Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Macedonia National Olympiad
In a trapezoid $ABCD$ with a base $AD$, point $L$ is the orthogonal projection of $C$ on $AB$, and $K$ is the point on $BC$ such that $AK$ is perpendicular to $AD$. Let $O$ be the circumcenter of triangle $ACD$. Suppose that the lines $AK , CL$ and $DO$ have a common point. Prove that $ABCD$ is a parallelogram.
2007 Macedonia National Olympiad
Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy $$a | b+c+bc, b | c+a+ca, c | a+b+ab.$$

Prove that at least one of the numbers $a, b, c$ is not prime.
2007 Macedonia National Olympiad
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy

$$ f (x^{3} + y^{3}) = x^{2}f (x) + yf (y^{2})

$$

for all $ x, y \in\mathbb R.$
2007 Macedonia National Olympiad
Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$
2006 Macedonia National Olympiad
A natural number is written on the blackboard. In each step, we erase the units digit and add four times the erased digit to the remaining number, and write the result on the blackboard instead of the initial number. Starting with the number $13^{2006}$, is it possible to obtain the number $2006^{13}$ by repeating this step finitely many times?
2006 Macedonia National Olympiad
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x, y, z,$

$$f(x+y^2+z)=f(f(x))+yf(y)+f(z). $$
2006 Macedonia National Olympiad
Let $a,b,c$ be real numbers distinct from $0$ and $1$, with $a+b+c=1$. Prove that

$$8\left(\frac{1}{2}-ab-bc-ca\right)\left(\frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} \right)\ge 9 $$
2006 Macedonia National Olympiad
Let $M$ be a point on the smaller arc $A_1A_n$ of the circumcircle of a regular $n$-gon $A_1A_2\ldots A_n$ .



$(a)$ If $n$ is even, prove that $\sum_{i=1}^n(-1)^iMA_i^2=0$.



$(b)$ If $n$ is odd, prove that $\sum_{i=1}^n(-1)^iMA_i=0$.
2006 Macedonia National Olympiad
All segments joining $n$ points (no three of which are collinear) are coloured in one of $k$ colours. What is the smallest $k$ for which there always exists a closed polygonal line with the vertices at some of the $n$ points, whose sides are all of the same colour?
2001 Macedonia National Olympiad
Prove that if $m$ and $s$ are integers with $ms=2000^{2001}$, then the equation $mx^2-sy^2=3$ has no integer solutions.
2001 Macedonia National Olympiad
Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that

$$f(f(n-1)=f(n+1)-f(n)\quad\text{for all}\ n\ge 2\text{?} $$
2001 Macedonia National Olympiad
Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$, $CA\cap t_B$, and $BC\cap t_A$ exist, and that they are collinear.
2001 Macedonia National Olympiad
Let $\Omega$ be a family of subsets of $M$ such that:



$(\text{i})$ If $|A\cap B|\ge 2$ for $A,B\in\Omega$, then $A=B$;

$(\text{ii})$ There exist different subsets $A,B,C\in\Omega$ with $|A\cap B\cap C|=1$;

$(\text{iii})$ For every $A\in\Omega$ and $a\in M \ A$, there is a unique $B\in\Omega$ such that $a\in B$ and $A\cap B=\emptyset$.



Prove that there are numbers $p$ and $s$ such that:



$(1)$ Each $a\in M$ is contained in exactly $p$ sets in $\Omega$;

$(2)$ $|A|=s$ for all $A\in\Omega$;

$(3)$ $s+1\ge p$.
2000 Macedonia National Olympiad
Let $AB$ be a diameter of a circle with centre $O$, and $CD$ be a chord perpendicular to $AB$. A chord $AE$ intersects $CO$ at $M$, while $DE$ and $BC$ intersect at $N$. Prove that $CM:CO=CN:CB$.
2000 Macedonia National Olympiad
If $a_1,a_2,a_3\ldots a_n$ are positive numbers, find the maximum value of

$$\frac{a_1a_2\ldots a_{n-1}a_n}{(1+a_1)(a_1+a_2)\ldots (a_{n-1}+a_n)(a_n+2^{n+1})} $$
2000 Macedonia National Olympiad
In a triangle with sides $a,b,c,t_a,t_b,t_c$ are the corresponding medians and $D$ the diameter of the circumcircle. Prove that

