Found problems: 40
2024 Mexico National Olympiad, 3
Let $ABCDEF$ a convex hexagon, and let $A_1,B_1,C_1,D_1,E_1$ y $F_1$ be the midpoints of $AB,BC,CD,$ $DE,EF$ and $FA$, respectively. Construct points $A_2,B_2,C_2,D_2,E_2$ and $F_2$ in the interior of $A_1B_1C_1D_1E_1F_1$ such that both
1. The sides of the dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ are all equal and
2. $\angle A_1B_2B_1+\angle C_1D_2D_1+\angle E_1F_2F_1=\angle B_1C_2C_1+\angle D_1E_2E_1+\angle F_1A_2A_1=360^\circ$, where all these angles are less than $180 ^\circ$,
Prove that $A_2B_2C_2D_2E_2F_2$ is cyclic.
[b]Note:[/b] Dodecagon $A_2A_1B_2B_1C_2C_1D_2D_1E_2E_1F_2F_1$ is shaped like a 6-pointed star, where the points are $A_1,B_1,C_1,D_1,E_1$ y $F_1$.
2024 Mexican Girls' Contest, 5
Consider the acute-angled triangle \(ABC\). The segment \(BC\) measures 40 units. Let \(H\) be the orthocenter of triangle \(ABC\) and \(O\) its circumcenter. Let \(D\) be the foot of the altitude from \(A\) and \(E\) the foot of the altitude from \(B\). Additionally, point \(D\) divides the segment \(BC\) such that \(\frac{BD}{DC} = \frac{3}{5}\). If the perpendicular bisector of segment \(AC\) passes through point \(D\), calculate the area of quadrilateral \(DHEO\).
2023 Mexican Girls' Contest, 5
Mia has $2$ green sticks of $\textbf{3cm}$ each one, $2$ blue sticks of $\textbf{4cm}$ each one and $2$ red sticks of $\textbf{5cm}$ each one. She wants to make a triangle using the $6$ sticks as it´s perimeter, all at once and without overlapping them. How many non-congruent triangles can make?
2024 Mexican Girls' Contest, 6
On a \(4 \times 4\) board, each cell is colored either black or white such that each row and each column have an even number of black cells. How many ways can the board be colored?
2024 Regional Olympiad of Mexico Southeast, 4
Let \(n\) be a non-negative integer and define \(a_n = 2^n - n\). Determine all non-negative integers \(m\) such that \(s_m = a_0 + a_1 + \dots + a_m\) is a power of 2.
2021 Regional Olympiad of Mexico Center Zone, 6
The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions:
[list]
[*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$
[*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$
[/list]
Prove that $a_n=n$ for all positive integers $n$.
[i]Proposed by José Alejandro Reyes González[/i]
2023 Mexican Girls' Contest, 6
Alka finds a number $n$ written on a board that ends in $5.$ She performs a sequence of operations with the number on the board. At each step, she decides to carry out one of the following two operations:
$1.$ Erase the written number $m$ and write it´s cube $m^3$.
$2.$ Erase the written number $m$ and write the product $2023m$.
Alka performs each operation an even number of times in some order and at least once, she finally obtains the number $r$. If the tens digit of $r$ is an odd number, find all possible values that the tens digit of $n^3$ could have had.
2023 Mexican Girls' Contest, 3
In the country Máxico are two islands, the island "Mayor" and island "Menor". The island "Mayor" has $k>3$ states, with exactly $n>3$ cities each one. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapantla" offer direct flights from every city in Máxico to any other city in Máxico. "Aerocenzontle" only offers direct flights from every city of the island "Mayor" to any other city of the island "Mayor".
If the percentage of flights that connect two cities in the same state it´s the same for the flights of each airline, What is the least number of cities that can be in the island "Mayor"?
2021 Regional Olympiad of Mexico Center Zone, 1
Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a [i]ghost[/i] if it doesn’t appear in $S$.
What is the smallest ghost that is not a multiple of $p$?
[i]Proposed by Guerrero[/i]
2021 Regional Olympiad of Mexico Center Zone, 2
The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple?
[i]Proposed by CDMX[/i]
2016 Mexico National Olmypiad, 6
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 $CQ$.
2023 Mexican Girls' Contest, 1
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?
2016 Mexico National Olmypiad, 5
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.
