## Task 4 - 240624

Rita is experimenting with a balance scale.

(With such a scale, it can be determined whether the content of one pan weighs as much as the content of the other pan or which of the two contents weighs more than the other.)

Rita has 17 balls, 6 cubes, and 1 pyramid. She observes:

(1) Each ball weighs as much as each of the other balls.

(2) Each cube weighs as much as each of the other cubes.

(3) The pyramid and 5 cubes together weigh as much as 14 balls.

(4) A cube and 8 balls together weigh as much as the pyramid.

Rolf asks Rita, after she has made these observations:

"How many balls weigh as much as the pyramid?"

Prove that Rolf's question can already be answered uniquely using observations (1), (2), (3), (4), without the need for any further weighing! What is the answer?
If $k, w$ and $p$ are the weights of a sphere, a cube, and the pyramid, respectively, then from (1) and (2) it follows that each sphere has the weight $k$ and each cube has the weight $w$. From (3) and (4) it further follows that

\[
\begin{aligned}
& p+5 w=14 k \\
& w+8 k=p
\end{aligned}
\]

Due to (6), (5) states $w+8 k+5 w=14 k$, thus $6 w=6 k$ and consequently $w=k$. According to this, (6) yields $9 k=p$.

Thus, it is proven that Rolf's question can be answered uniquely using the observations (1), (2), (3), (4). The answer is: 9 spheres weigh as much as the pyramid.

Solutions of the II. Round 1984 taken from [5]

### 3.26 XXV. Olympiad 1985

### 3.26.1 I. Round 1985, Class 6