1993 UNCO Math Contest II Problems/Problem 1


How many times must one shoot at this target, and which rings must one hit in order to score exactly $100$ points?

[asy] draw(circle((0,0),1),black); draw(circle((0,0),2),black); draw(circle((0,0),3),black); draw(circle((0,0),4),black); draw(circle((0,0),5),black); draw(circle((0,0),6),black); MP("40",(0,0-.3),N); MP("39",(0,1-.1),N); MP("24",(0,2-.1),N); MP("23",(0,3-.1),N); MP("17",(0,4-.1),N); MP("16",(0,5-.1),N); [/asy]


Solution 1

We can try combinations of numbers that add to $100$. We soon find that one must shoot at the target $6$ times and hit the $16$ ring $2$ times and hit the $17$ ring $4$ times.

Solution 2

Since $100=2^2\cdot5^2$, we can take $\pmod25$ and $\pmod4$ of the $6$ numbers. From $\pmod25$ we have the set (in order) ${15,14,-1,-2,-8,16}$, and from $\pmod4$ we have the set ${0,-1,0,-1,1,0}$. We notice that as $8$ is $1/2$ times $16$, we probably have two times more $17$s than $16$s, and it works perfectly, so one has to shoot the $16$ ring $2$ times and the $17$ ring $2$ times.

See also

1993 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
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All UNCO Math Contest Problems and Solutions