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Workings

Hint
The word JANUARY appears 10 times.

Answer
The word JANUARY appears 10 times.

Workings

Hint
This is a well known nursery rhyme.

Answer
AJCTA.

These are the initial letters from the nursery rhyme 'Jack and Jill went up the hill'. The letters represent 'and Jill came tumbling after'.

Workings

Hint
Look a the letters.

Answer
Michael.

Each name begins with the last letter of the preceding name.

Workings

Hint
Start by looking at clue (5), and then at the first part of clue (6).

Answer
# Name   Wore  Colour
1 Gary   2     red
2 Larry  3     green
3 Barry  1     yellow
4 Harry  4     blue

Reasoning
By (5), the winner wore #2, and Harry finished last:
   # Name   Wore  Colour
   1        2
   2
   3
   4 Harry

By (6), the person who finished second wore green, and by (4), wore #3:
   # Name   Wore  Colour
   1        2
   2        3     green
   3
   4 Harry

By (1), only the person in last place could have worn the same number as the position they finished:
   # Name   Wore  Colour
   1        2
   2        3     green
   3        1
   4 Harry  4

By (3), Gary didn't wear green (so can't have finished second) and beat Barry (so can't have finished third), which means that he must have finished first.
   # Name   Wore  Colour
   1 Gary   2
   2        3     green
   3        1
   4 Harry  4

By (6), Barry wore yellow, so must have finished third:
   # Name   Wore  Colour
   1 Gary   2
   2        3     green
   3 Barry  1     yellow
   4 Harry  4

By (6) the person wearing red beat the person wearing blue, giving:
   # Name   Wore  Colour
   1 Gary   2     red
   2 Larry  3     green
   3 Barry  1     yellow
   4 Harry  4     blue