N=sum of primes between smallest and largest prime factor of N (inclusive).
Find all possible values for the composite number N below 300.
The square ABCD has side length 2*sqrt2.
A circle with centre A and radius 1 is drawn. A second circle with centre C is drawn so that it just
touches the 1st circle at point P on AC .
Determine the total area of the
regions inside the square but outside the two circles.
Source: one of Mayhem problems
The number 1987 can be written as a three digit number xyz in some base
b.
If x + y + z = 1 + 9 + 8 + 7=25, determine all possible values of x, y, z, b.
Source: 1987 Canadian Mathematical Olympiad.
A box contains p white balls and q black balls. Beside the box
there is a pile of black balls. Two balls are taken out from the
box.
If they are of the same colour, a black ball from the pile is put into
the box.
If they are of different colours, the white ball is put back into the box.
This procedure is repeated until the last pair of balls are removed from the box and one last ball is put in.
What is the probability that this last ball is white?
Source: Australian Olympiad 1983
Dinner would have been splendid … if the wine had been as cold as the XXXX, the beef as rare as the XXXXXXX, the brandy as old as the XXXX, and the maid as willing as the XXXXXXX.
Attributed to Winston Churchill.
Try to insert the missing words, taking into account the immense wit of W.C.
A sphenic number S.N. is a product of three distinct prime numbers.
a. Clearly each S.N. has 8 divisors. Show that not only S.N.s claim this feature.
b. Find the smallest consecutive pair n,n+1 of sphenic numbers.
c. Same for the smallest triplet.
d. Prove that sphenic quadruplet n,n+1,n+2,n+3 is "mission impossible".
