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Problems
Contests
National and Regional Contests
Canada Contests
Canadian Open Math Challenge
2024 Canadian Open Math Challenge
C1
C1
Part of
2024 Canadian Open Math Challenge
Problems
(1)
2024 COMC C1
Source:
11/4/2024
Let the function
f
(
x
,
y
,
t
)
=
x
2
−
y
2
2
−
(
x
−
y
t
)
2
1
−
t
2
f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}
f
(
x
,
y
,
t
)
=
2
x
2
−
y
2
−
1
−
t
2
(
x
−
y
t
)
2
for all real values
x
,
y
x,y
x
,
y
and
t
≠
±
1
t\not=\pm1
t
=
±
1
a) Evaluate
f
(
2
,
0
,
3
)
f(2,0,3)
f
(
2
,
0
,
3
)
and
f
(
0
,
2
,
3
)
f(0,2,3)
f
(
0
,
2
,
3
)
. b) Show that
f
(
x
,
y
,
0
)
=
f
(
y
,
x
,
0
)
f(x,y,0)=f(y,x,0)
f
(
x
,
y
,
0
)
=
f
(
y
,
x
,
0
)
for any values of
(
x
,
y
)
(x,y)
(
x
,
y
)
. c) Show that
f
(
x
,
y
,
t
)
=
f
(
y
,
x
,
t
)
f(x,y,t)=f(y,x,t)
f
(
x
,
y
,
t
)
=
f
(
y
,
x
,
t
)
for any values of
(
x
,
y
)
(x,y)
(
x
,
y
)
and
t
≠
±
1
t\not=\pm1
t
=
±
1
. d) Given
g
(
x
,
y
,
s
)
=
(
x
2
−
y
2
)
(
1
+
sin
(
s
)
)
2
−
(
x
−
y
sin
(
s
)
)
2
1
−
sin
(
s
)
g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}
g
(
x
,
y
,
s
)
=
2
(
x
2
−
y
2
)
(
1
+
sin
(
s
))
−
1
−
sin
(
s
)
(
x
−
y
sin
(
s
)
)
2
for all real values
x
,
y
x,y
x
,
y
and
s
≠
π
2
+
2
π
k
s\not=\frac{\pi}{2}+2\pi k
s
=
2
π
+
2
πk
, where
k
k
k
is an integer number, show that
g
(
x
,
y
,
s
)
=
g
(
y
,
x
,
s
)
g(x,y,s)=g(y,x,s)
g
(
x
,
y
,
s
)
=
g
(
y
,
x
,
s
)
for any values of
(
x
,
y
)
(x,y)
(
x
,
y
)
and
s
s
s
in the domain of
g
(
x
,
y
,
s
)
g(x,y,s)
g
(
x
,
y
,
s
)
.
Comc