This is a problem I proposed for the Final round of the Peruvian Mathematical Olympiad 2013, about 1 month ago. I hope you will enjoy this problem.

**Problem** Let \(x, y, z\) be real numbers such that \(x^2+y^2+z^2\leq 100\), find the maximum and minimum values of \(2xy+2yz+7xz.\).

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestI hate to do this, but I feel inclined to bash using Lagrange multipliers. For any \(x^2+y^2+z^2=k\), \(0\le k\le100\), we want \(\left<2y+7z,2x+2z,2y+7x\right>=\lambda\left<2x,2y,2z\right>\). Solving for \(\lambda\), we get \(2y+7z=2x\lambda\), \(2x+2z=2y\lambda\), and \(2y+7x=2z\lambda\). Eliminate \(x\) by substituting \(x=y\lambda-z\) to get \(2y+7z=2y\lambda^2-2z\lambda\) and \(2y+7(y\lambda-z)=2z\lambda\). Plugging in the second as \(2z\lambda\), we get the quadratic \(0=2y\lambda^2-7y\lambda-4y=y(2\lambda+1)(\lambda-4)\), so \(\lambda=-\frac{1}{2}\text{ or }4\), or \(y=0\). Whew, \(\lambda\) is somewhat easy to work with (thanks Jorge!).

In the case that \(y=0\), we have \(7xz\) to optimize. Assume WLOG that \(x,z\ge0\), so \(7xz\le7\left(\sqrt{\frac{x^2+z^2}{2}}\right)^2\le7\frac{100}{2}=350\). That means \(y=0\) gives a maximum of \(350\) (\(x=z=5\sqrt{2}\)) and minimum of \(-350\) (\(x=5\sqrt{2},z=-5\sqrt{2}\)).

In the case that \(\lambda=4\), \(2y+7z=4(2x)\) and \(2x+2z=4(2y)\). Solving for \(y\) and \(z\), \(y=\frac{1}{2}x\) and \(z=x\), which plugs in as \(2xy+2yz+5zx=9x^2\) and \(x^2+y^2+z^2=\frac{9}{4}x^2\le100\). The minimum here is \(0\) (\(x=y=z=0\)) clearly, but the maximum is \(9\frac{4}{9}100=400\) (\(x=z=\frac{20}{3},y=\frac{10}{3}\)).

In the case that \(\lambda=-\frac{1}{2}\), \(2y+7z=-\frac{1}{2}(2x)\) and \(2x+2z=-\frac{1}{2}(2y)\). Solving for \(y\) and \(z\), we get \(y=-4x\) and \(z=x\), which plugs in as \(2xy+2yz+5zx=-11x^2\) and \(x^2+y^2+z^2=18x^2\le100\). The maximum here is clearly \(0\) (\(x=y=z=0\)), but the minimum is \(-4\frac{100}{18}=-\frac{550}{9}\) (\(x=z=\frac{5\sqrt{2}}{3},y=-\frac{20\sqrt{2}}{3}\)).

In effect, the maximum value is \(\boxed{400}\) at \(\left(\frac{20}{3},\frac{10}{3},\frac{20}{3}\right)\), and the minimum value is \(\boxed{-350}\) at \(\left(5\sqrt{2},0,-5\sqrt{2}\right)\).

Now the real question is what is the non-LM solution?

Log in to reply

I was reading your solution very carefully trying to spot the mistake because your answers were wrong, then I realized that you edited while I was reading... now your answers are correct!

Of course there is a elementary solution, in fact this problem was proposed to students with no knowledge of Calculus (grades 9 and 10 in Peru) and only one student managed to solve both parts of the problem.

Log in to reply

Is it Jensen's?

Log in to reply

This is exactly how I would've done it, great solution, regardless of any bashing

Log in to reply

If we want the greatest possible value of \(2xy+2yz+2xz\) (instead of \(2xy+2yz+7xz\) ) we can use the inequalities \(2xy\leq x^2+y^2\), \(2yz\leq y^2+z^2\) and \(2xz\leq x^2+z^2\), in this case the maximum occurs when \(x=y=z\).

