2006 Seniors Pancyprian/2nd grade/Problem 2

Problem

Find all three digit numbers $\overline{xyz}$ ( $=100x+10y+z$ ) for which $\frac {7}{4}(\overline{xyz})=\overline{zyx}$ .

Solution

$175x+\frac{35}{2}y+\frac{7}{4}z=100z+10y+x$

$174x+\frac{15}{2}y-\frac{393}{4}z=0$

$232x+10y-131z=0$

We can now make some conclusions. z is even, so z can be only 2, 4, 6, or 8, since 0 is an extraneous solution. Now if $z=2$ , then $x=1$ and $y=3$ . If $z=4$ , then $x=2$ and $y=6$ . If $z=6$ , then $x=3$ and $y=9$ . If $z=8$ , then $x=4$ , but there are no solutions for $y$ . Thus only $132$ , $264$ , and $396$ work.

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2006 Seniors Pancyprian/2nd grade/Problem 2

Problem

Solution

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