2026 AMC 8 Problem 25

Question

In an equiangular hexagon, all interior angles measure

120^\circ.

An example of such a hexagon with side lengths of

2,

3,

1,

3,

2,

and

2

is shown below, inscribed in equilateral triangle

ABC.

Consider all equiangular hexagons with positive integer side lengths that can be inscribed in

\triangle ABC,

with all six vertices on the sides of the triangle. What is the total number of such hexagons? Hexagons that differ only by a rotation or a reflection are considered the same.

4

5

6

7

8

Answer: E

Concepts:equiangular polygon equilateral triangle systematic listing

Difficulty rating: 2000

Small hint:

Let

x,y,z

be the small corner lengths at

A,B,C

Big hint:

The triangle side is

6

, so

x+y

,

y+z

, and

z+x

are all less than

6

Video solution:

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Written solution:

The example shows that each side of

\triangle ABC

has length

6

. Let

x,y,z

be the small corner lengths cut off at

A,B,C

, respectively. Then the other three side lengths of the hexagon are

6-x-y

,

6-y-z

, and

6-z-x

. All six side lengths are positive integers exactly when

x,y,z

are positive integers and each pair sum is less than

6

. Up to rotation and reflection, this means counting unordered triples

x\le y\le z

with

y+z\lt6

:

(1,1,1)

,

(1,1,2)

,

(1,1,3)

,

(1,1,4)

,

(1,2,2)

,

(1,2,3)

,

(2,2,2)

, and

(2,2,3)

. There are

8

such hexagons.

2026 AMC 8 Problem 25

Answer: E

Difficulty rating: 2000

Small hint:

Big hint:

Video solution:

Written solution:

Problem 25 in other years