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RSM olympiad question, need help pls

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Five friends are all of different heights. The average height of the three tallest friends is exactly the height of one of them. The average height of the four tallest friends is exactly the height of one of them. The average height of all five friends is exactly the height of one of them. The second-tallest friend is 16 cm taller than the secondshortest one. The tallest friend is taller than the shortest one by how many centimeters?

Apr 18, 2022

#1
+23242
+2

Let their heights be a, b, c, d, and e with a > b > c > d > e

Since the tallest three friends have an average height exactly equal to the height of one of them, the average height of these

three must be b.

Also, a must be as much greater than b as c is less than b; call this amount x.

Thus:  a = b + x

b = b

c = b - x

Since the average of the tallest four is the height of one of them; this amount must be c.

So d = b - 4x.                 [Check:  ( (b + x) + (b) + (b -x) + (b - 4x) ) / 4  =  (4b - 4x) / 4  =  b - x.]

The average height of all five is the height of one of them; this amount must be b - 4x.

So e = b - 16x     [Check: ( (b + x) + (b) + (b - x) + (b - 4x) + (b - 16x) ) / 5  =  (5b - 20x) / 5  =  b - 4x.]

The second-tallest friend is 16 cm taller than the second-shortest one:

b  =  16 + (b - 4x)     --->     0 = 16 - 4x     --->     4x  = 16     --->     x  =  4cm

How much taller is the tallest friend than the shortest one?

tallest = b + x  =  b + 4     shortest = b - 16x  = b - 64 cm

Thus, the tallest friend is 68 cm taller than the shortest friend.

Apr 18, 2022

#1
+23242
+2

Let their heights be a, b, c, d, and e with a > b > c > d > e

Since the tallest three friends have an average height exactly equal to the height of one of them, the average height of these

three must be b.

Also, a must be as much greater than b as c is less than b; call this amount x.

Thus:  a = b + x

b = b

c = b - x

Since the average of the tallest four is the height of one of them; this amount must be c.

So d = b - 4x.                 [Check:  ( (b + x) + (b) + (b -x) + (b - 4x) ) / 4  =  (4b - 4x) / 4  =  b - x.]

The average height of all five is the height of one of them; this amount must be b - 4x.

So e = b - 16x     [Check: ( (b + x) + (b) + (b - x) + (b - 4x) + (b - 16x) ) / 5  =  (5b - 20x) / 5  =  b - 4x.]

The second-tallest friend is 16 cm taller than the second-shortest one:

b  =  16 + (b - 4x)     --->     0 = 16 - 4x     --->     4x  = 16     --->     x  =  4cm

How much taller is the tallest friend than the shortest one?

tallest = b + x  =  b + 4     shortest = b - 16x  = b - 64 cm

Thus, the tallest friend is 68 cm taller than the shortest friend.

geno3141 Apr 18, 2022
#2
+65
-4

I think that correct

sincerely:𝓘𝓽𝔃_¥𝓪 𝓑Ø¥_Ð@®𝓴 𝓟®ØÐ𝓘𝓖¥

Kakashi  Apr 18, 2022