Algebra 3933 =x 12 +x 22 +···+xn^2 ≥nn√
x^21 x 22 ···xn^2 =n.Therefore,n≤3. Eliminating case by case, we find among linear polynomialsx+1 and
x−1, and among quadratic polynomialsx^2 +x−1 andx^2 −x−1. As for the cubic
polynomials, we should have equality in the AM–GM inequality. So all zeros should
have the same absolute values. The polynomial should share a zero with its derivative.
This is the case only forx^3 +x^2 −x−1 andx^3 −x^2 −x+1, which both satisfy the
required property. Together with their negatives, these are all desired polynomials.
(Indian Olympiad Training Program, 2005)
160.The first Viète relation gives
r 1 +r 2 +r 3 +r 4 =−b
a,
sor 3 +r 4 is rational. Also,
r 1 r 2 +r 1 r 3 +r 1 r 4 +r 2 r 3 +r 2 r 4 +r 3 r 4 =c
a.
Therefore,
r 1 r 2 +r 3 r 4 =c
a
−(r 1 +r 2 )(r 3 +r 4 ).Finally,
r 1 r 2 r 3 +r 1 r 2 r 4 +r 1 r 3 r 4 +r 2 r 3 r 4 =−
d
a,
which is equivalent to
(r 1 +r 2 )r 3 r 4 +(r 3 +r 4 )r 1 r 2 =−d
a.
We observe that the productsr 1 r 2 andr 3 r 4 satisfy the linear system of equations
αx+βy=u,
γx+δy=v,whereα=1,β=1,γ =r 3 +r 4 ,δ=r 1 +r 2 ,u=ac−(r 1 +r 2 )(r 3 +r 4 ),v=−da.
Becauser 1 +r 2 =r 3 +r 4 , this system has a unique solution; this solution is rational.
Hence bothr 1 r 2 andr 3 r 4 are rational, and the problem is solved.
(64th W.L. Putnam Mathematical Competition, 2003)
161.First solution: Letα=arctanu,β=arctanv, and arctanw. We are required to
determine the sumα+β+γ. The addition formula for the tangent of three angles,