2023 Mexican Girls' Contest, 2
In the city of $\textrm{Las Cobayas}$, the houses are arranged in a rectangular grid of $3$ rows and $n\geq 2$ columns, as illustrated in the figure. $\textrm{Mich}$ plans to move there and wants to tour the city to visit some of the houses in a way that he visits at least one house from each column and does not visit the same house more than once. During his tour, $\textrm{Mich}$ can move between adjacent houses, that is, after visiting a house, he can continue his journey by visiting one of the neighboring houses to the north, south, east, or west, which are at most four. The figure illustrates one $\textrm{Mich´s}$ position (circle), and the houses to which he can move (triangles). Let $f(n)$ be the number of ways $\textrm{Mich}$ can complete his tour starting from a house in the first column and ending at a house in the last column. Prove that $f(n)$ is odd.
[asy]size(200);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((2,0)--(3,0)--(3,1)--(2,1)--cycle);
draw((4,0)--(5,0)--(5,1)--(4,1)--cycle);
draw((0,2)--(1,2)--(1,3)--(0,3)--cycle);
draw((2,2)--(3,2)--(3,3)--(2,3)--cycle);
draw((4,2)--(5,2)--(5,3)--(4,3)--cycle);
draw((0,4)--(1,4)--(1,5)--(0,5)--cycle);
draw((2,4)--(3,4)--(3,5)--(2,5)--cycle);
draw((4,4)--(5,4)--(5,5)--(4,5)--cycle);
fill(circle((0.5,2.5), 0.4), black);
fill((0.1262,4.15)--(0.8738,4.15)--(0.5,4.7974)--cycle, black);
fill((0.1262,0.15)--(0.8738,0.15)--(0.5,0.7974)--cycle, black);
fill((2.1262,2.15)--(2.8738,2.15)--(2.5,2.7974)--cycle, black);
fill(circle((6,0.5), 0.07), black);
fill(circle((6.3,0.5), 0.07), black);
fill(circle((6.6,0.5), 0.07), black);
fill(circle((6,2.5), 0.07), black);
fill(circle((6.3,2.5), 0.07), black);
fill(circle((6.6,2.5), 0.07), black);
fill(circle((6,4.5), 0.07), black);
fill(circle((6.3,4.5), 0.07), black);
fill(circle((6.6,4.5), 0.07), black);
draw((8,0)--(9,0)--(9,1)--(8,1)--cycle);
draw((10,0)--(11,0)--(11,1)--(10,1)--cycle);
draw((8,2)--(9,2)--(9,3)--(8,3)--cycle);
draw((10,2)--(11,2)--(11,3)--(10,3)--cycle);
draw((8,4)--(9,4)--(9,5)--(8,5)--cycle);
draw((10,4)--(11,4)--(11,5)--(10,5)--cycle);
draw((0,-0.2)--(0,-0.5)--(5.5,-0.5)--(5.5,-0.8)--(5.5,-0.5)--(11,-0.5)--(11,-0.5)--(11,-0.2));
label("$n$", (5.22,-1.15), dir(0), fontsize(10));
label("$\textrm{West}$", (-2,2.5), dir(0), fontsize(10));
label("$\textrm{East}$", (11.1,2.5), dir(0), fontsize(10));
label("$\textrm{North}$", (4.5,5.7), dir(0), fontsize(10));
label("$\textrm{South}$", (4.5,-2), dir(0), fontsize(10));
draw((0.5,2.5)--(2,2.5)--(1.8,2.7)--(2,2.5)--(1.8,2.3));
draw((0.5,2.5)--(0.5,4)--(0.3,3.7)--(0.5,4)--(0.7,3.7));
draw((0.5,2.5)--(0.5,1)--(0.3,1.3)--(0.5,1)--(0.7,1.3));
[/asy]
2024 Regional Olympiad of Mexico Southeast, 1
Find all pairs of positive integers \(a, b\) such that the numbers \(a+1\), \(b+1\), \(2a+1\), \(2b+1\), \(a+3b\), and \(b+3a\) are all prime numbers.
2023 Mexico National Olympiad, 3
Let $ABCD$ be a convex quadrilateral. If $M, N, K$ are the midpoints of the segments $AB, BC$, and $CD$, respectively, and there is also a point $P$ inside the quadrilateral $ABCD$ such that, $\angle BPN= \angle PAD$ and $\angle CPN=\angle PDA$. Show that $AB \cdot CD=4PM\cdot PK$.
