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7.4. Problems on Inequalities 225

7.4 Problems on Inequalities

1. Find all real triples (x, y, z) for which (l-~)~+(x-y)~+(y-~)~+z~ =
   l/4.


2. Prove that, for real x, y, Z,

(x + y)z 5 (1/2)(Z2 + y2) + Z2.

3. Suppose that a and b are nonzero real numbers and that all the zeros
   of the real polynomial

at” - at”-’ + u,-2tn-2 +... + azt2 - n2bt + b = 0

are real and positive. Prove that all the zeros are equal.

4. Show that the real polynomial t” + at”-’ + btns2 + +.. + k has at
   least one nonreal zero if a2 - 2b < n(k”/“).


5. If z, y, z 1 0, prove that

x3 + y3 + z3 2 y2z + z2x + x2y

and determine when there is equality.

6. Prove, that, if x, y,z > 0, then

x(y - 2)” + Y(” - *)2 1 (x - *)(y - %)(X + y - *).

7. Show that, if all the zeros of at4 - bt3 + ct2 - t + 1 are positive, then
   c >80a + b.


8. Prove that, for x 1 0 and n a positive integer,

x”+l - (n + 1)x + n 1 0.

9. Let o, b, c, d > 0 and c2 + d2 = (a” + b2)3. Show that

a3/c + b3/d 2 1

with equality iff ad = bc.

10. Suppose N(x) = 0 for x negative, N(0) = 1 and N(x) = N(x - 6) +x
    for x positive. Show that, for each positive integer x, that

(x + 1)(x + 5)/12 5 N(x) 5 (x2 + 6x + 12)/12.

11. Let A, B, C be the angles of a triangle. Prove that

tan2(A/2) + tan2(B/2) + tan2(C/2) 2 1.