BdMO National Junior 2019/6

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
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samiul_samin
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BdMO National Junior 2019/6

Unread post by samiul_samin » Mon Mar 04, 2019 8:55 am

$ABCD$ is a trapezium and $AB$ parallel $CD$.$P,Q,R,S$ are the midpoint of $AB,BD,CD,AC$. $AC$ & $BD$ intersect at $O$.The area of $\triangle AOB=2019$ & $\triangle COD=2020$. Find the area of $PQRS$.

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samiul_samin
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Re: BdMO National Junior 2019/6

Unread post by samiul_samin » Wed Mar 13, 2019 10:16 am

Diagram
Screenshot_2019-03-12-22-38-58-1.png
Screenshot_2019-03-12-22-38-58-1.png (38.11 KiB) Viewed 6763 times
Solution

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math_hunter
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Re: BdMO National Junior 2019/6

Unread post by math_hunter » Fri Mar 15, 2019 7:45 pm

I don't understand the solution.
How the area of APQD is (3/4)ADB, PBCS is (3/4)ABC, CSR is (1/4)ACD, DQR is (1/4)DBC???
Can you please narrate the solution?

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Arifa
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Re: BdMO National Junior 2019/6

Unread post by Arifa » Sun Jan 19, 2020 10:21 pm

$Solution:$ Let x be the area of $BOC$ ; $(OQR) = (SOR)$ & $(POS) = (PQO)$ then,
$2(OQR) = 2(1010 - (2020+x)/4 ) = 2020 -1010 - x/2$
$2(POS) = 2 ( (2019+x)/4 - 2019/2) = x/2 + 2019/2$
Adding those 2 equations,
$2(OQR) + 2(POS)= 2020-1010- (2019/2) = 1/2$
Implies, $(PQRS) = 1/2$
Oops, I forgot

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Arifa
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Re: BdMO National Junior 2019/6

Unread post by Arifa » Thu Jan 23, 2020 2:49 pm

Arifa wrote:
Sun Jan 19, 2020 10:21 pm
$2(POS) = 2 ( (2019+x)/4 - 2019/2) = x/2 + 2019/2$
There is a typo , $sorry$ :( the $2nd$ equation should be-
$2(POS) = x/2 - 2019/2$
Oops, I forgot

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