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Number theory Difficulty 3.8 AMC 10/12 Find the answer Canada

In any triangle, the length of the longest side is less than half of the perimeter. All triangles with perimeter 57 and integer side lengths $x$ , $y$ , $z$ , such that $\text{x lt;y lt;z}$ are constructed. How many such triangles are there?

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Solution

We begin by numbering the checkerboard squares from 1 to 16, as shown, so that we may refer to each of them specifically.

We denote a move “up" by the letter $U$ and a move “right" by $R$ .

We will begin by determining which of the 16 squares will not be touched by the face with the circle and then proceed to show that the remaining squares will be touched by the face with the circle.

Since the cube begins on square 1 and the circle is facing out, square 1 will not be touched by the face with the circle.

Similarly, each of the squares 2, 3, 4 can only be reached by moving the cube right (the sequence of moves to reach each of these three squares is $R$ , $RR$ and $RRR$ , respectively), and in each case the circle remains facing out.

Squares 2, 3 and 4 will not be touched by the face with the circle.

Squares 5 and 9 can only be reached by moving the cube up (the sequence of moves to reach each of these two squares is $U$ and $UU$ , respectively).

In either case, the face with the circle will not touch squares 5 and 9.

Square 6 can be reached with two different sequences of moves, $RU$ or $UR$ .

In both cases, the face with the circle will not touch square 6.

Square 10 can be reached with three different sequences of moves, $UUR$ , $URU$ or $RUU$ .

In all three cases, the face with the circle will not touch square 10.

In turns out that these eight squares ( $1,2,3,4,5,6,9,10$ ) are the only squares that will not be touched by the face with the circle on any path.

The table below lists sequences of moves that demonstrate how each of the remaining eight squares will be touched by the face with the circle.

The second column lists the sequence of moves, while the third column lists the position of the face with the circle as the cube progresses through the sequence of moves.
We have used the letters $F$ for front, $B$ for back, $T$ for top, $O$ for bottom, $L$ for left, and $R$ for right to indicate the location of the face containing the circle.

Square	Sequence of Moves	Position of the Circle
7	$URR$	$TRO$
8	$RURR$	$FTRO$
11	$URUR$	$TRRO$
12	$RURUR$	$FTRRO$
13	$UUU$	$TBO$
14	$RUUU$	$FTBO$
15	$RRUUU$	$FFTBO$
16	$RRRUUU$	$FFFTBO$

Therefore, the number of different squares that will not be contacted by the face with the circle on any path is 8.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.