a b c d e f g h idet A = 0 and the cofactor of each element is its square (for example the cofactor of b is fg - di = b2). Show that all elements of A are zero.
| A1. The sequence an of real numbers satisfies an+1 = 1/(2 - an). Show that limn→∞an = 1. |
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| A2. R is the reals. f : R → R is continuous and satisfies f(r) = f(x) f(y) for all x, y, where r = √(x2 + y2). Show that f(x) = f(1) to the power of x2. |
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| A3. ABC is a triangle and P an interior point. Show that we cannot find a piecewise linear path K = K1K2 ... Kn (where each KiKi+1 is a straight line segment) such that: (1) none of the Ki do not lie on any of the lines AB, BC, CA, AP, BP, CP; (2) none of the points A, B, C, P lie on K; (3) K crosses each of AB, BC, CA, AP, BP, CP just once; (4) K does not cross itself. |
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| A4. Take the x-axis as horizontal and the y-axis as vertical. A gun at the origin can fire at any angle into the first quadrant (x, y ≥ 0) with a fixed muzzle velocity v. Assuming the only force on the pellet after firing is gravity (acceleration g), which points in the first quadrant can the gun hit? |
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| A5. The sequences an, bn, cn of positive reals satisfy: (1) a1 + b1 + c1 = 1; (2) an+1 = an2 + 2bncn, bn+1 = bn2 + 2cnan, cn+1 = cn2 + 2anbn. Show that each of the sequences converges and find their limits. |
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A6. A is the matrix
a b c d e f g h idet A = 0 and the cofactor of each element is its square (for example the cofactor of b is fg - di = b2). Show that all elements of A are zero. |
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| B1. Let R be the reals. f : [1, ∞) → R is differentiable and satisfies f '(x) = 1/(x2 + f(x)2) and f(1) = 1. Show that as x → ∞, f(x) tends to a limit which is less than 1 + π/4. |
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| B2. R is the reals. f :(0, 1) → R is differentiable and has a bounded derivative: |f '(x)| <= k. Prove that : |∫01 f(x) dx - ∑1n f(i/n) /n| ≤ k/n. |
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| B3. Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2? |
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| B4. p(z) = z2 + az + b has complex coefficients. |p(z)| = 1 on the unit circle |z| = 1. Show that a = b = 0. |
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| B5. Let p(x) be the polynomial (x - a)(x - b)(x - c)(x - d). Assume p(x) = 0 has four distinct integral roots and that p(x) = 4 has an integral root k. Show that k is the mean of a, b, c, d. |
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| B6. P is a variable point in space. Q is a fixed point on the z-axis. The plane normal to PQ through P cuts the x-axis at R and the y-axis at S. Find the locus of P such that PR and PS are at right angles. |
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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.
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© John Scholes
jscholes@kalva.demon.co.uk
5 Mar 2002