

A1. p(x) is a polynomial with integral coefficients. The leading coefficient, the constant term, and p(1) are all odd. Show that p(x) has no rational roots.


A2. Show that the solutions of the differential equation (9  x^{2}) (y')^{2} = 9  y^{2} are conics touching the sides of a square.


A3. Let the roots of the cubic p(x) = x^{3} + ax^{2} + bx + c be α, β, γ. Find all (a, b, c) so that p(α^{2}) = p(β^{2}) = p(γ^{2}) = 0.


A4. A map represents the polar cap from latitudes 45^{o} to 90^{o}. The pole (latitude 90^{o}) is at the center of the map and lines of latitude on the globe are represented as concentric circles with radii proportional to (90^{o}  latitude). How are eastwest distances exaggerated compared to northsouth distances on the map at a latitude of 30^{o}?


A5. a_{i} are reals ≠ 1. Let b_{n} = 1  a_{n}. Show that a_{1} + a_{2}b_{1} + a_{3}b_{1}b_{2} + a_{4}b_{1}b_{2}b_{3} + ... + a_{n}b_{1}b_{2} ... b_{n1} = 1  b_{1}b_{2} ... b_{n}.


A6. Prove that there are only finitely many cuboidal blocks with integer sides a x b x c, such that if the block is painted on the outside and then cut into unit cubes, exactly half the cubes have no face painted.


A7. Let O be the center of a circle C and P_{0} a point on the circle. Take points P_{n} on the circle such that angle P_{n}OP_{n1} = +1 for all integers n. Given that π is irrational, show that given any two distinct points P, Q on C, the (shorter) arc PQ contains a point P_{n}.


B1. ABC is a triangle with, as usual, AB = c, CA = b. Find necessary and sufficient conditions for b^{2}c^{2}/(2bc cos A) = b^{2} + c^{2}  2bc cos A.


B2. Find the surface comprising the curves which satisfy dx/(yz) = dy/(zx) = dz/(xy) and which meet the circle x = 0, y^{2} + z^{2} = 1.


B3. Let A(x) = be the matrix
0 ax bx
ax 0 cx
bx cx 0
For which (a, b, c) does det A(x) = 0 have a repeated root in x?


B4. The solid S consists of a circular cylinder radius r, height h, with a hemispherical cap at one end. S is placed with the center of the cap on the table and the axis of the cylinder vertical. For some k, equilibrium is stable if r/h > k, unstable if r/h < k and neutral if r/h = k. Find k and show that if r/h = k, then the body is in equilibrium if any point of the cap is in contact with the table.


B5. The sequence a_{n} is monotonic and ∑ a_{n} converges. Show that ∑ n(a_{n}  a_{n+1}) converges.


B6. A, B, C are points of a fixed ellipse E. Show that the area of ABC is a maximum iff the centroid of ABC is at the center of E.


B7. Let R be the reals. Define a_{n} by a_{1} = α ∈ R, a_{n+1} = cos a_{n}. Show that a_{n} converges to a limit independent of α.

