can you solve this math riddle?

Question

The midrange of a set of numbers is the average of the greatest and least value of a set. for a set of six distinct non-negative numbers, the mean, the median and the midrange are all 5. Find as many such sets as you can. the winner will be chosen as best answer. good luck.

Answer 1

Let numbers be A, B, C, D, E, F such that A
A< F, B< E and C < D → A + B + C < D + E + F

Midrange = 5 → ½(A + F) = 5

ie A + F = 10
ie F = 10 - A

Also since A < F
A + A < A + F = 10
So 2A < 10
A < 5*
Since the numbers are non-negative 0 <= A <5
(NOTE: A can be 0 as it is the least of the six integers)

Median = 5 → ½(C + D) = 5

ie C + D = 10
ie D = 10 - C
Similarly (using argument of above*) C < 5 and since A < C, 0 < C < 5

Mean = 5 → (A + B + C + D + E + F)/6 = 5

ie A + B + C + D + E + F = 30

whence (A + F) + (C + D) + (B + E) = 30 (by commutative and associative laws)

ie 10 + 10 + (B + E) = 30

Thus B + E = 10
ie E = 10 - B
Similarly (using argument of above*) B < 5 and since A < B, 0 < B < 5
Finally since A + B + C< D + E + F
then A + B + C < 10 - A + 10 - B + 10 - C

ie A + B + C < 30 + (A + B + C)
ie 2 (A + B + C) < 30
ie A + B + C < 15
To fulfill this condition and the condition of distinctness 0<= A < B < C < 5

So the conditions are:

1. A 2. F = 10 - A
3. D = 10 - C
4. E = 10 - B
5. 0 <= A < B < C < 5

Thus all integer possibilities for sets are
(0, 1, 2, 8, 9, 10}
{0, 1, 3, 7, 9, 10}
{0, 1, 4, 6, 9, 10}
(0, 2, 3, 7, 8, 10}
{0, 2, 4, 6, 8, 10}
(0, 3, 4, 6, 7, 10}
{1, 2, 3, 7, 8, 9}
{1, 2, 4, 6, 8, 9}
(1, 3, 4, 6, 7, 9}
{2, 3, 4, 6, 7, 8}

Of course if real numbers are permitted then the number of sets is infinite, provided these condtions are fulfilled

eg { φ, e, π, 10 - π, 10 - e, 10 - φ} fulfills the conditions in real numbers (where φ = golden ratio, e = Euler's number and π = ratio of circumference to diameter of a circle)

Answer 2

The other people didn't make those numbers distinct

Distinct means none of the numbers in the set are repeated.

6 integers
A 5 = (A + F)/2 (definition of midrange)
5 = (C + D)/2 (definition of median)

5 = (A + B + C + D + E + F) / 6
Therefore 5 = (B + E) / 2

possible combinations of #s are

#1 = 0 1 2 3 4 5
#2 = 10 9 8 7 6 5

{0, 1, 2, 8, 9, 10}
{0, 1, 3, 7, 9, 10}
{0, 2, 3, 7, 8, 10}

{1, 2, 3, 7, 8, 9}
{1, 2, 4, 6, 8, 9}
{1, 3, 4, 6, 7, 9}

{2, 3, 4, 6, 7, 8}
{2, 3, 4, 6, 7, 8}

Answer 3

As you asked for sets of distinct non-negative numbers, there are infinite answers.
{ 3 - 1/n , 3, 4, 6, 7, 7 + 1/n} for all n = 1, 2, 3, ... is one such infinite set of answers.

Of course, if you meant non-negative integers, there are 10 sets:
{0,1,2,8,9,10}
{0,1,3,7,9,10}
{0,1,4,6,9,10}
{0,2,3,7,8,10}
{0,2,4,6,8,10}
{0,3,4,6,7,10}
{1,2,3,7,8,9}
{1,2,4,6,8,9}
{1,3,4,6,7,9}
{2,3,4,6,7,8}

Answer 4

You have 6 positive numbers, A B C D E F

The median is 5, that tells us that the average of C and D are 5 (so C&D could be 4,6 or 3,7 or 2,8 or 1,9)

The mean is 5 tells us that (A+B+C+D+E+F)/6 = 5

The midrange is 5, that tells us that the average of A and F are 5 (so A&F could be 4,6 or 3,7 or 2,8 or 1,9)

Some possible sets that satisfy this are:

{1,3,4,6,7,9}

{1,2,4,6,8,9}

{2,3,4,6,7,8}

There's a few more sets .... I'll leave it to you and other folks to construct others.

Answer 5

1, 3,4,6,7,9
2,3,4,6,7,8
0,1,3,7,9,10
0,2,3,7,8,10
0,1,4,6,9,10

You didn't say they had to be integers.
1.1,3.1,4.1,5.9,6.9,8.9

Ok I'm a little tired of this so I'm done. But I solved your riddle. Yay for me. :)
____________________________________________________
Pito16places did not use the fact that these have to be distinct numbers. More yay for me. :)

Answer 6

obviously one is 5,5,5,5,5,5
10,0,10,0,10,0
im tired right now. more later
ps: dont put mine as best, give the credit to someone else if by some crazy chance i win