Difference between revisions of "1982 USAMO Problems/Problem 2"

Revision as of 17:31, 5 March 2020

Problem

Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$ ,

$(*)$ $\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$

for $(m,n)=(2,3),(3,2),(2,5)$ , or $(5,2)$ . Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$ .

Solution 1

This problem needs a solution. If you have a solution for it, please help us out by adding it. Claim Both $m,n$ can not be even.

Proof $x+y+z=0$ , $\implies x=-(y+z)$ .

Since $\frac{S_{m+n}}{m+n} = \frac{S_m S_n}{mn}$ ,

by equating cofficient of $y^{m+n}$ on LHS and RHS ,get

$\frac{2}{m+n}=\frac{4}{mn}$ .

$\implies \frac{m}{2} +\frac {n}{2} = \frac{m.n}{2.2}$ .

So we have, $\frac{m}{2} | \frac{n}{2}$ and $\frac{n}{2} | \frac{m}{2}$ .

$\implies m=n=4$ .

So we have $S_8=2.(S_4)^2$ .

Now since it will true for all real $x,y,z,x+y+z=0$ . So choose $x=1,y=-1,z=0$ .

$S_8=2$ and $S_4=2$ so $S_8 \neq 2 S_4^2$ .

This is contradiction !! So, atlest one of $m,n$ must be odd. WLOG assume $n$ is odd and m is even . The cofficient of $y^{m+n-1}$ in $\frac{S_{m+n}}{m+n}$ is $\frac{\binom{m+n}{1} }{m+n} =1$

The cofficient of $y^{m+n-1}$ in $\frac{S_m .S_n}{m.n}$ is $\frac{2}{m}$ .

So get $\boxed{m=2}$

Now choose $x=y=\frac1,z=(-2)$ .

Since $\frac{S_{n+2}}{2+n}=\frac{S_2}{2}\frac{S_n}{n}$ holds for all real $x,y,z ,x+y+z=0$ .

We have , $\frac{2^{n+2}-2}{n+2} = 3 . \frac{2^n-2}{n}$ .

$\implies \frac{2^{n+1}-1}{n+2} =3.\frac{2^{n-1}-1}{n} \cdots (**)$ .

Clearly $(**)$ holds for $n=5,3$ . Even one can say that for $n\ge 6$ , $\text{RHS of (**)} <\text{LHS of (**)}$ .

So our answer is $(m,n)=(5,2),(2,5),(3,2),(2,3)$ .

-ftheftics

@@ Line 6: / Line 6: @@
 for <math>(m,n)=(2,3),(3,2),(2,5)</math>, or <math>(5,2)</math>. Determine ''all'' other pairs of integers <math>(m,n)</math> if any, so that <math>(*)</math> holds for all real numbers <math>x,y,z</math> such that <math>x+y+z=0</math>.
-== Solution ==
+== Solution 1 ==
 {{solution}}
+Claim Both <math>m,n</math> can not be even.
+Proof
+<math>x+y+z=0</math> ,<math>\implies x=-(y+z)</math>.
+Since <math>\frac{S_{m+n}}{m+n} = \frac{S_m S_n}{mn}</math>,
+by equating cofficient of <math>y^{m+n}</math> on LHS and RHS ,get
+<math>\frac{2}{m+n}=\frac{4}{mn}</math>.
+<math>\implies  \frac{m}{2} +\frac {n}{2} = \frac{m.n}{2.2}</math>.
+So we have, <math>\frac{m}{2} | \frac{n}{2} </math> and <math>\frac{n}{2} | \frac{m}{2}</math>.
+<math>\implies m=n=4</math>.
+So we have <math>S_8=2.(S_4)^2</math>.
+Now since it will true for all real <math>x,y,z,x+y+z=0</math>.
+So choose <math>x=1,y=-1,z=0</math>.
+<math>S_8=2</math> and <math>S_4=2</math> so <math>S_8 \neq 2 S_4^2</math>.
+This is contradiction !!
+So, atlest one of <math>m,n</math> must be odd. WLOG assume <math>n</math> is odd and m is even .
+The cofficient of <math>y^{m+n-1}</math> in <math>\frac{S_{m+n}}{m+n}</math> is <math> \frac{\binom{m+n}{1} }{m+n} =1</math>
+The cofficient of <math>y^{m+n-1}</math> in <math>\frac{S_m .S_n}{m.n}</math> is <math>\frac{2}{m}</math>.
+So get <math>\boxed{m=2}</math>
+Now choose <math>x=y=\frac1,z=(-2)</math>.
+Since <math>\frac{S_{n+2}}{2+n}=\frac{S_2}{2}\frac{S_n}{n}</math> holds for all real <math>x,y,z ,x+y+z=0</math>.
+We have ,<math>\frac{2^{n+2}-2}{n+2} = 3 . \frac{2^n-2}{n}</math>.
+<math>\implies \frac{2^{n+1}-1}{n+2} =3.\frac{2^{n-1}-1}{n} \cdots (**)</math>.
+Clearly <math>(**)</math> holds for <math>n=5,3</math> .
+Even one can say that for <math>n\ge 6</math> ,
+<math>\text{RHS of (**)} <\text{LHS of (**)}</math>.
+So our answer is <math>(m,n)=(5,2),(2,5),(3,2),(2,3)</math>.
+-ftheftics
 == See Also ==

1982 USAMO (Problems • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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Difference between revisions of "1982 USAMO Problems/Problem 2"

Revision as of 17:31, 5 March 2020

Problem

Solution 1

See Also