$$\frac{a^2+b^2}{t_c}+\frac{b^2+c^2}{t_a}+\frac{c^2+a^2}{t_b}\le 6D$$
2000 Macedonia National Olympiad
Let $a,b$ be coprime positive integers. Show that the number of positive integers $n$ for which the equation $ax+by=n$ has no positive integer solutions is equal to $\frac{(a-1)(b-1)}{2}-1$.
2010 Malaysia National Olympiad
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.
2010 Malaysia National Olympiad
Adam has RM2010 in his bank account. He donates RM10 to charity every day. His first donation is on Monday. On what day will he donate his last RM10?
2010 Malaysia National Olympiad
Find the number of different pairs of positive integers $(a,b)$ for which $a+b\le100$ and $$\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10$$
2010 Malaysia National Olympiad
Let $ABC$ be a triangle in which $AB=AC$. A point $I$ lies inside the triangle such that $\angle ABI=\angle CBI$ and $\angle BAI=\angle CAI$. Prove that $$\angle BIA=90^o+\dfrac{\angle C}{2}$$
2010 Malaysia National Olympiad
Find the last digit of $$7^1\times 7^2\times 7^3\times \cdots \times 7^{2009}\times 7^{2010}.$$
2010 Malaysia National Olympiad
A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished behind her. How many runners were there in the race?
2010 Malaysia National Olympiad
Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12.
2010 Malaysia National Olympiad
A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?
2010 Malaysia National Olympiad
Let $\gamma=\alpha \times \beta$ where $$\alpha=999 \cdots 9$$ (2010 '9') and $$\beta=444 \cdots 4$$ (2010 '4')

Find the sum of digits of $\gamma$.
2010 Malaysia National Olympiad
Find the number of triples of nonnegative integers $(x,y,z)$ such that $$x^2+2xy+y^2-z^2=9.$$
2010 Malaysia National Olympiad
A two-digit integer is divided by the sum of its digits. Find the largest remainder that can occur.
2010 Malaysia National Olympiad
Let $ABC$ be a triangle in which $AB=AC$ and let $I$ be its incenter. It is known that $BC=AB+AI$. Let $D$ be a point on line $BA$ extended beyond $A$ such that $AD=AI$. Prove that $DAIC$ is a cyclic quadrilateral.
2010 Malaysia National Olympiad
For any number $x$, let $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. A sequence $a_1,a_2,\cdots$ is given, where $$a_n=\left\lfloor{\sqrt{2n}+\dfrac{1}{2}}\right\rfloor.$$

How many values of $k$ are there such that $a_k=2010$?
2010 Malaysia National Olympiad
Let $m$ and $n$ be positive integers such that $2^n+3^m$ is divisible by $5$. Prove that $2^m+3^n$ is divisible by $5$.
2010 Malaysia National Olympiad
Find $x$ such that $$2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}$$
2010 Malaysia National Olympiad
Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isoceles triangle with $a,b$ and $c$ as the length of sides. Determine how many possible three-digit number $N$ there are.



($N=\overline{abc}$ means that $a,b$ and $c$ are digits of $N$, and not $N=a\times b\times c$.)
2010 Malaysia National Olympiad
A semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$, $U$ on the semicircle, and $V$ on $XY$. What is the area of $STUV$?
2010 Malaysia National Olympiad
Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.
2010 Malaysia National Olympiad
Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.
2010 Malaysia National Olympiad
A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.
2010 Malaysia National Olympiad
Show that $$\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)$$ for all $a,b,c$ greater than 1.
2010 Malaysia National Olympiad
Show that there exist integers $m$ and $n$ such that $$\dfrac{m}{n}=\sqrt[3]{\sqrt{50}+7}-\sqrt[3]{\sqrt{50}-7}.$$
2019 Mexico National Olympiad
An integer number $m\geq 1$ is mexica if it's of the form $n^{d(n)}$, where $n$ is a positive integer and $d(n)$ is the number of positive integers which divide $n$. Find all mexica numbers less than $2019$.



Note. The divisors of $n$ include $1$ and $n$; for example, $d(12)=6$, since $1, 2, 3, 4, 6, 12$ are all the positive divisors of $12$.



