Problem 11. An athlete has a collection of 2 gold and 5 silver medals. They are numbered from 1 to 7 in some order. The figure shows black and white photos of the medals. It's known that in each photo exactly one of the medals is gold. What is the sum of the numbers on the two gold medals? So we know that in each of these pictures there's exactly one gold medal. So that means that in the first picture there's exactly one gold medal. And so either 1 is gold, 2 is gold, or 3 is gold. And we can look at each of those three cases. So first let's look at the case where 1 is gold. So we have two pictures where 1 is the one gold medal, and there are four pictures left. And we need some other medal to be gold such that all those four pictures that right now don't have any gold medals will have exactly one. But if we look at these four, right, we can see that there's no medal that appears in all four pictures. So no matter which medal we pick for our second gold medal, one of these remaining four pictures that doesn't have a gold medal right now still won't have a gold medal. And so this case is impossible. Now we can look at the case where 2 is gold. Again, we have four pictures left. And again, there is no one medal that appears in all four of these pictures. So for example, if we pick 4 to be another gold medal, then these bottom two still won't have any gold medals. If we pick 5, then these three will have a gold medal, but this one won't. If we pick 6 or 7, then this one here won't have a gold medal. So either way, no matter what we pick, there's a picture that doesn't have any gold medals. So this case also doesn't work. So now our final case is where 3 is gold. So when 3 is gold, we already have three pictures that have exactly one gold medal in it. Now if we look at the remaining three pictures, we see that we have 6 appear exactly once in each of these pictures. So if we make 6 also gold, then it will look like this, and each picture has exactly one gold medal. So the two gold medals are 3 and 6, and then we have 3 plus 6 is 9, so the answer is C.

athlete

gold medals

silver medals

photo conditions

medal numbers