Hints 241
- Multiply by 4 and complete the square on both sides.
- Find a quadratic equation for y in terms of z.
- Multiply by a and find two factors whose product contains the sum
of the first three terms. - Write y = x + 21 and apply a discriminant condition to the quadratic
in x to obtain an equation in u. - Obtain two rational equations and use them to find a homogeneous
quadratic equation in x and y. - Let the equation u + v = Z. Cube this equation and show that uv is
an integer p. Determine expressions for x and y. - Set the polynomial equal to u2 and examine the discriminant of the
polynomial in 2. - If there is a rational solution, it must be an integer. Let (u, v, w) be
such. What can be said about divisibility by 3? - Apply the AGM inequality to lrlr2... rkl to obtain an inequality for
the coefficients oi. - Add the three equations to determine x + y + Z. Take the difference
of the second and third equations to determine other simple relations
among the variables. Be careful about dividing by a quantity which
might be zero. - Q(z) - P(x) = (x - u)F(x). Wh a t can be said about the degree and
sign of F(x)? - What can be said about the convexity of the graph which can be
intersected by a line in four points? What implication does this have
for the second derivative of the quartic? - The condition 2b = a + c can be used to derive an equation for b
which does not involve p. - The hypothesis implies that p(t) - q(t) never changes in sign. Let
t = p(x), t = q(x). - Let y = mx + k be a side of the triangle. Then P(x, mx + k) has at
least three distinct zeros. Use the Factor Theorem. - Use the Factor Theorem.
- Write the number in the form v = u + u-l and determine u7 + up7
in terms of v.