Found problems: 40
2021 Regional Olympiad of Mexico Center Zone, 4
Two types of pieces, bishops and rooks, are to be placed on a $10\times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to [i]attack[/i] a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column.
A piece $P$ is [i]chocolate[/i] if no other piece $Q$ attacks $P$.
What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?
[i]Proposed by José Alejandro Reyes González[/i]
2024 Mexico National Olympiad, 2
Determine all pairs $(a, b)$ of integers that satisfy both:
1. $5 \leq b < a$
2. There exists a natural number $n$ such that the numbers $\frac{a}{b}$ and $a-b$ are consecutive divisors of $n$, in that order.
[b]Note:[/b] Two positive integers $x, y$ are consecutive divisors of $m$, in that order, if there is no divisor $d$ of $m$ such that $x < d < y$.
2024 Mexican Girls' Contest, 4
There are 6 squares in a row. Each one is labeled with the name of Ana or Beto and with a number from 1 to 6, using each number without repetition. Ana and Beto take turns painting each square according to the order of the numbers on the labels. Whoever paints the square will be the person whose name is on the label. When painting, the person can choose to paint the square either red or blue. Beto wins if at the end there are the same number of blue squares as red squares, and Ana wins otherwise. In how many of all the possible ways of labeling the squares can Beto ensure his victory?
The following is an example of a labeling of the labels.
[asy]
size(12cm);
draw((0,0)--(6,0)--(6,-1)--(0,-1)--cycle);
for (int i=1; i<6; ++i) {
draw((i,0)--(i,-1));
}
for (int i=1; i<6; ++i) {
draw((i,0)--(i,-1.25));
}
draw((0,0)--(6,0)--(6,-1.25)--(0,-1.25)--cycle);
for (int i=1; i<7; ++i) {
draw((i-0.5,-1)--(i-0.5,-1.25));
}
label("Ana", (0.25, -1.125));
label("Beto", (1.25, -1.125));
label("Ana", (2.25, -1.125));
label("Beto", (3.25, -1.125));
label("Ana", (4.25, -1.125));
label("Beto", (5.25, -1.125));
label("1", (0.75, -1.125));
label("3", (1.75, -1.125));
label("5", (2.75, -1.125));
label("2", (3.75, -1.125));
label("4", (4.75, -1.125));
label("6", (5.75, -1.125));
[/asy]
First Ana paints the first square, then Beto paints the fourth square, then Beto paints the second square, and so on.
2021 Regional Olympiad of Mexico Southeast, 4
Hernan wants to paint a $8\times 8$ board such that every square is painted with blue or red. Also wants to every $3\times 3$ subsquare have exactly $a$ blue squares and every $2\times 4$ or $4\times 2$ rectangle have exactly $b$ blue squares. Find all couples $(a,b)$ such that Hernan can do the required.
2024 Mexico National Olympiad, 5
Let $A$ and $B$ infinite sets of positive real numbers such that:
1. For any pair of elements $u \ge v$ in $A$, it follows that $u+v$ is an element of $B$.
2. For any pair of elements $s>t$ in $B$, it follows that $s-t$ is an element of $A$.
Prove that $A=B$ or there exists a real number $r$ such that $B=\{2r, 3r, 4r, 5r, \dots\}$.
2023 Mexican Girls' Contest, 1
Let $\triangle ABC$ such that $AB=AC$, $D$ and $E$ points on $AB$ and $BC$, respectively, with $DE\parallel AC$. Let $F$ on line $DE$ such that $CADF$ it´s a parallelogram. If $O$ is the circumcenter of $\triangle BDE$, prove that $O,F,A$ and $D$ lie on a circle.
2023 Mexican Girls' Contest, 8
There are $3$ sticks of each color between blue, red and green, such that we can make a triangle $T$ with sides sticks with all different colors. Dana makes $2$ two arrangements, she starts with $T$ and uses the other six sticks to extend the sides of $T$, as shown in the figure. This leads to two hexagons with vertex the ends of these six sticks. Prove that the area of the both hexagons it´s the same.
