

A1. Show that (2/3) n^{3/2} < ∑_{1}^{n} √r < (2/3) n^{3/2} + (1/2) √n.


A2. The complete graph with 6 points and 15 edges has each edge colored red or blue. Show that we can find 3 points such that the 3 edges joining them are the same color.


A3. a, b, c are real, and the sum of any two is greater than the third. Show that 2(a + b + c)(a^{2} + b^{2} + c^{2})/3 > a^{3} + b^{3} + c^{3} + abc.


A4. Using sin x = 2 sin x/2 cos x/2 or otherwise, find ∫_{0}^{π/2} ln sin x dx.


A5. S is a parabola with focus F and axis L. Three distinct normals to S pass through P. Show that the sum of the angles which these make with L less the angle which PF makes with L is a multiple of π.


A6. Show that √7, √(7  √7), √(7  √(7 + √7)), √(7  √(7 + √(7  √7))), ... converges and find its limit.


A7. p(x) = x^{3} + ax^{2} + bx + c has three positive real roots. Find a necessary and sufficient condition on a, b, c for the roots to be cos A, cos B, cos C for some triangle ABC.


B1. Does ∑_{1}^{∞} 1/n^{1 + 1/n} converge?


B2. p(x) is a real polynomial of degree n such that p(m) is integral for all integers m. Show that if k is a coefficient of p(x), then n! k is an integer.


B3. k is real. Solve the differential equations y' = z(y + z)^{k}, z' = y(y + z)^{k} subject to y(0) = 1, z(0) = 0.


B4. R is the reals. S is a surface in R^{3} containing the point (1, 1, 1) such that the tangent plane at any point P ∈ S cuts the axes at three points whose orthocenter is P. Find the equation of S.


B5. The coefficients of the complex polynomial z^{4} + az^{3} + bz^{2} + cz + d satisfy a^{2}d = c^{2} ≠ 0. Show that the ratio of two of the roots equals the ratio of the other two.


B6. A and B are equidistant from O. Given k > OA, find the point P in the plane OAB such that OP = k and PA + PB is a minimum.


B7. Show that we can express any irrational number α ∈ (0, 1) uniquely in the form ∑_{1}^{∞} (1)^{n+1} 1/(a_{1}a_{2} ... a_{n}), where a_{i} is a strictly monotonic increasing sequence of positive integers. Find a_{1}, a_{2}, a_{3} for α = 1/√2.