2023 Mexican Girls' Contest, 3
Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies:
\begin{eqnarray} a^4+b^2c^2=16a\nonumber \\ b^4+c^2a^2=16b \nonumber\\ c^4+a^2b^2=16c \nonumber \end{eqnarray}
2023 Mexican Girls' Contest, 7
Suppose $a$ and $b$ are real numbers such that $0 < a < b < 1$. Let
$$x= \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{b+a}},\hspace{1cm}
y= \frac{1}{b-a} - \frac{1}{b}\hspace{0.5cm}\textrm{and}\hspace{0.5cm}
z= \frac{1}{\sqrt{b-a}} - \frac{1}{\sqrt{b}}.$$
Show that $x$, $y$, $z$ are always ordered from smallest to largest in the same way, regardless of the choice of $a$ and $b$. Find this order among $x$, $y$, $z$.
2021 Mexico National Olympiad, 6
Determine all non empty sets $C_1, C_2, C_3, \cdots $ such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers $n$ and $m$, the number of elements in the set $C_n$ plus the number of elements in the set $C_m$ equals the sum of the elements in the set $C_{m + n}$.
[i]Note:[/i] We denote $\lvert C_n \lvert$ the number of elements in the set $C_n$, and $S_k$ as the sum of the elements in the set $C_n$ so the problem's condition is that for every $n$ and $m$:
\[\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}\]
is satisfied.
2021 Regional Olympiad of Mexico Center Zone, 3
Let $W,X,Y$ and $Z$ be points on a circumference $\omega$ with center $O$, in that order, such that $WY$ is perpendicular to $XZ$; $T$ is their intersection. $ABCD$ is the convex quadrilateral such that $W,X,Y$ and $Z$ are the tangency points of $\omega$ with segments $AB,BC,CD$ and $DA$ respectively. The perpendicular lines to $OA$ and $OB$ through $A$ and $B$, respectively, intersect at $P$; the perpendicular lines to $OB$ and $OC$ through $B$ and $C$, respectively, intersect at $Q$, and the perpendicular lines to $OC$ and $OD$ through $C$ and $D$, respectively, intersect at $R$. $O_1$ is the circumcenter of triangle $PQR$. Prove that $T,O$ and $O_1$ are collinear.
[i]Proposed by CDMX[/i]
2024 Regional Olympiad of Mexico Southeast, 2
Let \(ABC\) be an acute triangle with circumradius \(R\). Let \(D\) be the midpoint of \(BC\) and \(F\) the midpoint of \(AB\). The perpendicular to \(AC\) through \(F\) and the perpendicular to \(BC\) through \(B\) intersect at \(N\). Prove that \(ND = R\).
2021 Regional Olympiad of Mexico Center Zone, 5
Let $ABCD$ be a parallelogram. Half-circles $\omega_1,\omega_2,\omega_3$ and $\omega_4$ with diameters $AB,BC,CD$ and $DA$, respectively, are erected on the exterior of $ABCD$. Line $l_1$ is parallel to $BC$ and cuts $\omega_1$ at $X$, segment $AB$ at $P$, segment $CD$ at $R$ and $\omega_3$ at $Z$. Line $l_2$ is parallel to $AB$ and cuts $\omega_2$ at $Y$, segment $BC$ at $Q$, segment $DA$ at $S$ and $\omega_4$ at $W$. If $XP\cdot RZ=YQ\cdot SW$, prove that $PQRS$ is cyclic.
[i]Proposed by José Alejandro Reyes González[/i]
2024 Mexican Girls' Contest, 2
There are 50 slips of paper numbered from 1 to 50. It is desired to pick 3 slips such that for any of the three numbers, divided by the greatest common divisor of the other two, the square root of the result is a rational number.
How many unordered triples of slips satisfy this condition?
2022 Mexican Girls' Contest, 2
Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying
$$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$
Find all possible values of $\angle BCP$.
2024 Mexican Girls' Contest, 3
Let \( ABC \) be a triangle and \( D \) the foot of the altitude from \( A \). Let \( M \) be a point such that \( MB = MC \). Let \( E \) and \( F \) be the intersections of the circumcircle of \( BMD \) and \( CMD \) with \( AD \). Let \( G \) and \( H \) be the intersections of \( MB \) and \( MC \) with \( AD \). Prove that \( EG = FH \).