| A1. A cone has circular base radius 1, and vertex a height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What length is a side of the cube? |
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| A2. Let C be the circle center (0, 0), radius 1. Let X, Y be two points on C with positive x and y coordinates. Let X1, Y1 be the points on the x-axis with the same x-coordinates as X and Y respectively, and let X2, Y2 be the points on the y-axis with the same y-coordinates. Show that the area of the region XYY1X1 plus the area of the region XYY2X2 depends only on the length of the arc XY, and not on its position. |
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| A3. Let R be the reals. Let f : R → R have a continuous third derivative. Show that there is a point a with f(a) f '(a) f ''(a) f '''(a) ≥ 0. |
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| A4. Define the sequence of decimal integers an as follows: a1 = 0; a2 = 1; an+2 is obtained by writing the digits of an+1 immediately followed by those of an. When is an a multiple of 11? |
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| A5. A finite collection of disks covers a subset X of the plane. Show that we can find a pairwise disjoint subcollection S, such that X ⊆ ∪{3D : D ∈ S}, where 3D denotes the disk with the same center as D and 3 times the radius. |
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| A6. P, Q, R are three (distinct) lattice points in the plane. Prove that if (PQ + QR)2 < 8 area PQR + 1, then P, Q, R are vertices of a square. |
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| B1. Find the minimum of { (x + 1/x)6 - (x6 + 1/x6) - 2 }/{ (x + 1/x)3 + (x3 + 1/x3) }, for x > 0. |
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| B2. Let P be the point (a, b) with 0 < b < a. Find Q on the x-axis and R on y = x, so that PQ + QR + RP is as small as possible. |
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| B3. Let S be the sphere center the origin and radius 1. Let P be a regular pentagon in the plane z = 0 with vertices on S. Find the surface area of the part of the sphere which lies above (z > 0) P or its interior. |
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| B4. For what m, n > 0 is ∑0mn-1 (-1)[i/m] + [i/n] = 0? |
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| B5. Let n be the decimal integer 11...1 (with 1998 digits). What is the 1000th digit after the decimal point of √n? |
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| B6. Show that for any integers a, b, c we can find a positive integer n such that n3 + a n2 + b n + c is not a perfect square. |
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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 of the problems and the official solutions are in American Mathematical Monthly 106 (1999) .
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© John Scholes
jscholes@kalva.demon.co.uk
4 Nov 1999