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National and Regional Contests
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National High School Mathematics League
1994 National High School Mathematics League
4
Order the Numbers
Order the Numbers
Source: 1994 National High School Mathematics League, Exam One, Problem 4
March 1, 2020
Problem Statement
0
<
b
<
1
,
0
<
a
<
π
4
0<b<1,0<a<\frac{\pi}{4}
0
<
b
<
1
,
0
<
a
<
4
π
,
x
=
(
sin
a
)
log
b
sin
a
,
y
=
(
cos
a
)
log
b
cos
a
,
z
=
(
sin
a
)
log
b
cos
a
x=(\sin a)^{\log_{b}\sin a},y=(\cos a)^{\log_{b}\cos a},z=(\sin a)^{\log_{b}\cos a}
x
=
(
sin
a
)
l
o
g
b
s
i
n
a
,
y
=
(
cos
a
)
l
o
g
b
c
o
s
a
,
z
=
(
sin
a
)
l
o
g
b
c
o
s
a
. Then the order of
x
,
y
,
z
x,y,z
x
,
y
,
z
is
(A)
x
<
z
<
y
(B)
y
<
z
<
x
(C)
z
<
x
<
y
(D)
x
<
y
<
z
\text{(A)}x<z<y\qquad\text{(B)}y<z<x\qquad\text{(C)}z<x<y\qquad\text{(D)}x<y<z
(A)
x
<
z
<
y
(B)
y
<
z
<
x
(C)
z
<
x
<
y
(D)
x
<
y
<
z
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