Problem: y = x 2 y = x 2 2 2 x x m / n m / n m m n n m + n m + n
Answer Choices:
A. 14 1 4 15 1 5 16 1 6 17 1 7 18 1 8 Solution:
Suppose that the triangle has vertices A ( a , a 2 ) , B ( b , b 2 ) A ( a , a 2 ) , B ( b , b 2 ) C ( c , c 2 ) C ( c , c 2 ) A B ‾ A B
b 2 − a 2 b − a = b + a b − a b 2 − a 2 = b + a
so the slopes of the three sides of the triangle have a sum
( b + a ) + ( c + b ) + ( a + c ) = 2 ⋅ m n ( b + a ) + ( c + b ) + ( a + c ) = 2 ⋅ n m
The slope of one side is 2 = tan θ 2 = tan θ θ θ
tan ( θ ± π 3 ) = tan θ ± tan ( π / 3 ) 1 ∓ tan θ tan ( π / 3 ) = 2 ± 3 1 ∓ 2 3 = − 8 ± 5 3 11 tan ( θ ± 3 π ) = 1 ∓ tan θ tan ( π / 3 ) tan θ ± tan ( π / 3 ) = 1 ∓ 2 3 2 ± 3 = − 1 1 8 ± 5 3
Therefore
m n = 1 2 ( 2 − 8 + 5 3 11 − 8 − 5 3 11 ) = 3 11 n m = 2 1 ( 2 − 1 1 8 + 5 3 − 1 1 8 − 5 3 ) = 1 1 3
and m + n = 14 m + n = 1 4
Such a triangle exists. The x x ( 11 ± 5 3 ) / 11 ( 1 1 ± 5 3 ) / 1 1 − 19 / 11 − 1 9 / 1 1
\section*{OR}a < b a < b A B ‾ A B
2 = b 2 − a 2 b − a = b + a , so a = 1 − k and b = 1 + k 2 = b − a b 2 − a 2 = b + a , so a = 1 − k and b = 1 + k
for some k > 0 k > 0 D D A B ‾ A B
D = ( 1 , ( 1 − k ) 2 + ( 1 + k ) 2 2 ) = ( 1 , 1 + k 2 ) D = ( 1 , 2 ( 1 − k ) 2 + ( 1 + k ) 2 ) = ( 1 , 1 + k 2 )
The slope of the altitude C D ‾ C D − 1 / 2 − 1 / 2
1 − c = 2 ( c 2 − 1 − k 2 ) 1 − c = 2 ( c 2 − 1 − k 2 )
Therefore
C D 2 = ( 1 − c ) 2 + ( c 2 − 1 − k 2 ) 2 = 5 4 ( 1 − c ) 2 C D 2 = ( 1 − c ) 2 + ( c 2 − 1 − k 2 ) 2 = 4 5 ( 1 − c ) 2
Because △ A B C △ A B C
C D 2 = 3 4 A B 2 = 3 4 ( ( 2 k ) 2 + ( 4 k ) 2 ) = 15 k 2 C D 2 = 4 3 A B 2 = 4 3 ( ( 2 k ) 2 + ( 4 k ) 2 ) = 1 5 k 2
Hence
5 4 ( 1 − c ) 2 = 15 k 2 , so k 2 = ( 1 − c ) 2 12 4 5 ( 1 − c ) 2 = 1 5 k 2 , so k 2 = 1 2 ( 1 − c ) 2
Substitution into the equation 1 − c = 2 ( c 2 − 1 − k 2 ) 1 − c = 2 ( c 2 − 1 − k 2 ) c = 1 c = 1 c = − 19 / 11 c = − 1 9 / 1 1 c < 1 c < 1
a + b + c = 2 − 19 11 = 3 11 = m n , so m + n = 14 a + b + c = 2 − 1 1 1 9 = 1 1 3 = n m , so m + n = 1 4
The problems on this page are the property of the MAA's American Mathematics Competitions