location:
 PROBLEMS FOR JULY AND AUGUST

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

 a3a2 + b2 + b3b2 + c2 + c3c2 + a2 ³ 12 .

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 Ri and ri re the respective circumradius and inradius of triangle Ai Bi Ci (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 £ 2n-2.

165.
Let n be a positive integer. Determine all n-tples { a1, a2, ¼, an } of positive integers for which a1 + a2 + ¼+ an = 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,

 12n + 121/n < 1 .

170.
Solve, for real x,

 x ·21/x + 1x ·2x = 4 .

 top of page | contact us | privacy | site map |