x1 + x2 + x3 + x4 = 1
a1x1 + a2x2 + a3x3 + a4x4 = k
a12x1 + a22x2 + a32x3 + a42x4 = k2
have a solution in non-negative reals xi?
(2) The sequence a1 = 1, a2, a3, ... satisfies an = (4n - 6)an-1/n. Show that all terms are integers.
| 1. The non-negative reals a, b, c, d satisfy a5 + b5 ≤ 1, c5 + d5 ≤ 1. Show that a2c3 + b2d3 ≤ 1. |
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| 2. Find all primes p, q such that p(p+1) + q(q+1) = n(n+1) for some positive integer n. |
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| 3. A finite set of closed disks in the plane cover an area A (some of the disks may overlap). Show that we can find a subset of non-overlapping disks which cover an area of at least A/9. | |
| 4. The disjoint sets A and B together contain all the positive integers. Show that given any integer n, we can find integers a > b > n such that a, b and a+b are all in A or all in B. | |
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5. Given reals 0 < a1 < a2 < a3 < a4, for which real k do the equations
x1 + x2 + x3 + x4 = 1 a1x1 + a2x2 + a3x3 + a4x4 = k a12x1 + a22x2 + a32x3 + a42x4 = k2 have a solution in non-negative reals xi? |
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| 6. There are six lines in space such that given any three we can find two which are perpendicular. Show that we can divide the lines into two groups of three, so that the lines in each group are all perpendicular to each other. | |
| 7. Let C1, C2, C3, C4, C5, C6 be closed line segments in the plane. Each pair has at most one common point. S is the union of the six segments. There are four distinct points P1, P2, P3, P4, such that any straight line through at least one of the points Pi intersects S in exactly two points. Is S necessarily a hexagon? | |
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8. (1) Show that (2n+1 - 1) (2n - 1)2(2n-1 - 1)4(2n-2 - 1)8 ... (22 - 1)N divides (2n+1 - 1)! (where N = 2n-1).
(2) The sequence a1 = 1, a2, a3, ... satisfies an = (4n - 6)an-1/n. Show that all terms are integers. |
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9. A regular p-gon (p prime) has 1 x k rectangles on the outside of each edge. Each rectangle is divided into k unit squares, so that the figure is divided into pk + 1 regions. How many ways can the figure be colored with three colors, so that adjacent regions have different colors and there is no symmetry axis?
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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