But in our problem we can try to use similar inequalities like \(2xy\leq Ax^2+By^2\), where \(A\) and \(B\) are suitable constants.

Log in to reply

For minimum: \[ 2xy+2yz+7xz = (x+y+z)^2 - (x^2+y^2+z^2) + \frac{5}{2} (x+z)^2 - \frac{5}{2}(x ^2 + z^2) \\ \ge - (x^2+y^2+z^2) - \frac{5}{2}(x ^2 + z^2) \ge -100 - \frac{5}{2} (100) = -350. \] The equality holds when \(x+y+z=0, x+z=0, x^2+y^2+z^2=100, x^2+z^2=100\). There is an obvious solution, that is \(y=0, x=-z, x^2=z^2=50\).

Log in to reply

For me is very nice to see new solutions of this problem. Let me show my approach:

First note that \(x\) and \(z\) have a symmetric role. Then, to obtain \(x^2+y^2+z^2\) I will work with a expression of the form \((x+\lambda y+z)^2\) [note that \(x\) and \(z\) have the same coefficient]]. If we want a multiple of \(2xy+2yz+7xz\) we can take \(\lambda=\frac{2}{7}\).

Thus we work with the following inequality which is true: \[\left(x+\frac{2}{7}y+z\right)^2+\frac{45}{49}y^2 \geq 0,\] this inequality is equivalent to: \[\frac{2}{7}(2xy+2yz+7xz)\geq -x^2-y^2-z^2\] and we are done! [Of course, we have to complete some details]

Log in to reply

Very nice, interesting approach.

Log in to reply

What was your motivation behind breaking it up like this?....

Log in to reply

Really nice approach.

Log in to reply

2xy+2yz+7xz=what?

Log in to reply

Let \(k = 2xy+2yz+ 7zx\). Then \(k \leq \dfrac{100( 2xy+2yz+ 7zx)}{x^2+y^2+z^2}\), which becomes \[\begin{aligned} kx^2-x(200y+700z)+ky^2-200yz+kz^2\leq 0. \end{aligned}\] Since the solutions of \(kx^2-x(200y+700z)+ky^2-200yz+kz^2 = 0\) are real (solutions are \(x\) and \(\dfrac{200y+700z}{k}-x\)), its discriminant \(\Delta_x\) is nonnegative: \[\begin{aligned} \Delta_x = (200y+700z)^2-4k(ky^2-200yz+kz^2) &\geq 0\\ \implies (40000-4k^2)y^2 + (280000+800k)yz+(490000-4k^2)z^2&\geq 0.\qquad(\star) \end{aligned}\] If \(z = 0\), then \((\star)\) becomes \((40000-4k^2)y^2 \geq 0\) and since \(y^2\geq 0\) for all \(y\in\mathbb{R}\) we must have \(40000-4k^2\geq 0\implies |k|\leq 100\). If \(z\neq 0\), then dividing both sides of \((\star)\) and taking \(t=\dfrac{y}{z}\) gives \[\begin{aligned} (40000-4k^2)t^2 + (280000+800k)t+(490000-4k^2)\geq 0. \end{aligned}\] Since the roots of the equation \((40000-4k^2)t^2 + (280000+800k)t+(490000-4k^2)=0\) are real (note that \(t = \dfrac{y}{z}\in\mathbb{R}\), its discriminant \(\Delta_t\) is nonnegative: \[\begin{aligned} \Delta_t = (280000+800k)^2-4(40000-4k^2)(490000-4k^2)\geq 0\\ \implies k^3-142500k-7000000\leq 0. \end{aligned}\] (I used Wolfram Alpha to obtain this cubic inequality). Since the roots of \(k^3-142500k-7000000=0\) are \(k=-350, -50, 400\) (also obtained from Wolfram Alpha), the minimum value of \(k = 2xy+2yz+7zx\) is \(\boxed{-350}\) at \((x,y,z)=(-5\sqrt{2}, 0, 5\sqrt{2})\) and the maximum value is \(\boxed{400}\) at \((x,y,z)=\left(\dfrac{20}{3}, \dfrac{10}{3}, \dfrac{20}{3}\right)\).

