8
Miscellaneous Problems
- Solve x(3y - 5) = y2 + 1 in integers.
- Solve
a3 - b3 - c3 = 3abc
a2 = 2(b + c)
simultaneously in positive integers. - Find all integer solutions (x, y, z) of the system
3=x+y+z=x3+y3+t3.- Let f(x) = x2+x. Show that 4f(a) = f(b) has no solutions in positive
integers. - Solve the equation (x2 + y)(x + y2) = (x - y)” for integers x, y.
- Consider the diophantine equation x3 = y2 + 4. Observing that
y2 + 4 = (y + 2a’)(y - Pi), solve first the equation (U + ~i)~ = y + 2i
for integers u, V, y and use this to obtain solutions (x, y) in integers
to the given equation. - Let Q, b, c, d be integers with a # 0. Can axy + bx + cy + d = 0 have
infinitely many solutions in integers x and y? - Solve for integers I, y the equation
x3-y3=2xy+8.- Determine infinitely many solutions in rational numbers x, y, .z, t of
the equation:
(x + y&)2 + (z + t&i)’ = 27 + lOti.- Determine all integer solutions (x, y, z) of
~ZZ+f/L--&.- Find ten rational values of x such that 3x2 - 5x + 4 is the square of
a rational number.