Let $n$ be an integer greater than $3$. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of length $1$. Find the number of convex trapezoids which have vertices among the vertices of the $n^2$ squares of side length $1$, have side lengths less than or equal $3$ and have area equal to $2$ Note: Parallelograms are trapezoids.
Problem
Source: 2021 Pan-African Mathematics Olympiad, Problem 1
Tags: geometry, trapezoid, PAMO
SerdarBozdag
25.05.2021 14:54
Wrong
Ejaifeobuks
25.05.2021 17:01
I got $5n^2-8n+1$
SerdarBozdag
25.05.2021 18:43
Ejaifeobuks wrote: I got $5n^2-8n+1$ Solution please. Which part is wrong in my solution?
JuniorPerelman
27.05.2021 19:11
I think that the solution that you find is not correct
Ejaifeobuks
28.05.2021 02:11
Yeah it isn't
SerdarBozdag
28.05.2021 16:04
Ejaifeobuks wrote: Yeah it isn't Which part? I think we found different answers for (3,1) part.
Ejaifeobuks
28.05.2021 18:08
SerdarBozdag wrote: If $a\ge b$ are side lengths of parallel sides, possibilities are $(a,b)=(1,1),(2,1),(3,1),(2^{1/2},2^{1/2})$ and $(2,2)$. First and the last are the same and $(2,1)$ gives $h=4/3$ and side with length 1 must be parallel to the sides of the square which contradicts with $h=4/3$. Thus $(1,1)$ gives $2n^2-2n$, $(3,1)$ gives $12n^2-24n^2$, $(2^{1/2},2^{1/2})$ gives $(n-1)^2$ . Answer: $15n^2 -28n+1$. In the official solution, there are still more cases