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?

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][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][/asy] $\textbf{Note:}$ The above figure illustrates the first two quadrilaterals that Matilda drew.

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"?

A function $g$ is such that for all integer $n$: $$g(n)=\begin{cases} 1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\ 0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 & \end{cases}$$ A function $f$ is such that for all integers $n\geq 0$ and $m\geq 0$: $$f(0,m)=0 \hspace{0.5cm} \textrm{and}$$$$f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)$$ Find all the possible functions $f(m,n)$ that satisfies the above for all integers $n\geq0$ and $m\geq 0$

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?

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.

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 $x$, $y$, $z$.

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][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][/asy]

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.

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][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][/asy]

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}