Difference between revisions of "2015 AMC 8 Problems/Problem 24"
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We see by inspection that <math>(M,N)=(1,24)</math> is a solution. By the above work, we can easily deduce that <math>(4,20)</math> and <math>(7,16)</math> are solutions. The last one is the intended answer (the next solution fails <math>N>2M</math>) so our answer is <math>3N=\boxed{48\textbf{ (B)}}</math>. | We see by inspection that <math>(M,N)=(1,24)</math> is a solution. By the above work, we can easily deduce that <math>(4,20)</math> and <math>(7,16)</math> are solutions. The last one is the intended answer (the next solution fails <math>N>2M</math>) so our answer is <math>3N=\boxed{48\textbf{ (B)}}</math>. | ||
+ | |||
+ | ===Solution 2=== | ||
+ | On one team they play <math>\binom{3}{2}N</math> games in their division and <math>4(M)</math> games in the other. This gives <math>3N+4M=76</math> | ||
+ | |||
+ | Since <math>M>4</math> we start by trying <math>M=5</math>. This doesn't work because <math>56</math> is not divisible by <math>3</math>. | ||
+ | |||
+ | Next <math>M=6</math>, does not work because <math>52</math> is not divisible by <math>3</math> | ||
+ | |||
+ | We try <math>M=7</math> this does work giving <math>N=16,~M=7</math> and thus <math>3\times 16=\boxed{\textbf{B},~48}</math> games in their division. |
Revision as of 17:03, 25 November 2015
A baseball league consists of two four-team divisions. Each team plays every other team in its division games. Each team plays every team in the other division
games with
and
. Each team plays a 76 game schedule. How many games does a team play within its own division?
Solution 1
Note that the equation rewrites to .
Now remark that if is a solution to this equation, then so is
. This is because
Thus, we can now take an "edge case" solution and work upward until both conditions (
and
) are met.
We see by inspection that is a solution. By the above work, we can easily deduce that
and
are solutions. The last one is the intended answer (the next solution fails
) so our answer is
.
Solution 2
On one team they play games in their division and
games in the other. This gives
Since we start by trying
. This doesn't work because
is not divisible by
.
Next , does not work because
is not divisible by
We try this does work giving
and thus
games in their division.