Problem 3. Marco wrote 6 different numbers on 3 cards, one on each side of the cards. The sums of the two numbers written on each of the three cards are equal. Marco placed the cards so that only three numbers are visible, as shown in the diagram. It is known that the hidden numbers are prime. Which numbers did Marco hide.

![](https://cdn.mathpix.com/cropped/2024_06_05_a1dcf8241dfa57377282g-09.jpg?height=156&width=310&top_left_y=1883&top_left_x=1172)
Solution. Let $S$ be the sum of the numbers on each of the cards individually. If $S$ is an even number, then the hidden numbers on the second and third cards must be even numbers, which contradicts the condition of the problem that the hidden numbers are different prime numbers. Therefore, this case is not possible (there is only one even prime number, 2). Therefore, $S$ is an odd number, i.e., the hidden number on the card with the number 99 is even and prime, i.e., 2. Then $S=101$ and therefore the hidden number on the second card is $101-78=23$, and on the third card is $101-60=41$.