Proposed by Cuauhtémoc Gómez
2019 Mexico National Olympiad
Let $H$ be the orthocenter of acute-angled triangle $ABC$ and $M$ be the midpoint of $AH$. Line $BH$ cuts $AC$ at $D$. Consider point $E$ such that $BC$ is the perpendicular bisector of $DE$. Segments $CM$ and $AE$ intersect at $F$. Show that $BF$ is perpendicular to $CM$.



Proposed by Germán Puga
2019 Mexico National Olympiad
Let $n\geq 2$ be an integer. Consider $2n$ points around a circle. Each vertex has been tagged with one integer from $1$ to $n$, inclusive, and each one of these integers has been used exactly two times. Isabel divides the points into $n$ pairs, and draws the segments joining them, with the condition that the segments do not intersect. Then, she assigns to each segment the greatest integer between its endpoints.



a) Show that, no matter how the points have been tagged, Isabel can always choose the pairs in such a way that she uses exactly $\lceil n/2\rceil$ numbers to tag the segments.



b) Can the points be tagged in such a way that, no matter how Isabel divides the points into pairs, she always uses exactly $\lceil n/2\rceil$ numbers to tag the segments?



Note. For each real number $x$, $\lceil x\rceil$ denotes the least integer greater than or equal to $x$. For example, $\lceil 3.6\rceil=4$ and $\lceil 2\rceil=2$.



Proposed by Victor Domínguez
2019 Mexico National Olympiad
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list $10,34,34,22,30,22$ the sublist $22,30,22$ is good and $10,34,34,22$ is not. A list is very good if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values on it.
2019 Mexico National Olympiad
Let $a > b$ be relatively prime positive integers. A grashopper stands at point $0$ in a number line. Each minute, the grashopper jumps according to the following rules:



If the current minute is a multiple of $a$ and not a multiple of $b$, it jumps $a$ units forward.

If the current minute is a multiple of $b$ and not a multiple of $a$, it jumps $b$ units backward.

If the current minute is both a multiple of $b$ and a multiple of $a$, it jumps $a - b$ units forward.

If the current minute is neither a multiple of $a$ nor a multiple of $b$, it doesn't move.



Find all positions on the number line that the grasshopper will eventually reach.
2019 Mexico National Olympiad
Let $ABC$ be a triangle such that $\angle BAC = 45^{\circ}$. Let $H,O$ be the orthocenter and circumcenter of $ABC$, respectively. Let $\omega$ be the circumcircle of $ABC$ and $P$ the point on $\omega$ such that the circumcircle of $PBH$ is tangent to $BC$. Let $X$ and $Y$ be the circumcenters of $PHB$ and $PHC$ respectively. Let $O_1,O_2$ be the circumcenters of $PXO$ and $PYO$ respectively. Prove that $O_1$ and $O_2$ lie on $AB$ and $AC$, respectively.
2018 Mexico National Olympiad
Let $A$ and $B$ be two points on a line $\ell$, $M$ the midpoint of $AB$, and $X$ a point on segment $AB$ other than $M$. Let $\Omega$ be a semicircle with diameter $AB$. Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$. Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$. The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$. Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$. Show that $MX > XY$
2018 Mexico National Olympiad
For each positive integer $m$, we define $L_m$ as the figure that is obtained by overlapping two $1 \times m$ and $m \times 1$ rectangles in such a way that they coincide at the $1 \times 1$ square at their ends, as shown in the figure.



[asy]

pair h = (1, 0), v = (0, 1), o = (0, 0);

for(int i = 1; i < 5; ++i)

{

  o = (i*i/2 + i, 0);

  draw(o -- o + i*v -- o + i*v + h -- o + h + v -- o + i*h + v -- o + i*h -- cycle);

  string s = "$L_" + (string)(i) + "$";

  label(s, o + ((i / 2), -1));

  for(int j = 1; j < i; ++j)

  {

    draw(o + j*v -- o + j*v + h);

    draw(o + j*h -- o + j*h + v);

  }

}

label("...", (18, 0.5));

[/asy]



Using some figures $L_{m_1}, L_{m_2}, \dots, L_{m_k}$, we cover an $n \times n$ board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of $m_1 + m_2 + \dots + m_k$.



Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.
2018 Mexico National Olympiad
A sequence $a_2, a_3, \dots, a_n$ of positive integers is said to be campechana, if for each $i$ such that $2 \leq i \leq n$ it holds that exactly $a_i$ terms of the sequence are relatively prime to $i$. We say that the size of such a sequence is $n - 1$. Let $m = p_1p_2 \dots p_k$, where $p_1, p_2, \dots, p_k$ are pairwise distinct primes and $k \geq 2$. Show that there exist at least two different campechana sequences of size $m$.
2018 Mexico National Olympiad
Let $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$.



Proposed by Misael Pelayo
2018 Mexico National Olympiad
Let $n\geq 5$ an integer and consider a regular $n$-gon. Initially, Nacho is situated in one of the vertices of the $n$-gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$, Nacho will have put a flag in each of the $n$ vertices?
2018 Mexico National Olympiad
Let $ABC$ be an acute-angled triangle with circumference $\Omega$. Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$. Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$. Prove that the center of the circle passing through $I$, $M'$, $N'$ lies on the altitude of triangle $ABC$ from $A$.



Proposed by Victor Domínguez and Ariel García
2017 Mexico National Olympiad
A knight is placed on each square of the first column of a $2017 \times 2017$ board. A move consists in choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers $k$ with $1 \leq k \leq 2017$ such that it is possible for each square in the $k$-th column to contain one knight after a finite number of moves.



Note: Two squares are a knight-step away if they are opposite corners of a $2 \times 3$ or $3 \times 2$ board.
2017 Mexico National Olympiad
A set of $n$ positive integers is said to be balanced if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$.
2017 Mexico National Olympiad
Let $ABC$ be an acute triangle with orthocenter $H$. The circle through $B, H$, and $C$ intersects lines $AB$ and $AC$ at $D$ and $E$ respectively, and segment $DE$ intersects $HB$ and $HC$ at $P$ and $Q$ respectively. Two points $X$ and $Y$, both different from $A$, are located on lines $AP$ and $AQ$ respectively such that $X, H, A, B$ are concyclic and $Y, H, A, C$ are concyclic. Show that lines $XY$ and $BC$ are parallel.
2017 Mexico National Olympiad
A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain.
2017 Mexico National Olympiad
On a circle $\Gamma$, points $A, B, N, C, D, M$ are chosen in a clockwise order in such a way that $N$ and $M$ are the midpoints of clockwise arcs $BC$ and $AD$ respectively. Let $P$ be the intersection of $AC$ and $BD$, and let $Q$ be a point on line $MB$ such that $PQ$ is perpendicular to $MN$. Point $R$ is chosen on segment $MC$ such that $QB = RC$, prove that the midpoint of $QR$ lies on $AC$.
2017 Mexico National Olympiad
Let $n \geq 2$ and $m$ be positive integers. $m$ ballot boxes are placed in a line. Two players $A$ and $B$ play by turns, beginning with $A$, in the following manner. Each turn, $A$ chooses two boxes and places a ballot in each of them. Afterwards, $B$ chooses one of the boxes, and removes every ballot from it. $A$ wins if after some turn of $B$, there exists a box containing $n$ ballots. For each $n$, find the minimum value of $m$ such that $A$ can guarantee a win independently of how $B$ plays.
2016 Mexico National Olmypiad
Let $C_1$ and $C_2$ be two circumferences externally tangents at $S$ such that the radius of $C_2$ is the triple of the radius of $C_1$. Let a line be tangent to $C_1$ at $P \neq S$ and to $C_2$ at $Q \neq S$. Let $T$ be a point on $C_2$ such that $QT$ is diameter of $C_2$. Let the angle bisector of $\angle SQT$ meet $ST$ at $R$. Prove that $QR=RT$
2016 Mexico National Olmypiad
A pair of positive integers $m, n$ is called $guerrera$, if there exists positive integers $a, b, c, d$ such that $m=ab$, $n=cd$ and $a+b=c+d$. For example the pair $8, 9$ is $guerrera$ cause $8= 4 \cdot 2$, $9= 3 \cdot 3$ and $4+2=3+3$. We paint the positive integers if the following order:



We start painting the numbers $3$ and $5$.



If a positive integer $x$ is not painted and a positive $y$ is painted such that the pair $x, y$ is $guerrera$, we paint $x$.



