A1. Show that if an (infinite) arithmetic progression includes a square, then it must include infinitely many squares. | |
A2. Take three-dimensional coordinates with origin O. C is the point (0,0,c). P is a point on the x-axis, and Q is a point on the y-axis such that OP + OQ = k, where k is fixed. Let W be the center of the sphere through O, C, P, Q. Let W' be the projection of W on the xy-plane. Find the locus of W' as P and Q vary. Find also the locus of W as P and Q vary. | |
A3. The tourism office is collecting figures on the number of sunny days and the number of rainy days in the regions A, B, C, D, E, F.
sunny/rainy unclassifiable A 336 29 B 321 44 C 335 30 D 343 22 E 329 36 F 330 35If one region is excluded then the total number of rainy days in the other regions is one-third of the total number of sunny days in those regions. Which region is excluded? |
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B1. The triangle ABC has ∠A = 36o, ∠B = 72o, ∠C = 72o. The bisector of ∠C meets AB at D. Find the angles of BCD. Express the length BC in terms of AC, without using any trigonometric functions. | |
B2. 21 counters are arranged in a 3 x 7 grid. Some of the counters are black and some white. Show that one can always find 4 counters of the same color at the vertices of a rectangle. | |
B3. A convex n-gon is divided into m triangles, so that no two triangles have interior points in common, and each side of a triangle is either a side of the polygon or a side of another triangle. Show that m + n must be even. Given m, n, find the number of triangle sides in the interior of the polygon and the number of vertices in the interior of the polygon. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
25 March 2004
Last corrected/updated 25 Mar 04