| 1. One face of a pyramid with square base and all edges 2 is glued to a face of a regular tetrahedron with edge length 2 to form a polyhedron. What is the total edge length of the polyhedron? |
|
| 2. P is a point on the circumcircle of the square ABCD between C and D. Show that PA2 - PB2 = PB·PD - PA·PC. |
|
| 3. E is the midpoint of the arc BC of the circumcircle of the triangle ABC (on the opposite side of the line BC to A). DE is a diameter. Show that ∠DEA is half the difference between the ∠B and ∠C. |
|
| 4. The triangle ABC (with sidelengths a, b, c as usual) satisfies log(a2) = log(b2) + log(c2) - log(2bc cos A). What can we say about the triangle? |
|
| 5. Let X = {1, 2, 3, 4, 5, 6, 7}. How many 7-tuples (X1, X2, X3, X4, X5, X6, X7) are there such that each Xi is a different subset of X with three elements and the union of the Xi is X? |
|
| 6. Each member of the sequence a1, a2, ... , an belongs to the set {1, 2, ... , n-1} and a1 + a2 + ... + an < 2n. Show that we can find a subsequence with sum n. |
|
| 7. Put f(x) = x3 - x. Show that the set of positive real A such that for some real x we have f(x + A) = f(x) is the interval (0, 2]. |
|
| 8. The sequence of nonzero reals x1, x2, x3, ... satisfies xn = xn-2xn-1/(2xn-2 - xn-1) for all n > 2. For which (x1, x2) does the sequence contain infinitely many integral terms? |
|
| 9. The year 1978 had the property that 19 + 78 = 97. In other words the sum of the number formed by the first two digits and the number formed by the last two digits equals the number formed by the middle two digits. Find the closest years either side of 1978 with the same property. |
|
| 10. Show that (1 + x)n ≥ (1 - x)n + 2nx(1 - x2)(n-1)/2 for all 0 ≤ x ≤ 1 and all positive integers n. | |
| 11. Given a positive real k, for which real x0 does the sequence x0, x1, x2, ... defined by xn+1 = xn(2 - k xn) converge to 1/k? |
|
| 12. Show that for n a positive integer we have cos4k + cos42k + ... cos4nk = 3n/8 - 5/16 where k = π/(2n+1). |
|
| 13. ABC is a triangle with AB = 2·AC and E is the midpoint of AB. The point F lies on the line EC and the point G lies on the line BC such that A, F, G are collinear and FG = AC. Show that AG3 = AB·CE2. | |
| 14. a1, a2, ... , an are integers and m < n is a positive integer. Put Si = ai + ai+1 + ... + ai+m, and Ti = am+i + am+i+1 + ... + an-1+i, for where we use the usual cyclic subscript convention, whereby subscripts are reduced to the range 1, 2, ... , n by subtracting multiples of n as necessary. Let m(h, k) be the number of elements i in {1, 2, ... , n} for which Si = h mod 4 and Ti = b mod 4. Show that m(1, 3) = m(3, 1) mod 4 iff m(2, 2) is even. | |
| 15. X is a finite set. X1, X2, ... , Xn are distinct subsets of X (n > 1), each with 11 elements, such that the intersection of any two subsets has just one element and given any two elements of X, there is an Xi containing them both. Find n. |
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 June 2003
Last updated/corrected 11 Jun 03