Montmorency Contest 2004 2005

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13 2



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I bn

W lmlGONd

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2 2 + · · · +

√ 13

1 − 1 2

I bn

W lmlGONd

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2K1"%;'-7:<;,JQ08619)E

m = n = 2RF@0G2)2H;)0BE°&6@AC0@Ea0

x 2619*

y 2-I9}2H;682$AC*08AbK"%;)0@AC0

0 < x < yRZ8-I9:H0fa)$D2H;Eb°'&61AC08E;6[y0

4*-7O[-.2FEGK‰"%0};6Gy0

y 2 = x 2 +3131KD$A

(y −x)(y +x) = 31×101RZ8-79:40

RZSN`

n K

lmlGONd

I b n

W lmlGONd

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x 6@9*

y ;6[y02H;)0,E<61J>05?6@AC-2\MsRZ8-79:40

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l + w = 5 6@9*

lw = 2rT2H;)0@AC0G{l$AC0KT2H;)0Q?L08AC-IJQ0[2408AV$'{2H;)0AC01:C2F619O[(70Ž-7E

2(l + w) = 2(5) = 10:HJK6@9*52H;)0V6@AC0@6Ž-IE

lw = 2:HJ

2R

x 3 − 70x 2 + 1629x − 12600

= x 3 − (l + w + h)x 2 + (lh + lw + wh)x − lhwRS#;'-IEf-IJ>?(I-70@Ev2H;6G2

lh + lw + wh = 1629619*

lhw = 12600r12H;'&EGKD2H;)0ŽE<&AF{+6D:4061AC0865$'{2H;)0V-I9'2408AC-7$Af$'{2H;)0%AC$)$J -7E

2(lw + lh + wh) = 2(1629) = 3258 J

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p = 56@9*

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M = p + q − 1 = 2p − 1R

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p+q p

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P X = P Y + P ZR

ABKzAC08E?01:C2C-y08(MR  $D2C02H;682

DR = DS 619*

ET = EURF@0G2

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y = z = 1R

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x = 1K6@9*52H;)0%O[-y0@9>0@°&6G2C-+$9M-708(7*)E

n = 3R/l ck R

yz = 2K"%;'-+:<;Q-IJ>?(I-70@E 2H;682

y = 2 6@9*

z = 1R5!+2 {l$'(I(7$1"VE 2H;682

x ≥ 2K619*¤2H;)0>O[-y089u0@°&6G2C-+$9wAC0@*&L:40@Ez2C$

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f (c) = c 2 (2b + 1 − c){l$A

2 ≤ c ≤ 2b-7E

f (2) = 4(2b − 1)RS#;)08AC08{l$AC0'K

BE 2 − AD 2 > 2K6:H$9'24AF61*-7:C2C-+$9R}¯Ž0@9:40K

a = br2H;682-IEGK

AC = BCR

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