Problem 6. Find all positive integers d for which there exists a degree d polynomial P with real coefficients such that there are at most d different values among P (0) , P (1) , P (2) , . . . , P ( d2− d ).
Solution
We claim that the answer is $d=1,2$ only which clearly work, now assume $d \geq 3$. Firstly notice that if there are $\leq d-1$ distinct values then by lagrange interpolation the polynomial must be constant polynomial, and hence there will be exactly $d-1$ distinct values each appearing $d-1$ times while one value appearing $d$ times, now notice that scaling has no effect on the statement of the problem, WLOG $$P(0)=P(a_1)=P(a_2)= \cdots = P(a_{d-2})=P(a_{d-1})=0$$,(because we have $d$ equal values) so $$P = x(x-a_1)(x-a_2)\cdots (x-a_{d-2})(x-a_{d-1})$$for non-negative integers $a_i$. Let the other $d-1$ sequences be $B_1,B_2 \cdots B_{d-1}$ where $B_i = \{{B_i}_j\}$ where $j$ varies from $1$ to $d-1$ and let $P(B_i)=C_i$ then we have $\prod (x-{B_i}_j) \mid P(x)-C_i$, let $S= a_1+a_2+ \cdots a_{d-1}$, and therefore we have $$P(x)=(\prod (x-{B_i}_j))(x-(S-\sum {B_i}_j))+C_i $$due to vieta's formulas and because $P$ is monic polynomial. Now, again by vieta's we have $(-1)^{d-1}(\prod {B_i}_j)(S-\sum {B_i}_j)=C_i$ now the idea is to put $x=S- \sum {B_i}_j$ and then again compare and do bounding to get $d=1,2$.
 
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