[asy]size(300);
pair A, B, C, D, M, N, P, Q, R, S, T, U, V, W, X, Y, Z, K;
A = (0, 0);
B = (1, 0);
C=(-0.5,2);
D=(-1.1063,4.4254);
M=(-1.7369,3.6492);
N=(3.5,0);
P=(-2.0616,0);
Q=(0.2425,-0.9701);
R=(1.6,-0.8);
S=(7.5164,0.8552);
T=(8.5064,0.8552);
U=(7.0214,2.8352);
V=(8.1167,-1.546);
W=(9.731,-0.7776);
X=(10.5474,0.8552);
Y=(6.7813,3.7956);
Z=(6.4274,3.6272);
K=(5.0414,0.8552);
draw(A--B, blue);
label("$b$", (A + B) / 2, dir(270), fontsize(10));
label("$g$", (B+C) / 2, dir(10), fontsize(10));
label("$r$", (A+C) / 2, dir(230), fontsize(10));
draw(B--C,green);
draw(D--C,green);
label("$g$", (C + D) / 2, dir(10), fontsize(10));
draw(C--A,red);
label("$r$", (C + M) / 2, dir(200), fontsize(10));
draw(B--N,green);
label("$g$", (B + N) / 2, dir(70), fontsize(10));
draw(A--P,red);
label("$r$", (A+P) / 2, dir(70), fontsize(10));
draw(A--Q,blue);
label("$b$", (A+Q) / 2, dir(540), fontsize(10));
draw(B--R,blue);
draw(C--M,red);
label("$b$", (B+R) / 2, dir(600), fontsize(10));
draw(Q--R--N--D--M--P--Q, dashed);
draw(Y--Z--K--V--W--X--Y, dashed);
draw(S--T,blue);
draw(U--T,green);
draw(U--S,red);
draw(T--W,red);
draw(T--X,red);
draw(S--K,green);
draw(S--V,green);
draw(Y--U,blue);
draw(U--Z,blue);
label("$b$", (Y+U) / 2, dir(0), fontsize(10));
label("$b$", (U+Z) / 2, dir(200), fontsize(10));
label("$b$", (S+T) / 2, dir(100), fontsize(10));
label("$r$", (S+U) / 2, dir(200), fontsize(10));
label("$r$", (T+W) / 2, dir(70), fontsize(10));
label("$r$", (T+X) / 2, dir(70), fontsize(10));
label("$g$", (U+T) / 2, dir(70), fontsize(10));
label("$g$", (S+K) / 2, dir(70), fontsize(10));
label("$g$", (V+S) / 2, dir(30), fontsize(10));
[/asy]
2023 Mexican Girls' Contest, 2
Matilda drew $12$ quadrilaterals. The first quadrilateral is an rectangle of integer sides and $7$ times more width than long. Every time she drew a quadrilateral she joined the midpoints of each pair of consecutive sides with a segment. It´s is known that the last quadrilateral Matilda drew was the first with area less than $1$. What is the maximum area possible for the first quadrilateral?
[asy]size(200);
pair A, B, C, D, M, N, P, Q;
real base = 7;
real altura = 1;
A = (0, 0);
B = (base, 0);
C = (base, altura);
D = (0, altura);
M = (0.5*base, 0*altura);
N = (0.5*base, 1*altura);
P = (base, 0.5*altura);
Q = (0, 0.5*altura);
draw(A--B--C--D--cycle); // Rectángulo
draw(M--P--N--Q--cycle); // Paralelogramo
dot(M);
dot(N);
dot(P);
dot(Q);
[/asy]
$\textbf{Note:}$ The above figure illustrates the first two quadrilaterals that Matilda drew.
2024 Mexican Girls' Contest, 7
Consider the quadratic equation \(x^2 + a_0 x + b_0\) for some real numbers \((a_0, b_0)\). Repeat the following procedure as many times as possible:
Let \(c_i = \min \{r_i, s_i\}\), with \(r_i, s_i\) being the roots of the equation \(x^2 + a_i x + b_i\). The new equation is written as \(x^2 + b_i x + c_i\). That is, for the next iteration of the procedure, \(a_{i+1} = b_i\) and \(b_{i+1} = c_i\).
We say that \((a_0, b_0)\) is an $\textit{interesting}$ pair if, after a finite number of steps, the equation we obtain after one step is the same, so that \((a_i, b_i) = (a_{i+1}, b_{i+1})\). Find all $\textit{interesting}$ pairs.
2024 Mexico National Olympiad, 1
The figure shows all 6 colorings with for different colors of a $1\times 1$ square divided in four $\tfrac{1}{2} \times \tfrac{1}{2}$ cells (two colorings are considered equal if one is the result of rotating the other). Each of the $1\times 1$ colorings will be used as a piece for a puzzle. The pieces can be rotated but not reflected. Two pieces [i]fit[/i] if when sharing a side, the touching $\tfrac{1}{2} \times \tfrac{1}{2}$ cells are the same color respectively (see examples). ¿Is it possible to assemble a $3 \times 2$ puzzle using each of the 6 pieces exactly once and such that every pair of adjacent pieces fit?
[img]https://imagizer.imageshack.com/img922/6019/ZUKcED.jpg[/img]
2023 Regional Olympiad of Mexico Southeast, 1
Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.
2024 Regional Olympiad of Mexico Southeast, 3
A large cube of size \(4 \times 4 \times 4\) is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition:
In each block of \(1 \times 1 \times 4\), \(1 \times 4 \times 1\), and \(4 \times 1 \times 1\) cubes, there must be exactly one red cube.
Determine how many different ways it is possible to choose the 16 small cubes to be colored red.
Note: Two colorings are considered different even if one can be obtained from the other by rotations or symmetries of the cube.
2016 Mexico National Olmypiad, 4
We say a non-negative integer $n$ "[i]contains[/i]" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ [i]contains[/i] $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not [i]contain[/i] a multiple of $7$.
2022 Mexican Girls' Contest, 3
Consider a set $S$ of $16$ lattice points. The $16$ points of $S$ are divided into $8$ pairs in such a way that
[i]for every point $A$ and any of the $7$ pairs of points $(B,C)$ where $A$ is not included, $A$ is at a distance of at most $\sqrt{5}$ from either $B$ or $C$[/i]
Prove that any two points in the set $S$ are at a distance of at most $3\sqrt5$.
2024 Mexico National Olympiad, 4
Let $ABC$ an acute triangle with orthocenter $H$. Let $M$ be a point on segment $BC$. The line through $M$ and perpendicular to $BC$ intersects lines $BH$ and $CH$ in points $P$ and $Q$ respectively. Prove that the orthocenter of triangle $HPQ$ lies on the line $AM$.