| 1. Given a positive integer n, let s(n) be the sum of the positive divisors of n. For example s(5) = 6. Given any three consecutive integers, show that at least one has s(n) even. | |
| 2. Each point on the boundary of a square is to be colored with one of n colors. Find the smallest n such that we can color the points in such a way that there is no right-angled triangle with its vertices on the boundary and all the same color. | |
| 3. Show that 2(xy + yz + zx)1/2 ≤ 31/2(x + y)1/3(y + z)1/3(z + x)1/3 for all positive reals x, y, z. | |
| 4. The cubic x3 + ax2 + bx + c has roots uv, uk, vk, where u and v are real and k is a positive integer. Show that if a, b, c are rational and k = 2, then uv is rational. Is the same true if k = 3? | |
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5. K is a point on the diameter AB closer to A than B. CD is a variable chord through K. The lines BC and BD meet the tangent at A at P and Q respectively. Show that AP·AQ is constant.
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| 6. Does there exist a function f: Z → Z (where Z is the set of integers) such that: (1) f(Z) includes the values 1, 2, 4, 23, 92; (2) f(92 + n) = f(92 - n) for all n; (3) f(1748 + n) = f(1748 - n) for all n; (4) f(1992 + n) = f(1992 - n) for all n? | |
| 7. What conditions must the angles of the triangle ABC satisfy if there is a point X in space such that the angles AXB, BXC, CXA are all 90o? If the point X exists, d = max(XA, XB, XC), and h = length of longest altitude of ABC, show that (√(2/3) ) h ≤ d ≤ h. | |
| 8. x1, x2, ... , xn are non-zero real numbers with sum s such that (s - 2x1 - 2x2)/x1 = (s - 2x2 - 2x3)/x2 = ... = (s - 2xn-1 - 2xn)/xn-1 = (s - 2xn - 2x1)/xn. What possible values can be taken by (s - x1)(s - x2) ... (s - xn)/(x1 ... xn) ? | |
| 9. n is an integer > 1. A word is a sequence X1, X2, ... , X2n of 2n symbols, n of which are A and n of which are B. Let r(n) be the number of words such that only one of the sequences X1, X2, ... , Xk have equal numbers of As and Bs (namely the sequence with k = 2n). Let s(n) be the number of words such that just two of the sequences have equal numbers of As and Bs. Find s(n)/r(n). |
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
18 Dec 2002
Last corrected/updated 18 Dec 2002