Show that $x^2 + y^2 + z^2 \ge 35$ if $x+3y+5z \ge 35.$

You have an elegant solution using Cauchy Schwarz already in comments. Another way using AM-GM:

$$x^2+1^2 \geqslant 2x, \quad y^2+3^2 \geqslant 6y, \quad z^2+5^2 \geqslant 10 z$$ $$\implies x^2+y^2+z^2+(1+9+25) \geqslant 2(x+3y+5z)$$ $$\implies x^2+y^2+z^2 \geqslant 35$$