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40 1. Fundamentals

5. Prove that, if the roots of x2 +px + q = 0 are real, then the roots of
   x2 + px + q + (x + u)(2z + p) = 0 will be real for every real number a.


6. A mathematics teacher wrote the quadratic z2 + 101: + 20 on the
   board. Then each student either increased by 1 or decreased by 1
   either the constant or the linear coefficient. Finally x2 + 202 + 10
   appeared. Did a quadratic trinomial with integer zeros necessarily
   appear on the board in the process?


7. Suppose a < b and c < d. Solve the system

a2 + b2 = c2 + d2

a+b+c+d=O.

8. Find necessary and sufficient conditions on the real numbers a, b, c,
   d for the equation

z2 + (u + bi)z + (c + di) = 0

to have exactly one real and one nonreal root.

9. Show that if z2 + px + q = 0 and px2 + qx + 1 = 0 have a common
   root, theneitherp+q+1=Oorp2+q2+1=pq+p+q.


10. If p and q are real numbers which do not take simultaneously the
    values p = 0, q = 1, and if the roots of the equation

(1-q+$) x2 + p(1 + q)z + 4(4 - I) + f =^0

are equal, show that p2 = 4q.

11. Show that all the real values of x which satisfy the equation
    tan(7rcotx) = cot(n tan z) are given by

4tanr=2n+1fJ4n2+4n-15,

when n is a positive or negative integer different from -2, -1, 1.

12. Find all positive integers n for which the quadratic equation

an+1x2 - : + ug +... + Q;+~ + (al + u2 + f.. + a,) =^0

has real roots for all reals al, us,... , a,+~.

13. Let p(z) = r2 +a%+ b have complex coefficients and satisfy Ip( = 1
    whenever 1~1 = 1. Prove that a = b = 0.