21 < x1 + x2 < 22
22 < x2 + x3 < 23
...
2n < xn + xn+1 < 2n+1.
| 1. Find the number of rectangles with sides parallel to the axes whose vertices are all of the form (a, b) with a and b integers such that 0 ≤ a, b ≤ n. |
|
| 2. The sequence a1, a2, a3, ... is defined by a1 = 2, an is the largest prime divisor of a1a2 ... an-1 + 1. Show that 5 does not occur in the sequence. |
|
| 3. Does there exist a function f(n) on the positive integers which takes positive integer values and satisfies f(n) = f( f(n-1) ) + f( f(n+1) ) for all n > 1? |
|
|
4. Find the largest n for which we can find a real number x satisfying: 21 < x1 + x2 < 22 22 < x2 + x3 < 23 ... 2n < xn + xn+1 < 2n+1. |
|
| 5. In the triangle ABC, ∠A = 90o. X is the foot of the perpendicular from A, and D is the reflection of A in B. Y is the midpoint of XC. Show that DX is perpendicular to AY. |
|
| 6. If all an = ±1 and a1a2 + a2a3 + ... an-1an + ana1 = 0, show that n is a multiple of 4. |
|
| 7. Show that 1/33 + 1/43 + ... + 1/n3 < 1/12. |
|
| 8. p1 < p2 < ... < p15 are primes forming an arithmetic progression, show that the difference must be a multiple of 2·3·5·7·11·13. |
|
| 9. Let an = 2 cos(t/2n) - 1. Simplify a1a2 ... an and deduce that it tends to (2 cos t + 1)/3. |
|
| 10. Let T be the set of all (2k-1)-tuples whose entries are all 0 or 1. There is a subset S of T with 2k elements such that given any element x of T, there is a unique element of S which disagrees with x in at most 3 positions. If k > 5, show that it must be 12. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Irish home
© John Scholes
jscholes@kalva.demon.co.uk
11 July 2003
Last updated/corrected 11 Jul 03