2. A nonahedron was formed by gluing a cube and a regular quadrilateral pyramid. On each face of this nonahedron, a number is written. Their sum is 3003. For each face $S$ of the considered nonahedron, we sum the numbers on all faces that share exactly one edge with $S$. This gives us nine identical sums. Determine all the numbers written on the faces of the nonahedron.
2. Let $A, B, C, D, E, F, G, H$ be the vertices of the mentioned cube and $V$ the vertex of the attached pyramid (Fig. 1). The numbers written on the side faces $A B F E, B C G F, C D H G, D A E H$ are denoted by $a_{1}, a_{2}, a_{3}$, and $a_{4}$, respectively, and the numbers on the side faces $E F V, F G V, G H V$, and $H E V$ of the attached pyramid are denoted by $b_{1}, b_{2}, b_{3}$, and $b_{4}$, respectively. The number on the base $A B C D$ is denoted by $c$. Let $s$ be the given common sum.

By comparing the sums of the corresponding faces $E F V$ and $G H V$, we get the equality $a_{1}=a_{3}$, and similarly for the next pair of faces, we get $a_{2}=a_{4}$. By comparing the sums of the corresponding faces $A B F E$ and $C D H G$, we get $b_{1}=b_{3}$, and similarly for the next such pair of faces, we get $b_{2}=b_{4}$. By comparing the sums of the corresponding faces $C D H G$ and $H E V$, we get the equality $b_{1}=c+a_{2}$, and similarly for the pair of faces $D A H E$ and $G H V$, we get the equality $b_{2}=c+a_{1}$. By comparing the sums of two adjacent faces of the cube

![](https://cdn.mathpix.com/cropped/2024_04_17_fa4e6f4674eb91bf6d8dg-2.jpg?height=488&width=412&top_left_y=1618&top_left_x=1450)

Fig. 1, we get $a_{2}+a_{4}+b_{1}=a_{1}+a_{3}+b_{2}$, or $2 a_{2}+b_{1}=2 a_{1}+b_{2}$, which, by substituting from the last two obtained equalities, gives the equality $a_{2}=a_{1}$, and thus also $b_{2}=b_{1}=c+a_{1}$. Therefore, we can write $a_{1}=a_{2}=a_{3}=a_{4}=a, b_{1}=b_{2}=b_{3}=b_{4}=b=a+c$, and from the equality of the sums of the corresponding base and one of the side faces of the cube, we get $4 a=2 a+b+c=3 a+2 c$, so $a=2 c$, $b=3 c$, and the total sum of all numbers is $c+4 a+4 b=21 c$. From the equation $21 c=3003$, we get $c=143$. The numbers on the faces of the nonagon are 143, 286 (four times), and 429 (four times).

Note. The problem can also be solved by writing and solving a system of ten linear equations for the nine unknown numbers written on the faces of the body and the tenth unknown equal to the common sum.

Full solution is worth 6 points, of which 1 point for the equalities $a_{3}=a_{1}, a_{4}=a_{2}$ and 1 point for the equalities $b_{3}=b_{1}, b_{4}=b_{2}$. If these symmetries are only intuitively guessed, deduct 2 points.