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MATHEMATICAL CHALLENGE FOR FILIPINO KIDS TRAINING PROGRAMLEVEL 6 QUIZ # 1(Answer Key)

1. Find point Pinside ABCor on its side so that it makes the sum of distances from it to three vertexes

largest.Solution:

A

B C

'P

1P

2PP

First, it will e proved that point Pcannot e inside

ABC. !f point Pis inside ABC, then an ellipse with foci Band Cis made topass P.

"et the ellipse intersect ABand ACat P1and P2, respectivel#.$onnect APand extend it to intersect P1P2at point P, then PA% max &P1A,

P2A'. Suppose that P(A% P1A, then P1A) P1B) P1C* PA) +P1B) P1C -PA) PB) PC.

ote: an ellipse is the locus of points such that the sum of the distance to each

focus is constant.

2. "et ae each side length of e/uilateral triangle ABC. !f curve Ldivides this triangle into two portions with

e/ual areas, then its length is0

.2 1

al

roof:

A

CB

3ake Aas the center and Ras the radius to make arc L( so that

ABCis divided into two portions with e/ual areas, then

2 21 1 ,

4 2 0R a =

so

0 25.

2

aR

=

3he length of Lis0

1' 2 .4 2

al R

= =

ow it is necessar# to prove '.l l

6ake 7 copies of ABCand arrange them as shown, then Lmakes a closed curve and L( makes a circle, and

the areas enclosed # them are e/ual, so 4 4 '.l l 3hat is,0

.2 1

al

. ABCDis a trape8oid +AB//CD, Eis a point on segment AB, and Fis a point on segment CD. Segments CE

and BFintersect at point H, and segments EDand AFintersect at point G.

rove: 9.27 .EGFH ABCDS S

roof:

A B

D CF

G H

E "et AGE- , then

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2

9.7 sin , 9.7 sin ,

9.7 sin , 9.7 sin ,

so 9.27 sin

.

AGE ADG

DFG FGE

AGE DFG

ADG FGE

S AG GE S DG AG

S DG GF S FG GE

S S DG GF AG GE

S S

= =

= =

=

=

From the ine/ualit# of arithmetic means, then

2 2 .

,2 ,

0 2 .

AGE DFG AGE DFG DGA FGE

GEF AFE AGE ADE AGE ADG

AGE DGF FGE

FGE FGE AGE DGF AEFD

S S S S S S

S S S S S S

S S S

S S S S S

+ =

= = =

+

+ + =

Q

!n the same wa#,0 .

FHE EBCFS S

3herefore,0 0 0 , then 9.27 .

EGFH FHE FGE EBCF AEFD ABCD EGFH ABCDS S S S S S S S

= + + =

repared #:

3;: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> =rade "evel: >>>>>>>>>>>>>>>>> Score: >>>>>>>>>>>>>

School: >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 63= $enter: >>>>>>>>>>>>>>>> ?ate: >>>>>>>>>>>>>>

1. Find point Pinside ABCor on its side so that it makes the sum of distances from it to three vertexeslargest.

2. "et ae each side length of e/uilateral triangle ABC. !f curve Ldivides this triangle into two portions with

e/ual areas, then its length is0

.2 1

al

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. ABCDis a trape8oid +AB//CD, Eis a point on segment AB, and Fis a point on segment CD. Segments CE

and BFintersect at point H, and segments EDand AFintersect at point G.

rove:9.27 .

EGFH ABCDS S

repared #:

3;: