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National and Regional Contests
PEN Problems
PEN F Problems
1
1
Part of
PEN F Problems
Problems
(1)
F 1
Source:
5/25/2007
Suppose that a rectangle with sides
a
a
a
and
b
b
b
is arbitrarily cut into
n
n
n
squares with sides
x
1
,
…
,
x
n
x_{1},\ldots,x_{n}
x
1
,
…
,
x
n
. Show that
x
i
a
∈
Q
\frac{x_{i}}{a}\in\mathbb{Q}
a
x
i
∈
Q
and
x
i
b
∈
Q
\frac{x_{i}}{b}\in\mathbb{Q}
b
x
i
∈
Q
for all
i
∈
{
1
,
⋯
,
n
}
i\in\{1,\cdots, n\}
i
∈
{
1
,
⋯
,
n
}
.
geometry
rectangle
induction
rational numbers