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There are 100 people. On each front an integer from 1 to 100 is written (they can be repeated).
Each person sees everyone else's 99 numbers, but not their own. After thinking as much as they want, everyone simultaneously writes their own bet.
They win if at least one person gets their number exactly right. Before entering they can agree on a strategy.
Then there is no communication. Is there a strategy that always guarantees victory?
Back to the problem
Answer: Yes, there is a guaranteed winning strategy.
Number people like $0,1,\dots,99$ (mod 100).
Also represent front numbers modulo 100 (100 is interpreted as 0).
Person $i$:
$$ g_i\equiv i-s_i\pmod{100}. $$
Let $S$ be the real sum of the 100 numbers (mod 100).
Person $i$ sees $s_i\equiv S-x_i$, where $x_i$ is their real number.
So:
$$ g_i\equiv i-(S-x_i)\equiv x_i+(i-S)\pmod{100}. $$
Therefore, $g_i=x_i$ exactly when $i\equiv S\pmod{100}$.
Since among $0,\dots,99$ there is exactly one index equal to $S$, exactly one person is sure to get it right.
Conclusion: At least one person always wins.
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