| 1. A tetrahedron is such that the foot of the altitude from each vertex is the incenter of the opposite face. Show that the tetrahedron is regular. | |
| 2. Find the 4 digit number N which uses only two digits, neither of them 0, such that the greatest common divisor of N and the number obtained from N by interchanging the two digits is as large as possible. For example, if N was 2444, then the greatest common divisor of 2444 and 4222 is 2, which is not maximal. | |
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3. Show that for n > 1 and any positive real numbers k, x1, x2, ... , xn:
f(x1 - x2)/(x1 + x2) + f(x2 - x3)/(x2 + x3) + ... + f(xn - x1)/(xn + x1) ≥ n2/(2(x1 + ... + xn)), where f(x) = kx. When does equality hold? |
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| 4. A1A2A3A4A5A6A7 is a regular heptagon and the point P lies on the circumcircle between A7 and A1. Show that PA1 + PA3 + PA5 + PA7 = PA2 + PA4 + PA6. | |
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5. Given n > 2 distinct integers a1, a2, ... , an, find all solutions (x1, x2, ... , xn, y) in non-negative integers to:
a1x1 + a2x2 + ... + anxn = yx1 a2x1 + a3x2 + ... + a1xn = yx2 ... anx1 + a1x2 + ... + an-1xn = yxn such that the greatest common divisor of all the xi is 1. |
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| 6. The points of a graph are labeled A1, A2, ... , An and B1, B2, ... , Bn. From Ai we must draw a red arrow to one of Bi, Ai+1 or Bi+1. and a blue arrow to one of Bi, Ai-1 or Bi-1, but the two arrows must go to different points. From Bi we must draw a red arrow to one of Ai, Ai-1 or Bi-1 and a blue arrow to one of Ai, Ai+1 or Bi+1. Again the two arrows must go to different points. Note that A0, B0, An+1, Bn+1 do not exist, so (for example) the blue arrow from A1 must go to B1. How many different possible configurations are there? | |
| 7. An m x n array of real numbers, each with absolute value at most 1, has all its column sums zero. Show that we can rearrange the numbers in each column so that the absolute value of each resulting row sum is less than 2. | |
| 8. Let X be the set of real numbers > 1. Define f: X → X and g: X → X by f(x) = 2x and g(x) = x/(x-1). Show that given any real numbers 1 < A < B we can find a finite sequence x1 = 2, x2, ... , xn such that A < xn < B and xi = f(xi-1) or g(xi-1). | |
| 9. Find all functions f which are defined on the rationals, take real values and satisfy f(x + y) = f(x) f(y) - f(xy) + 1 for all x, y. |
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 December 2002
Last corrected/updated 14 Feb 2004