There are some examples that there are similar problems in math contests (I can't distinguish if it's deliberately copied from math contests or just a coincidence).

• 317E Princess and Her Shadow, which is almost the same as a problem from Canadian National Olympiad link

• 618F Double Knapsack, see the discussion here

• 1684H Hard Cut, see the discussion here

• K in Yuhao Du Contest 5. Yes, I admit the idea is from a problem in All-Russian MO link

Sometimes authors just learn the idea from MO problems. But sometimes, the problem is just identical (like 1684H).

I don't think there is much difference between copying problems from math contests and copying problems from an old opencup contest. However, the community seems much more tolerable of copying problems from math contests.

Maybe the difference is that the opencup problems are known to more participants. So copying problems from math contests has less impact.

Some updates:

For all problems mentioned above, solutions are almost the same as MO problems they correspond to.

The following are more examples that share the same setting with MO problems, but the solutions are not very similar. I feel these problems are ok.

• HDU 4660, commented by MinakoKojima. It shares the same setting with IMO 2011, Q2. But this problem is asking different things.

• I in Yuhao Du Contest 7. It shares the same setting with IMOSL 2009, C5. But it's much harder than that IMOSL problem. So I feel knowing this IMOSL problem doesn't help.

• A Chinese problem. It's definitely inspired by RMM 2019 Q3. And they share a similar idea. But I feel it's not bad since there are still several steps (however, basically implementation issues) to make the proof in the RMM problem work in this problem.

• D in Tianjin 2012. And this problem also happens in a recent Chinese team training contest. As far as I remember, there is a similar IMO problem in the 1990s. But I didn't remember the exact source.