regional mo 2015
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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
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this is a problem of regional math olympiad 2015
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Re: regional mo 2015
Disclose the first approach
- asif e elahi
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Re: regional mo 2015
Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$
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Re: regional mo 2015
$PA \times PD = 12 \times (PA + AB + BC + CD) = 12 \times (12 + AB + BC + 2AB)$asif e elahi wrote:Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$
$ = 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