Log in to reply

Woah! Amazing solution! Truly genius, in my opinion!

Log in to reply

For maximum: \[ 2xy+2yz+7xz = 4(x^2+y^2+z^2) - \frac{1}{2}((x-2y)^2+(z-2y)^2 + 7(x-z)^2) \le 400. \] The equality holds when \(x=2y=z\) and \(x^2+y^2+z^2=100\).

Or apply Cauchy-Schwarz to:

\[ (2xy+2yz+7xz)^2 = [(x)(2y) + (2y)(z) + (x)(z) + (x)(z)+ (x)(z) + (z)(x) + (z)(x)+(z)(x)+(z)(x)]^2\\ \le[x^2+(2y)^2+x^2+x^2+x^2+z^2+z^2+z^2+z^2][(2y)^2+z^2+z^2+z^2+z^2+x^2+x^2+x^2+x^2]\\ =(4x^2+4y^2+4z^2)^2 \]

Or simply as Jorge pointed out: \[ 2xy = x(2y)\le \frac{1}{2}(x^2+4y^2), 2yz = (2y)z\le \frac{1}{2}(z^2+4y^2), 7xz \le \frac{7}{2}(x^2+z^2) \] then add all three together.

Log in to reply

Why has the diagram of a sphere been given?

Log in to reply

It's only a point inside a sphere :) If \(x^2+y^2+z^2\leq 100\) then the point \((x,y,z)\) is inside the sphere of radius 10.

Log in to reply

I have no idea what to do here.I tried establishing a bound but I am not sure what to use here.I have tried to use some elementary inequalities such as Cauchy-Schwarz and AM-GM.Using AM-GM,we find that

\(\dfrac{x^2+y^2+z^2}{3}\ge [(xy)(yz)(zx)]^\dfrac{1}{3}\le \dfrac{xy+yz+zx}{3}\)

But I am not sure how that helps us.Also,

\(x^2+y^2+z^2\ge \dfrac{(x+y+z)^2}{3}\)

by Titu's lemma.I have also tried to rewrite the expression as

\(2xy+2yz+7zx=2(xy+yz+zx)+5zx\)

and was wondering whether I can apply AM-GM here.

Log in to reply

I think something like \(2xy+2yz+7zx\le (2^2+2^2+7^2)(x^2y^2+y^2z^2+z^2x^2)=57(x^2y^2+y^2z^2+z^2x^2)\le 57(x^4+y^4+z^4)\) would help, because \(x^2y^2\) (and similar terms) are nonnegative.

Log in to reply

That gives you an upper bound on the possible value of \( 2xy+2yz+7zx \). Another (easy) upper bound is that since \( |x|, |y|, |z| \leq 10 \), hence the value is at most \( 200+200+700=1100\). Hence, we also need to check and see if it is indeed the

least upper bound.For it to be the least upper bound, we must have equality hold throughout. What are the equality conditions for each of your inequalities?

Log in to reply

Hi Rahul! If I'm not mistaken for using AM. G.M the numbers on which the inequality is applied must be positive, here is there any condition that \(x,y,z\) is positive?

Log in to reply

No,but \(x^2,y^2,z^2\) are always non-negative.However,when I say

\( [(xy)(yz)(zx)]^\dfrac{1}{3}\le \dfrac{xy+yz+zx}{3}\)

I am possibly wrong,since we don't know if \(xy,yz,zx\) are non-negative.However,when looking for the maximum,we can safely assume that \(x,y,z\) are positive.

NOTE:AM-GM stipulates the values beingnon-negative,notpositive.Log in to reply

Log in to reply

how can i become as intelligent as you

Log in to reply

Definitely not by asking "how can i become as intelligent as you."

Log in to reply

If you think being sarcastic sounds clever, you're highly mistaken.

Log in to reply

Practice, practice, practice

Learn from others

Read the Techniques page. etc.

Log in to reply

How can you be so rude??

Log in to reply