| 1. A triangle has sides a > b > c and corresponding altitudes ha, hb, hc. Show that a + ha > b + hb > c + hc. |
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| 2. 6 ducks are swimming on a pond radius 5. Show that any moment there are two ducks a distance at most 5 apart. | |
| 3. xi are reals. Show that for n = 3, x1 + x2 + ... + xn = 0 implies x1x2 + x2x3 + ... + xn-1xn + xnx1 ≤ 0. For which n > 3 is this true? |
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| 4. p(x) is a polynomial of degree 3 with 3 distinct real zeros. How many real zeros does p'(x)2 - 2p(x)p''(x) have? | |
| 5. Show that there is a constant c > 1 such that if the positive integers m, n satisfy m/n < √7, then 7 - m2/n2 ≥ c/n2. What is the largest such c? | |
| 6. The sequence a1, a2, a3, ... is defined by a1 = 1, an+1 = √(an2 + 1/an). Show that 1/2 ≤ an/nc ≤ 2 for some c (and all 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
29 September 2003
Last corrected/updated 29 Sep 03