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 βΎ A B
b 2 β a 2 b β a = b + a b β a b 2 β a 2 β = b + a
so the slopes of the three sides of the triangle have a sum
( b + a ) + ( c + b ) + ( a + c ) = 2 β
m n ( b + a ) + ( c + b ) + ( a + c ) = 2 β
n m β
The slope of one side is 2 = tan β‘ ΞΈ 2 = tan ΞΈ ΞΈ ΞΈ
tan β‘ ( ΞΈ Β± Ο 3 ) = tan β‘ ΞΈ Β± tan β‘ ( Ο / 3 ) 1 β tan β‘ ΞΈ tan β‘ ( Ο / 3 ) = 2 Β± 3 1 β 2 3 = β 8 Β± 5 3 11 tan ( ΞΈ Β± 3 Ο β ) = 1 β tan ΞΈ tan ( Ο / 3 ) tan ΞΈ Β± tan ( Ο / 3 ) β = 1 β 2 3 β 2 Β± 3 β β = β 1 1 8 Β± 5 3 β β
Therefore
m n = 1 2 ( 2 β 8 + 5 3 11 β 8 β 5 3 11 ) = 3 11 n m β = 2 1 β ( 2 β 1 1 8 + 5 3 β β β 1 1 8 β 5 3 β β ) = 1 1 3 β
and m + n = 14 m + n = 1 4
Such a triangle exists. The x x ( 11 Β± 5 3 ) / 11 ( 1 1 Β± 5 3 β ) / 1 1 β 19 / 11 β 1 9 / 1 1
\section*{OR}a < b a < b A B βΎ A B
2 = b 2 β a 2 b β a = b + a , so a = 1 β k and b = 1 + k 2 = b β a b 2 β a 2 β = b + a , so a = 1 β k and b = 1 + k
for some k > 0 k > 0 D D A B βΎ A B
D = ( 1 , ( 1 β k ) 2 + ( 1 + k ) 2 2 ) = ( 1 , 1 + k 2 ) D = ( 1 , 2 ( 1 β k ) 2 + ( 1 + k ) 2 β ) = ( 1 , 1 + k 2 )
The slope of the altitude C D βΎ C D β 1 / 2 β 1 / 2
1 β c = 2 ( c 2 β 1 β k 2 ) 1 β c = 2 ( c 2 β 1 β k 2 )
Therefore
C D 2 = ( 1 β c ) 2 + ( c 2 β 1 β k 2 ) 2 = 5 4 ( 1 β c ) 2 C D 2 = ( 1 β c ) 2 + ( c 2 β 1 β k 2 ) 2 = 4 5 β ( 1 β c ) 2
Because β³ A B C β³ 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 β A B 2 = 4 3 β ( ( 2 k ) 2 + ( 4 k ) 2 ) = 1 5 k 2
Hence
5 4 ( 1 β c ) 2 = 15 k 2 , so k 2 = ( 1 β c ) 2 12 4 5 β ( 1 β c ) 2 = 1 5 k 2 , so k 2 = 1 2 ( 1 β c ) 2 β
Substitution into the equation 1 β c = 2 ( c 2 β 1 β k 2 ) 1 β c = 2 ( c 2 β 1 β k 2 ) c = 1 c = 1 c = β 19 / 11 c = β 1 9 / 1 1 c < 1 c < 1
a + b + c = 2 β 19 11 = 3 11 = m n , so m + n = 14 a + b + c = 2 β 1 1 1 9 β = 1 1 3 β = n m β , so m + n = 1 4 β
The problems on this page are the property of the MAA's American Mathematics Competitions