Algebra & Inequalities

By considering a suitable scalar product, prove that $$(ax+by+cz{)}^2⩽(a^2+b^2+c^2)(x^2+y^2+z^2)$$ for any real numbers $a$, $b$, $c$, $x$, $y$ and $z$. Deduce a necessary and sufficient condition on $a$, $b$, $c$, $x$, $y$ and $z$ for the following equation to hold: $$(ax+by+cz{)}^2=(a^2+b^2+c^2)(x^2+y^2+z^2).$$

1. Show that $(x+2y+2z{)}^2⩽9(x^2+y^2+z^2)$ for all real numbers $x$, $y$ and $z$.
2. Find real numbers $p$, $q$ and $r$ that satisfy both $$p^2+4q^2+9r^2=729\text{ and }8p+8q+3r=243.$$

5 (i) Given that $a>b>c>0$ are constants, and that $x,y,z$ are non-negative variables, show that$$ax+by+cz\le a(x+y+z).$$In the acute-angled triangle $ABC$, $a$, $b$ and $c$ are the lengths of sides $BC$, $CA$ and $AB$, respectively, with $a>b>c$. $P$ is a point inside, or on the sides of, the triangle, and $x$, $y$ and $z$ are the perpendicular distances from $P$ to $BC$, $CA$ and $AB$, respectively. The area of the triangle is $\Delta$.

(ii) (a) Find $\Delta$ in terms of $a$, $b$, $c$, $x$, $y$ and $z$.

(b) Find both the minimum value of the sum of the perpendicular distances from $P$ to the three sides of the triangle and the values of $x$, $y$ and $z$ which give this minimum sum, expressing your answers in terms of some or all of $a$, $b$, $c$ and $\Delta$.

(iii) (a) Show that, for all real $a$, $b$, $c$, $x$, $y$ and $z$,$$(a^2+b^2+c^2)(x^2+y^2+z^2)=(bx-ay{)}^2+(cy-bz{)}^2+(az-cx{)}^2+(ax+by+cz{)}^2.$$(b) Find both the minimum value of the sum of the squares of the perpendicular distances from $P$ to the three sides of the triangle and the values of $x$, $y$ and $z$ which give this minimum sum, expressing your answers in terms of some or all of $a$, $b$, $c$ and $\Delta$.

(iv) Find both the maximum value of the sum of the squares of the perpendicular distances from $P$ to the three sides of the triangle and the values of $x$, $y$ and $z$ which give this maximum sum, expressing your answers in terms of some or all of $a$, $b$, $c$ and $\Delta$.