3rd Putnam 1940


A1. p(x) is a polynomial with integer coefficients. For some positive integer c, none of p(1), p(2), ... , p(c) are divisible by c. Prove that p(b) is not zero for any integer b.
A2. y = f(x) is continuous with continuous derivative. The arc PQ is concave to the chord PQ. X is chosen on the arc PQ to maximize PX + XQ. Prove that XP and XQ are equally inclined to the tangent at X.
A3. α is a fixed real number. Find all functions f: R → R (where R is the reals) which are continuous, have a continuous derivative, and satisfy ∫_b^y f^α(x) dx = ( ∫_b^y f(x) dx )^α for all y and some b.
A4. p is a positive constant. Let R is the curve y² = 4px. Let S be the mirror image of R in the y-axis (y² = - 4px). R remains fixed and S rolls around it without slipping. O is the point of S initially at the origin. Find the equation for the locus of O as S rolls?
A5. Prove that the set of points satisfying x⁴ - x² = y⁴ - y² = z⁴ - z² is the union of 4 straight lines and 6 ellipses.
A6. p(x) is a polynomial with real coefficients and derivative r(x) = p'(x). For some positive integers a, b, r^a(x) divides p^b(x). Prove that for some real numbers A and α and for some integer n, we have p(x) = A(x - α)ⁿ.
A7. a_i and b_i are real, and ∑₁^∞ a_i² and ∑₁^∞ b_i² converge. Prove that ∑₁^∞ (a_i - b_i)^p converges for p ≥ 2.
A8. Show that the area of the triangle bounded by the lines a_ix + b_iy + c_i = 0 (i = 1, 2, 3) is Δ²/\|2(a₂b₃ - a₃b₂)(a₃b₁ - a₁b₃)(a₁b₂ - a₂b₁)\|, where Δ is the 3 x 3 determinant with columns a_i, b_i, c_i.
B1. A stone is thrown from the ground with speed v at an angle θ to the horizontal. There is no friction and the ground is flat. Find the total distance it travels before hitting the ground. Show that the distance is greatest when sin θ ln (sec θ + tan θ) = 1.
B2. C₁, C₂ are cylindrical surfaces with radii r₁, r₂ respectively. The axes of the two surfaces intersect at right angles and r₁ > r₂. Let S be the area of C₁ which is enclosed within C₂. Prove that S = 8r₂²A = 8r₁²C - 8(r₁² - r₂²)B, where A = ∫₀¹ (1 - x²)^1/2(1 - k²x²)^-1/2 dx, B = ∫₀¹ (1 - x²)^-1/2(1 - k²x²)^-1/2 dx, and C = ∫₀¹ (1 - x²)^-1/2(1 - k²x²)^1/2 dx, and k = r₂/r₁.
B3. Let p be a positive real, let S be the parabola y² = 4px, and let P be a point with coordinates (a, b). Show that there are 1, 2 or 3 normals from P to S according as 4(2p - a)² + 27 pb² >, = or < 0.
B4. Let S be the surface ax² + by² + cz² = 1 (a, b, c all non-zero), and let K be the sphere x² + y² + z² = 1/a + 1/b + 1/c (known as the director sphere). Prove that if a point P lies on 3 mutually perpendicular planes, each of which is tangent to S, then P lies on K. Comment. The original question also asked, apparently in error, for a proof that every point of K had this property, which is (a) false unless we allow planes which are asymptotes (or tangents at infinity), and (b) unreasonably hard - at least, I cannot see a neat proof.
B5. Find all rational triples (a, b, c) for which a, b, c are the roots of x³ + ax² + bx + c = 0.
B6. The n x n matrix (m_ij) is defined as m_ij = a_ia_j for i ≠ j, and a_i² + k for i = j. Show that det(m_ij) is divisible by k^n-1 and find its other factor.
B7. Given n > 8, let a = √n and b = √(n+1). Which is greater a^b or b^a?

The Putnam fellow (winner) was Andrew M Gleason, his (winning) mark is unknown. To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and solutions are available in: A M Gleason, R E Greenwood & L M Kelly, The William Lowell Putnam Mathematical Competition, Problems and Solutions, 1938-1964, MAA 1980. Out of print, but available in some university libraries.