MathDB

2024 COMC C1

Source:

November 4, 2024

Comc

Problem Statement

Let the function

f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}

for all real values

x,y

and

t\not=\pm1

a) Evaluate

f(2,0,3)

and

f(0,2,3)

. b) Show that

f(x,y,0)=f(y,x,0)

for any values of

(x,y)

. c) Show that

f(x,y,t)=f(y,x,t)

for any values of

(x,y)

and

t\not=\pm1

. d) Given

g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}

for all real values

x,y

and

s\not=\frac{\pi}{2}+2\pi k

, where

k

is an integer number, show that

g(x,y,s)=g(y,x,s)

for any values of

(x,y)

and

s

in the domain of

g(x,y,s)

.

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