A regular octagon $A B C D E F G H$ and a point $X$ are given such that point $A$ is the orthocenter (intersection of the altitudes) of triangle $B D X$. Calculate the sizes of the interior angles of this triangle.

( V. Žádník)
Let $A$ be the orthocenter of triangle $B D X$. This means that on the line connecting point $A$ with vertex $B$, there is a perpendicular to side $D X$, and similarly, on the line connecting $A$ with vertex $D$, there is a perpendicular to side $B X$. Therefore, side $D X$ is perpendicular to line $A B$, and similarly, side $B X$ is perpendicular to line $A D$. Given that points $A, B$, and $D$ are vertices of a regular octagon, the following holds:

1. The perpendicular to $A B$ passing through point $D$ is line $C D$. (The exterior angle of a regular octagon is $45^{\circ}$. The angle between $A B$ and $C D$ is the angle at vertex $P$ in triangle $B P C$, and thus it is a right angle.)
2. The perpendicular to $A D$ passing through point $B$ is line $B G$. (Lines $A D$ and $B G$ are diagonals parallel to sides $B C$ and $A H$. These are perpendicular to sides $A B$ and $C D$, and therefore the mentioned diagonals are perpendicular.)
3. Point $X$ is the intersection of lines $C D$ and $B G$.

![](https://cdn.mathpix.com/cropped/2024_04_17_b3bac5a705eebcae18d0g-4.jpg?height=1068&width=1037&top_left_y=157&top_left_x=515)

Now we can easily determine all the angles in triangle $B D X$. Triangle $B C D$ is isosceles with legs $B C$ and $C D$, and angle $B C D$ is an interior angle of the regular octagon. From this, we calculate

$$
|\angle C D B|=|\angle C B D|=\frac{1}{2}\left(180^{\circ}-135^{\circ}\right)=22.5^{\circ} .
$$

Lines $A D$ and $B C$ are parallel, the angle between $A D$ and $B G$ is a right angle, so angle $C B X$ is also a right angle. From this, we express

$$
|\angle D B X|=22.5^{\circ}+90^{\circ}=112.5^{\circ} .
$$

The last unknown interior angle in triangle $B D X$ has the size

$$
|\angle D X B|=180^{\circ}-22.5^{\circ}-112.5^{\circ}=45^{\circ} .
$$

Evaluation. 3 points for finding point $X$; 1 point each for the angles sought, including justification.

Note. We have a similar problem from the home round (Z9-I-4), so the initial analysis can be shortened: $X$ is the intersection of the altitudes in triangle $A B D$.

In the above, we use the crucial fact several times that the sum of the interior angles in any triangle is $180^{\circ}$. From this, in particular, it follows that the interior angle in a regular octagon is $135^{\circ}$.