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1975 USAMO Problems/Problem 2

Revision as of 22:35, 3 January 2009 by Minsoens (talk | contribs)

Problem

Let $A,B,C,D$ denote four points in space and $AB$ the distance between $A$ and $B$, and so on. Show that

$AC^2+BD^2+AD^2+BC^2\ge AB^2+CD^2$.

Solution

[asy] defaultpen(fontsize(8)); pair A=(2,4), B=(0,0), C=(4,0), D=(4,2); label("A",A,(0,1));label("D",D,(1,0));label("B",B,(-1,-1));label("C",C,(1,-1)); axialshade(A--C--D--cycle, lightgray, A, gray, D); draw(A--B--C--A--D--C);draw(B--D, linetype("8 8")); label("$m$",(A+B)/2,(-1,1));label("$n$",(C+D)/2,(1,0)); label("$c$",(B+C)/2,(0,-1));label("$b$",(A+C)/2,(-1,-1)); label("$a$",(A+D)/2,(1,1));label("$d$",(B+D)/2,(-1,1)); [/asy] If we project points $A,B,C,D$ onto the plane parallel to $AB$ and $CD$, $AB$ and $CD$ stay the same but $BC, AC, AD, BD$ all decrease, making the inequality sharper. Thus, it suffices to prove the inequality when $A,B,C,D$ are coplanar: [asy] size(200); defaultpen(fontsize(8)); pair A=(8,3), B=(4,-5), C=(10,0), D=(0,0); draw(A--C--B--D--A--B);draw(C--D);draw(anglemark(A,B,D,40));draw(anglemark(C,B,A,55,60)); label("A",A,(0,1));label("D",D,(-1,0));label("B",B,(0,-1));label("C",C,(1,0)); label("$m$",(A+B)/2,(1,0));label("$n$",(C+D)/2,(0,1)); label("$c$",(B+C)/2,(1,-1));label("$b$",(A+C)/2,(-1,-1)); label("$a$",(A+D)/2,(0,1));label("$d$",(B+D)/2,(-1,-1)); label("$\phi-\theta$",anglemark(A,B,D,40),(1,5));label("$\theta$",anglemark(C,B,A,55),(8,9)); [/asy]

Let $AD=a, AC=b, BC=c, BD=d, AB=m, CD=n$. We wish to prove that $a^2+b^2+c^2+d^2\ge m^2+n^2$. Let us fix $\triangle BCD$ and the length $AB$ and let $A$ vary on the circle centered at $B$ with radius $m$. If we find the minimum value of $a^2+b^2$, which is the only variable quantity, and prove that it is larger than $m^2+n^2-c^2-d^2$, we will be done.

First, we express $a^2+b^2$ in terms of $c,d,m,\theta,\phi$, using the Law of Cosines:

$a^2+b^2=c^2+d^2+2m^2-2cm\cos(\theta)-2dm\cos(\phi-\theta)$

$\implies (a^2+b^2-c^2-d^2-2m^2)^2$ $=4m^2(c^2\cos^2(\theta)+d^2\cos^2(\phi-\theta)+2cd\cos(\theta)\cos(\phi-\theta))$.

$a^2+b^2$ is a function of $\theta$, so we take the derivative with respect to $\theta$ and obtain that $a^2+b^2$ takes a minimum when

$c\sin(\theta)-d\sin(\phi-\theta)=0$

$\implies c^2\sin^2(\theta)+d^2\sin^2(\phi-\theta)-2cd\sin(\theta)\sin(\phi-\theta)=0$. $⟹(a2+b2−c2−d2−2m2)2=4m2(c2+d2+2cd(cos⁡(θ)cos⁡(ϕ−θ)−sin⁡(θ)sin⁡(ϕ−θ)))=4m2(c2+d2+2cdcos⁡ϕ)=4m2(2c2+2d2−n2)$ (Error compiling LaTeX. Unknown error_msg)

Define $p=a^2+b^2$ and $q=c^2+d^2$:

$(p-q-2m^2)^2=4m^2(2q-n^2)$

$\implies p^2+q^2+4m^4-4m^2p+4m^2q-2pq=8m^2q-4m^2n^2$

$\implies p^2+q^2+4m^4-4m^2p-4m^2q-2pq=-4m^2n^2$

$\implies p^2-2pq+q^2-4m^2(p+q)=-4m^2(m^2+n^2)$

$\implies \frac{(p-q)^2}{m^2}=p+q-m^2-n^2\ge 0$

$\implies a^2+b^2+c^2+d^2\ge m^2+n^2$.

See also

1975 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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