ but not to $\begin{tabular}{| c | c | c | c |}\hline 2 & 1 & 2 & 1 \\ \hline\end{tabular}$ because $0$, $1$, and $0$ are not distinct.



Changes can be applied as often as wanted, even to items already changed. Show that for $n<12$, it is not possible to perform a finite number of changes so that the sum of the elements in each column is equal.
2010 Mexico National Olympiad
Let $ABC$ be an acute triangle with $AB\neq AC$, $M$ be the median of $BC$, and $H$ be the orthocenter of $\triangle ABC$. The circumcircle of $B$, $H$, and $C$ intersects the median $AM$ at $N$. Show that $\angle ANH=90^\circ$.
2010 Mexico National Olympiad
Let $p$, $q$, and $r$ be distinct positive prime numbers. Show that if



$$pqr\mid (pq)^r+(qr)^p+(rp)^q-1,$$



then



$$(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).$$
2009 Mexico National Olympiad
In $\triangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. A circle centered at $D$ with radius $AD$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Show that $\triangle AQP\sim\triangle ABC$.
2009 Mexico National Olympiad
In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules:



$\bullet$ If $p$ is a prime number, we place it in box $1$.



$\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$.



Find all positive integers $n$ that are placed in the box labeled $n$.
2009 Mexico National Olympiad
Let $a$, $b$, and $c$ be positive numbers satisfying $abc=1$. Show that



$$\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1$$
2009 Mexico National Olympiad
Let $n>1$ be an odd integer, and let $a_1$, $a_2$, $\dots$, $a_n$ be distinct real numbers. Let $M$ be the maximum of these numbers and $m$ the minimum. Show that it is possible to choose the signs of the expression $s=\pm a_1\pm a_2\pm\dots\pm a_n$ so that



$$m<s<M$$
2009 Mexico National Olympiad
Consider a triangle $ABC$ and a point $M$ on side $BC$. Let $P$ be the intersection of the perpendiculars from $M$ to $AB$ and from $B$ to $BC$, and let $Q$ be the intersection of the perpendiculars from $M$ to $AC$ and from $C$ to $BC$. Show that $PQ$ is perpendicular to $AM$ if and only if $M$ is the midpoint of $BC$.
2009 Mexico National Olympiad
At a party with $n$ people, it is known that among any $4$ people, there are either $3$ people who all know one another or $3$ people none of which knows another. Show that the $n$ people can be separated into two rooms, so that everyone in one room knows one another and no two people in the other room know each other.
2008 Mexico National Olympiad
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$. Find all values of $n$ such that $n=d_2^2+d_3^3$.
2008 Mexico National Olympiad
Consider a circle $\Gamma$, a point $A$ on its exterior, and the points of tangency $B$ and $C$ from $A$ to $\Gamma$. Let $P$ be a point on the segment $AB$, distinct from $A$ and $B$, and let $Q$ be the point on $AC$ such that $PQ$ is tangent to $\Gamma$. Points $R$ and $S$ are on lines $AB$ and $AC$, respectively, such that $PQ\parallel RS$ and $RS$ is tangent to $\Gamma$ as well. Prove that $[APQ]\cdot[ARS]$ does not depend on the placement of point $P$.
2008 Mexico National Olympiad
Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.



$$\begin{tabular}{| c | c | c | c | c | c | c | c |}

\hline

1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline

9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\hline

17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\
\hline

25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\
\hline

33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\
\hline

41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 \\
\hline

49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 \\
\hline

57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 \\
\hline

\end{tabular}$$



Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain?



Note. For any $2\times3$ or $3\times2$ rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner.
2008 Mexico National Olympiad
A king decides to reward one of his knights by making a game. He sits the knights at a round table and has them call out $1,2,3,1,2,3,\dots$ around the circle (that is, clockwise, and each person says a number). The people who say $2$ or $3$ immediately lose, and this continues until the last knight is left, the winner.



Numbering the knights initially as $1,2,\dots,n$, find all values of $n$ such that knight $2008$ is the winner.
2008 Mexico National Olympiad
We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices.



a) Prove that $\frac23E\le V$.

b) Can $E=V$?
2008 Mexico National Olympiad
The internal angle bisectors of $A$, $B$, and $C$ in $\triangle ABC$ concur at $I$ and intersect the circumcircle of $\triangle ABC$ at $L$, $M$, and $N$, respectively. The circle with diameter $IL$ intersects $BC$ at $D$ and $E$; the circle with diameter $IM$ intersects $CA$ at $F$ and $G$; the circle with diameter $IN$ intersects $AB$ at $H$ and $J$. Show that $D$, $E$, $F$, $G$, $H$, and $J$ are concyclic.
2007 Mexico National Olympiad
Find all integers $N$ with the following property: for $10$ but not $11$ consecutive positive integers, each one is a divisor of $N$.
2007 Mexico National Olympiad
Given an equilateral $\triangle ABC$, find the locus of points $P$ such that $\angle APB=\angle BPC$.
2007 Mexico National Olympiad
Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that



$$\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2$$
2007 Mexico National Olympiad
The fraction $\frac1{10}$ can be expressed as the sum of two unit fraction in many ways, for example, $\frac1{30}+\frac1{15}$ and $\frac1{60}+\frac1{12}$.



