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I.S.I. and C.M.I. Entrance

Multi Variable Equations – CMI UG Entrance 2019 – Problem 5

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

Number System


Multi variable linear equations are equations that have two or more unknowns (generally represented by ‘x’ and ‘y’).

Try the problem


Three positive real numbers $x, y, z$ satisfy
[
\begin{aligned}
x^{2}+y^{2} &=3^{2} \
y^{2}+y z+z^{2} &=4^{2} \
x^{2}+\sqrt{3} x z+z^{2} &=5^{2}
\end{aligned}
]
Find the value of 2 x y+

CMI UG Entrance, 2019

Multi variable Equations

6 out of 10

AOPS Intermediate Algebra

Use some hints


Assume that the given inequality is true for ( a>0 , b>0 ) and ( a+b<2 ) . Then proceed. Note that the values on the RHS of the three equations are squares of the Pythagorean triplet (3,4,5).
So draw a right-angled triangle with sides 3, 4, & 5 and then proceed.
(Say we take a triangle ABC with sides AB = 3, BC = 4, & CA = 5)

In the triangle that you were asked to construct in (Hint 1), take a point O inside it and name OA = x, OB = y and OC = z.
Now try to relate these with the equations you are provided with in the question.

Try to predict the angles AOB, BOC, & COA to make use of the cosine formula.
EXAMPLE : From the first equation and the triangle you were asked to make, it is quite obvious that angle AOB = 90 degrees.
Hence, from triangle AOB, we obtain:
\(x^2 + y^2 – 2xy cos90^\circ = 3^2\), i.e., \(x^2 + y^2 = 3^2\)

\( (a-b)^2 \geq 0 \) as \( a,b \in{R}\) .

And to get ( (1-ab)>0 ) use the well known inequality for positive reals i.e. \( AM \geq GM \) and the still unused inequality i.e ( a+b <2 ) also .

\( a>0 , b>0 \Rightarrow \sqrt{ab}>0 \Rightarrow( 1+ \sqrt{ab})>0  \)  \(a>0 , b>0 , a+b <2 \Rightarrow  1 > \frac{a+b}{2} \geq \sqrt {ab} \ \Rightarrow 1 > \sqrt{ab} \ \Rightarrow ( 1 – \sqrt{ab}) >0 \ \Rightarrow (1 – \sqrt{ab}) (1+ \sqrt{ab}) >0 \ \Rightarrow (1 – ab)>0 \)

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