Because of the variability of summer plans, the usual ration of problems has been doubled and the deadline set later so that students can have a chance to organize their work conveniently. Send your solutions to Prof. E.J. Barbeau, Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3 no later than September 10, 2002. Please make sure that the front page of your solution contains your complete mailing address and your email address.

Notes. A composite integer is one that has positive divisors other than 1 and itself; it is not prime. A set of point in the plane is concyclic (or cyclic, inscribable) if and only if there is a circle that passes through all of them.

157.

Prove that if the quadratic equation x² + ax + b + 1 = 0 has nonzero integer solutions, then a² + b² is a composite integer.

158.

Let f(x) be a polynomial with real coefficients for which the equation f(x) = x has no real solution. Prove that the equation f(f(x)) = x has no real solution either.

159.

Let 0 £ a £ 4. Prove that the area of the bounded region enclosed by the curves with equations

y = 1 - |x - 1 |

and

y = |2x - a |

cannot exceed ¹/₃.

160.

Let I be the incentre of the triangle ABC and D be the point of contact of the inscribed circle with the side AB. Suppose that ID is produced outside of the triangle ABC to H so that the length DH is equal to the semi-perimeter of DABC. Prove that the quadrilateral AHBI is concyclic if and only if angle C is equal to 90^°.

161.

Let a, b, c be positive real numbers for which a + b + c = 1. Prove that

a3 a2 + b2 + b3 b2 + c2 + c3 c2 + a2 ³ 1 2  .

162.

Let A and B be fixed points in the plane. Find all positive integers k for which the following assertion holds:

among all triangles ABC with |AC | = k |BC |, the one with the largest area is isosceles.

163.

Let R_i and r_i re the respective circumradius and inradius of triangle A_i B_i C_i (i = 1, 2). Prove that, if ÐC₁ = ÐC₂ and R₁r₂ = r₁R₂, then the two triangles are similar.

164.

Let n be a positive integer and X a set with n distinct elements. Suppose that there are k distinct subsets of X for which the union of any four contains no more that n - 2 elements. Prove that k £ 2^n-2.

165.

Let n be a positive integer. Determine all n-tples { a₁, a₂, ¼, a_n } of positive integers for which a₁ + a₂ + ¼+ a_n = 2n and there is no subset of them whose sum is equal to n.

166.

Suppose that f is a real-valued function defined on the reals for which

f(xy) + f(y - x) ³ f(y + x)

for all real x and y. Prove that f(x) ³ 0 for all real x.

167.

Let u = (Ö5 - 2)^1/3 - (Ö5 + 2)^1/3 and v = (Ö[189] - 8)^1/3 - (Ö[189] + 8)^1/3. Prove that, for each positive integer n, uⁿ + vⁿ⁺¹ = 0.

168.

Determine the value of

cos5° + cos77° + cos149° +cos221° + cos293° .

169.

Prove that, for each positive integer n exceeding 1,

1 2n + 1 21/n < 1 .

170.

Solve, for real x,

x ·21/x + 1 x ·2x = 4 .