How many convex polygons are there, each of which has three consecutive vertices at the points $(0 ; 5)$, $(5 ; 5)$, and $(5 ; 0)$, and the coordinates of the other vertices are also non-negative integers?
Let the vertices of the convex polygon be $A(5 ; 0), B(5 ; 5), C(0 ; 5), D, \ldots$; the origin of the coordinate system is $O$. We will count the polygons that satisfy the conditions based on the number of their vertices.

There is one triangle, which is $A B C$. For a quadrilateral, the fourth vertex, different from $A, B, C$, can be any lattice point that lies inside the triangle $A C O$ or on the sides $C O$ or $O A$ (1st figure).

![](https://cdn.mathpix.com/cropped/2024_05_02_ae94851aa0d7d70b17e7g-1.jpg?height=542&width=501&top_left_y=350&top_left_x=801)

The number of such points is 15, so there are 15 quadrilaterals. The pentagon $A B C D E$ satisfies the conditions if the vertex $E$ is to the right of the line $C O$ - in the half-plane containing $A$ - and below the line $A D$ - in the half-plane containing $O$ - as shown in the 2nd figure.

![](https://cdn.mathpix.com/cropped/2024_05_02_ae94851aa0d7d70b17e7g-1.jpg?height=574&width=548&top_left_y=1106&top_left_x=775)

Depending on the choice of $D$, we get a different number of pentagons, as shown in the 2nd figure. In total, there are $10+8+4+6+4+2+4+3+2+1=44$ valid pentagons. Using the same method, we can count the hexagons (3rd figure), which number 16.

![](https://cdn.mathpix.com/cropped/2024_05_02_ae94851aa0d7d70b17e7g-1.jpg?height=547&width=504&top_left_y=1849&top_left_x=797)

There is only one heptagon, as shown in the 4th figure. Polygons with more vertices are not possible because among the at least 5 vertices different from $A, B, C$, there would either be two with the same first or second coordinate, making the polygon non-convex, or there would be three on the same line.

![](https://cdn.mathpix.com/cropped/2024_05_02_ae94851aa0d7d70b17e7g-2.jpg?height=548&width=523&top_left_y=127&top_left_x=779)

In total, there are $1+15+44+16+1=77$ polygons that satisfy the conditions.

Based on the work of Bálint Rákóczi (Fazekas M. Főv. Gyak. Gimn., II. o. t.)