Find all positive integers $x$ that can be painted.
2016 Mexico National Olmypiad
Find the minimum real x that satisfies [x]<[x^2]<[x^3)<...<[x^n]<[x^n+1]<...
2016 Mexico National Olmypiad
We say a non-negative integer $n$ "contains" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ contains $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not contain a multiple of $7$.
2016 Mexico National Olmypiad
The numbers from $1$ to $n^2$ are written in order in a grid of $n \times n$, one number in each square, in such a way that the first row contains the numbers from $1$ to $n$ from left to right; the second row contains the numbers $n + 1$ to $2n$ from left to right, and so on and so forth. An allowed move on the grid consists in choosing any two adjacent squares (i.e. two squares that share a side), and add (or subtract) the same integer to both of the numbers that appear on those squares.



Find all values of $n$ for which it is possible to make every squares to display $0$ after making any number of moves as necessary and, for those cases in which it is possible, find the minimum number of moves that are necessary to do this.
2016 Mexico National Olmypiad
Let $ABCD$ a quadrilateral inscribed in a circumference, $l_1$ the parallel to $BC$ through $A$, and $l_2$ the parallel to $AD$ through $B$. The line $DC$ intersects $l_1$ and $l_2$ at $E$ and $F$, respectively. The perpendicular to $l_1$ through $A$ intersects $BC$ at $P$, and the perpendicular to $l_2$ through $B$ cuts $AD$ at $Q$. Let $\Gamma_1$ and $\Gamma_2$ be the circumferences that pass through the vertex of triangles $ADE$ and $BFC$, respectively. Prove that $\Gamma_1$ and $\Gamma_2$ are tangent to each other if and only if $DP$ is perpendicular to $CQ$.
2015 Mexico National Olympiad
Let $ABC$ be an acuted-angle triangle and let $H$ be it's orthocenter. Let $PQ$ be a segment through $H$ such that $P$ lies on $AB$ and $Q$ lies on $AC$ and such that $ \angle PHB= \angle CHQ$. Finally, in the circumcircle of $\triangle ABC$, consider $M$ such that $M$ is the mid point of the arc $BC$ that doesn't contain $A$. Prove that $MP=MQ$



Proposed by Eduardo Velasco/Marco Figueroa
2015 Mexico National Olympiad
Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$. We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the top-right corner of the $n \times n$ board. Then, the $k$ rectangles are painted of black. This process leaves a white figure in the board.



How many different white figures are possible to do with $k$ rectangles that can't be done with less than $k$ rectangles?



Proposed by David Torres Flores
2015 Mexico National Olympiad
Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions:



$f(1)=1$

$f(a+b+ab)=a+b+f(ab)$



Find the value of $f(2015)$



Proposed by Jose Antonio Gomez Ortega
2015 Mexico National Olympiad
Let $n$ be a positive integer. Mary writes the $n^3$ triples of not necessarily distinct integers, each between $1$ and $n$ inclusive on a board. Afterwards, she finds the greatest (possibly more than one), and erases the rest. For example, in the triple $(1, 3, 4)$ she erases the numbers 1 and 3, and in the triple $(1, 2, 2)$ she erases only the number 1,



Show after finishing this process, the amount of remaining numbers on the board cannot be a perfect square.
2015 Mexico National Olympiad
Let $I$ be the incenter of an acute-angled triangle $ABC$. Line $AI$ cuts the circumcircle of $BIC$ again at $E$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $J$ be the reflection of $I$ across $BC$. Show $D$, $J$ and $E$ are collinear.
2015 Mexico National Olympiad
Let $n$ be a positive integer and let $d_1, d_2, \dots, d_k$ be its positive divisors. Consider the number

$$f(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k$$Assume $f(n)$ is a power of 2. Show if $m$ is an integer greater than 1, then $m^2$ does not divide $n$.
2014 Mexico National Olympiad
Each of the integers from 1 to 4027 has been colored either green or red. Changing the color of a number is making it red  if it was green and making it green if it was red. Two positive integers $m$ and $n$ are said to be cuates if either $\frac{m}{n}$ or $\frac{n}{m}$ is a prime number. A step consists in choosing two numbers that are cuates and changing the color of each of them. Show it is possible to apply a sequence of steps such that every integer from 1 to 2014 is green.
2014 Mexico National Olympiad
A positive integer $a$ is said to reduce to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$.

Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.