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Swedish Mathematical Competition
2010 Swedish Mathematical Competition
2
2
Part of
2010 Swedish Mathematical Competition
Problems
(1)
x_1 + x_4 =x_2+x_3, y_1y_4 =y_2y_3 wanted, points on y = mx-k^2
Source: 2010 Swedish Mathematical Competition p2
4/30/2021
Consider the four lines
y
=
m
x
−
k
2
y = mx-k^2
y
=
m
x
−
k
2
for different integer
k
k
k
. Let
(
x
i
,
y
i
)
(x_i,y_i)
(
x
i
,
y
i
)
,
i
=
1
,
2
,
3
,
4
i = 1,2,3,4
i
=
1
,
2
,
3
,
4
be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat
x
1
≤
x
2
≤
x
3
≤
x
4
x_1\leq x_2\leq x_3\leq x_4
x
1
≤
x
2
≤
x
3
≤
x
4
. Show that
x
1
+
x
4
=
x
2
+
x
3
x_1 + x_4 =x_2+x_3
x
1
+
x
4
=
x
2
+
x
3
and
y
1
y
4
=
y
2
y
3
y_1y_4 =y_2y_3
y
1
y
4
=
y
2
y
3
.
analytic geometry