In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of number of runs scored by X to the number of runs scored by Y is equal to ratio of number of runs scored by Y to number of runs scored by Z. Value-I = Runs scored by X Value-II = Runs scored by Y Value-III = Runs scored by Z Which one of the following is correct ?

• Let the runs scored by X, Y, and Z be $x$, $y$, and $z$ respectively.

• The condition 'ratio of X to Y is equal to ratio of Y to Z' implies $x/y = y/z$, which means $y^2 = xz$.

• This indicates that $x, y, z$ are in a Geometric Progression (GP).

• The total runs are given as $x + y + z = 37$.

• Since the common ratio $r$ of the GP is not specified, there are multiple integer solutions (e.g., if $r=1$, $x=y=z=12.33$ which is not integer, but if we consider non-integer or different integer sets like $1, 2, 4$ summing to 7, we can scale or find other integer triples).

• Specifically, for integer runs, we need $x, y, z$ to be integers in GP summing to 37. One possible set is $1, 2, 4$ (sum 7, not 37). Another is $4, 8, 16$ (sum 28). Let's check $1, 1, 1$ (sum 3). Actually, if $x, y, z$ are in GP, $y$ is the geometric mean. If $x=1, y=2, z=4$, sum=7. If we multiply by $k$, sum is $7k$. 37 is not divisible by 7. However, the GP doesn't have to be integers in the simplest form, but runs must be integers.

• Wait, if $x, y, z$ are integers in GP, they can be written as $a/r, a, ar$ or $a, ar, ar^2$. If $r$ is an integer, say $r=2$, terms are $a, 2a, 4a$. Sum $7a = 37$, no integer solution. If $r=1$, $3a=37$, no integer solution. If $r$ is a fraction, e.g., $1/2$, terms are $4a, 2a, a$, sum $7a=37$, no integer solution.

• Are there other GPs? $x, y, z$ integers. $y^2 = xz$. $x+z = 37-y$. $xz = y^2$. $x$ and $z$ are roots of $t^2 - (37-y)t + y^2 = 0$. Discriminant $D = (37-y)^2 - 4y^2 = 37^2 - 74y + y^2 - 4y^2 = 1369 - 74y - 3y^2$. For $x,z$ to be real, $D \ge 0$. For $x,z$ to be integers, $D$ must be a perfect square.

• Let's test values of $y$. If $y=12$, $D = 1369 - 888 - 432 = 49 = 7^2$. Roots: $(12 \pm 7)/2$? No, sum is $37-12=25$. Roots are $(25 \pm 7)/2 = 16, 9$. So $9, 12, 16$ is a valid integer GP ($12^2 = 144, 9 \times 16 = 144$). Here $x=9, y=12, z=16$ (Value-I < Value-II < Value-III) OR $x=16, y=12, z=9$ (Value-I > Value-II > Value-III).

• Since both increasing and decreasing GPs are possible with the same total, the order cannot be uniquely determined.