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CEMC Gauss (Grade 8), 2012

Combinatorics Difficulty 4.8 AIME Find the answer Canada

Stones are numbered $1,2,3,4,5,6,7,8,9,10$ . Three groups of stones can be selected so that the sum of each group is 11. For example, one arrangement is $\{1,10\}$ , $\{2,3,6\}$ , $\{4,7\}$ . Including the example, how many arrangements are possible?

Pick one

Solution

Diagonal $PR$ divides parallelogram $PQRS$ into two equal areas.

That is, the area of $\triangle PRS$ is one half of the area of parallelogram $PQRS$ , or 20.

In $\triangle PRS$ , we construct median $RT$ .

(A median is a line segment that joins a vertex of a triangle to the midpoint of its opposite side.)

Median $RT$ divides $\triangle PRS$ into two equal areas since its base, $PS$ , is halved while the height remains the same.
That is, the area of $\triangle TRS$ is one half of the area of $\triangle PRS$ , or 10.

Similarly, we construct median $TV$ in $\triangle TRS$ , as shown.

Median $TV$ divides $\triangle TRS$ into two equal areas since its base, $SR$ , is halved while the height remains the same.

That is, the area of $\triangle TVS$ is one half of the area of $\triangle TRS$ , or 5.
The area of $PRVT$ is equal to the area of $\triangle TVS$ subtracted from the area of $\triangle PRS$ , or $20-5=15$ .

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