Find the number of ways that $\frac1{2007}$ can be expressed as the sum of two distinct positive unit fractions.
2007 Mexico National Olympiad
In each square of a $6\times6$ grid there is a lightning bug on or off. One move is to choose three consecutive squares, either horizontal or vertical, and change the lightning bugs in those $3$ squares from off to on or from on to off. Show if at the beginning there is one lighting bug on and the rest of them off, it is not possible to make some moves so that at the end they are all turned off.
2007 Mexico National Olympiad
Let $ABC$ be a triangle with $AB>BC>CA$. Let $D$ be a point on $AB$ such that $CD=BC$, and let $M$ be the midpoint of $AC$. Show that $BD=AC$ and that $\angle BAC=2\angle ABM.$
2006 Mexico National Olympiad
Let $ab$ be a two digit number. A positive integer $n$ is a relative of $ab$ if:



The units digit of $n$ is $b$.

The remaining digits of $n$ are nonzero and add up to $a$.



Find all two digit numbers which divide all of their relatives.
2006 Mexico National Olympiad
Let $ABC$ be a right triangle with a right angle at $A$, such that $AB < AC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$. Let $E$ be the intersection of the parallel to $AC$ that passes through $M$, with the perpendicular on $BD$ passing through $B$. Show that the triangles  $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$.
2006 Mexico National Olympiad
Let $n$ be an integer greater than $1$. In how many ways can we fill all the numbers $1, 2,..., 2n$ in the boxes of a grid of $2\times n$, one in each box, so that any two consecutive numbers are they in squares that share one side of the grid?
2006 Mexico National Olympiad
For which positive integers $n$ can be covered a ladder like that of the figure (but with $n$ steps instead of $4$) with $n$ squares of integer sides, not necessarily the same size, without these squares overlapping and without standing out from the edge of the figure ?
2006 Mexico National Olympiad
Let $ABC$ be an acute triangle , with altitudes $AD,BE$ and $CF$. Circle of diameter $AD$ intersects the sides $AB,AC$ in $M,N$ respevtively. Let $P,Q$ be the intersection points of $AD$ with $EF$ and $MN$ respectively. Show that $Q$ is the midpoint of $PD$.
2006 Mexico National Olympiad
Let n be the sum of the digits in a natural number A. The number A it's said to be "surtido" if every number 1,2,3,4....,n can be expressed as a sum of digits in A.



a)Prove that, if 1,2,3,4,5,6,7,8 are sums of digits in A, then A is "Surtido"

b)If 1,2,3,4,5,6,7 are sums of digits in A, does it follow that A is "Surtido"?
2005 Mexico National Olympiad
Let $O$ be the center of the circumcircle of an acute triangle $ABC$, let $P$ be any point inside the segment $BC$. Suppose the circumcircle of triangle $BPO$ intersects the segment $AB$ at point $R$ and the circumcircle of triangle $COP$ intersects $CA$ at point $Q$.

(i) Consider the triangle $PQR$, show that it is similar to triangle $ABC$ and that $O$ is its orthocenter.

(ii) Show that the circumcircles of triangles $BPO$, $COP$, $PQR$ have the same radius.
2005 Mexico National Olympiad
Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$.

(i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices.

(ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.
2005 Mexico National Olympiad
Already the complete problem:



Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers:



$\rightarrow\qquad\frac{a + xy}{b}$, $\frac{a + xy^2}{b^2}$, $\frac{a + xy^3}{b^3}$, $\ldots$, $\frac{a + xy^n}{b^n}$, $\ldots$



One idea?



:arrow: View all the problems from XIX Mexican Mathematical Olympiad
2005 Mexico National Olympiad
A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$.

Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.
2005 Mexico National Olympiad
Let $N$ be an integer greater than $1$. A deck has $N^3$ cards, each card has one of $N$ colors, has one of $N$ figures and has one of $N$ numbers (there are no two identical cards). A collection of cards of the deck is "complete" if it has cards of every color, or if it has cards of every figure or of all numbers. How many non-complete collections are there such that, if you add any other card from the deck, the collection becomes complete?
2005 Mexico National Olympiad
Let $ABC$ be a triangle and $AD$ be the angle bisector of $<BAC$, with $D$ on $BC$. Let $E$ be a point on segment $BC$ such that $BD = EC$. Through $E$ draw $l$ a parallel line to $AD$ and let $P$ be a point in $l$ inside the triangle. Let $G$ be the point where $BP$ intersects $AC$ and $F$ be the point where $CP$ intersects $AB$. Show $BF = CG$.
2004 Mexico National Olympiad
Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.
2004 Mexico National Olympiad
Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$
2004 Mexico National Olympiad
Let $Z$ and $Y$ be the tangency points of the incircle of the triangle $ABC$ with the sides $AB$ and $CA$, respectively. The parallel line to $Y Z$ through the midpoint $M$ of $BC$, meets $CA$ in $N$. Let $L$ be the point in $CA$ such that $NL = AB$ (and $L$ on the same side of $N$ than $A$). The line $ML$ meets $AB$ in $K$. Prove that $KA = NC$.
2004 Mexico National Olympiad
At the end of a soccer tournament in which any pair of teams played between them exactly once, and in which there were not draws, it was observed that for any three teams $A, B$ and C, if $A$ defeated $B$ and $B$ defeated $C$, then $A$ defeated $C$. Any team calculated the difference (positive) between the number of games that it won and the number of games it lost. The sum of all these differences was $5000$. How many teams played in the tournament? Find all possible answers.