regional mo 2015

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kh ibrahim
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regional mo 2015

Unread post by kh ibrahim » Sat Dec 10, 2016 12:49 pm

this is a problem of regional math olympiad 2015
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rmo 2016
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kh ibrahim
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Joined:Mon May 09, 2016 11:18 am

Re: regional mo 2015

Unread post by kh ibrahim » Mon Dec 12, 2016 11:40 pm

Disclose the first approach

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asif e elahi
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Re: regional mo 2015

Unread post by asif e elahi » Tue Dec 13, 2016 12:53 am

Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$

Absur Khan Siam
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Re: regional mo 2015

Unread post by Absur Khan Siam » Mon Jan 30, 2017 11:17 pm

asif e elahi wrote:Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$
$PA \times PD = 12 \times (PA + AB + BC + CD) = 12 \times (12 + AB + BC + 2AB)$
$ = 36AB + 12BC + 144...(i)$
$PB \times PC = (PA+AB) \times (PA + AB + BC) = (12+AB) \times (12 + AB + BC)$
$ = AB^2 + 24AB + 12AB + AB \times BC ...(ii)$
$(i) = (ii) \rightarrow 36AB + 12BC + 144 = AB^2 + 24AB + 12AB + AB \times BC$
$ \rightarrow 12AB = AB(AB+BC)$
Thus,$AB+BC = 12$
And we get , $PC = PA + AB + BC = 24$ ;)
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid

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