We will assume that the ribbons can be blue, white and red.

Let

b = the number of blue ribbons

w = the number of white ribbons

r = the number of red ribbons

A music competition awarded 40 ribbons:

b + w + r = 40

The number of blue ribbons awarded was 2 less than the number of white ribbons:

b = w - 2

The number of red ribbons was 5 more than the number of blue ribbons:

r = 5 + b

We have the following system of equations:

b + w + r = 40

b = w - 2

r = 5 + b

........

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b = 11 blue ribbons

w = 13 white ribbons

r = 16 red ribbons

There were 11 blue, 13 white and 16 red ribbons awarded.