| 1. Show that (∑xiyizi)2 ≤ (∑ xi3) (∑ yi3) (∑ zi3) for any positive reals xi, yi, zi, where i runs from 1 to n. | |
| 2. Each point of the plane is colored red or blue. Show that there are either three blue points forming an equilateral triangle, or three red points forming an equilateral triangle. |
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| 3. Find all positive integers n with 4-digits n1n2n3n4 such that: (1) n1 = n2 and n3 = n4; (2) the three digit numbers n1n1n3 and n1n3n3 are both prime; (3) n is the product of a one-digit prime, a two-digit prime and a three-digit prime. | |
| 4. Show that for any convex polygon we can find a circle through three adjacent vertices such that all points of the polygon lie inside or on the circle. | |
| 5. C is a cube side 1 and S its inscribed sphere. X is a vertex of the cube. Let L be a line through X which intersects S. Let Y be the point belonging to S and L for which XY is a minimum, and let Z be the point belonging to C and L for which XZ is a maximum. Let d(L) be the product XY.XZ. Find the maximum value of d(L) for all such lines L. Which lines give the maximum value? | |
| 6. Find the longest strictly increasing sequence of squares such that the difference between any two adjacent terms is a prime or the square of a prime. | |
| 7. Define f(1) = 2, f(2) = 3f(1), f(3) = 2f(2), f(4) = 3f(3), f(5) = 2f(4) and so on. Similarly, define g(1) = 3, g(2) = 2g(1), g(3) = 3g(2), g(4) = 2g(3), g(5) = 3g(4) and so on. Which is larger, f(10) or g(10)? | |
| 8. ABC is an acute-angled triangle and P a point inside or on the boundary. The feet of the perpendiculars from P to BC, CA, AB are A', B', C' respectively. Show that if ABC is equilateral, then (AC' + BA' + CB')/(PA' + PB' + PC') is the same for all positions of P, but that for any other triangle it is not. | |
| 9. Call a positive integer blue if for some odd k > 1, it is a sum of the squares of k consecutive positive integers. For example, 14 = 12 + 22 + 32 and 415 = 72 + 82 + 92 + 102 + 112 are blue. Find the smallest blue integer which is also an odd square. |
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
17 Dec 2002
Last corrected/updated 17 Dec 2002