, 2023

Jasmin’s total score was $\$3\cdot2 + 4\cdot5=6+20\$$ or 26 points.
Suppose that Sam had $n$
throws that each scored 5 points; then Sam had $2n$ throws that each scored 2
points.

Since Sam’s total score was 36 points, then $5\cdot n+2\cdot2n=36$ or $9n=36$, and so $n=4$.

In total, Sam took $n+2n=3n$ throws,
which is $3\cdot4=12$
throws.
Since Tia’s total score was 37 points, then $2t+5f=37$.

Since $2t$ is even for all integer
values of $t$, then $5f$ must be odd since their sum is 37
(which is odd).

The value of $5f$ is odd exactly when
$f$ is odd.

When $f=1$, we get $2t+5=37$ or $2t=32$, and so $t=16$.

When $f=3$, we get $2t+15=37$ or $2t=22$, and so $t=11$.

When $f=5$, we get $2t+25=37$ or $2t=12$, and so $t=6$.

When $f=7$, we get $2t+35=37$ or $2t=2$, and so $t=1$.

When $f\geq 9$, $5f\geq45$ and so $2t+5f>37$.

The possible ordered pairs $(t,f)$
are $(16,1)$, $(11,3)$, $(6,5)$, and $(1,7)$.
If $a$ throws each score 6
points and $b$ throws each score 21
points, then $6a+21b$ or $3(2a+7b)$ points are scored.

Since $a$ and $b$ are non-negative integers, then $2a+7b$ is a non-negative integer, and so
the total number of points scored, $3(2a+7b)$, is a multiple of 3.

Since 182 is not a multiple of 3, then it is not possible to have a
total score of 182 points.