MathDB

2

Part of 2023 Mexican Girls' Contest

Problems(2)

quadrilateral with midpoints

Source: 2nd National Women's Contest of Mexican Mathematics Olympiad 2023 , level 1+2 p2

7/22/2023
Matilda drew 1212 quadrilaterals. The first quadrilateral is an rectangle of integer sides and 77 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 11. 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] <spanclass=′latex−bold′>Note:</span><span class='latex-bold'>Note:</span> The above figure illustrates the first two quadrilaterals that Matilda drew.
Mexicoalgebra
the number of paths is odd

Source: 2nd National Women&#039;s Contest of Mexican Mathematics Olympiad 2023 , problem 2 teams

7/23/2023
In the city of Las Cobayas\textrm{Las Cobayas}, the houses are arranged in a rectangular grid of 33 rows and n≥2n\geq 2 columns, as illustrated in the figure. Mich\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, Mich\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 Mich´s\textrm{Mich´s} position (circle), and the houses to which he can move (triangles). Let f(n)f(n) be the number of ways Mich\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)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("nn", (5.22,-1.15), dir(0), fontsize(10)); label("West\textrm{West}", (-2,2.5), dir(0), fontsize(10)); label("East\textrm{East}", (11.1,2.5), dir(0), fontsize(10)); label("North\textrm{North}", (4.5,5.7), dir(0), fontsize(10)); label("South\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]
pathsMexico