Film Academy "Oscar" A committee of $3366$ film critics are voting for the Oscars. Every critic voted just an actor and just one actress. After the voting, it was found that for every positive integer $n \in \left\\{1; 2; ... ; 100\right\\}$, there is some actor or some actress who was voted exactly $n$ times. Prove that there are two critics who voted the same actor and the same actress. **My work so far:** All actors and actresses got together at least $5050$ votes from critics. Since $3366$ the critics, it is not less than $5050-3366 = 1684$ critics counted twice.

Source Serbian Mathematical Olympiad 2015

* * *

Assume the contrary. For each $i = 1,..., 100$ choose a candidate $A_i$ who was voted for exactly $i$ times.

The number of judges who gave both their votes for candidates in the set

$A = \\{A_{34}, A_{35},..., A_{100}\\}$

does not exceed the number of pairs actor-actress in $A$, and the number of such pairs is at most $33 \cdot 34 = 1122$.

On the other hand, of the $2 \cdot 3366 = 6732$ votes, exactly

$34 + 35 +···+ 100 = 4489$

were given to the candidates in $A$.

Therefore at most $6732 − 4489 = 2243$ judges could have given a vote to a candidate not in $A$.

Thus, there were at most $1122 + 2243 = 3365$ judges, a contradiction.