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[email protected]. MATHEMATICAL MAYHEM Vol. 2 * 10 40 www.mandelbrot.org * 180 2 30 * x = r 6 7 x + 1 x 2 * a + b + c a b c abc = 50

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Page 1: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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A

B C

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1

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tan θ = 1xr0uD1:O0�246=J

d = 1y 03*w*7>-0�:B2/5s0uD-246W8�24512/8w.Q8~T

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w2 = 1Jx1 = λ1 cos θ

J:+5QI

x2 = sin θJz0h*�d7P3*B|1?;0uD1:O0t0uD7?c8�24512/8w.Q8�d7:O0uDi,U?O5Qk�0uD�2/6 (

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2 ·A ·B − 9 = 10 ·A+B

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A246B = 7

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T _ D-.Q6=J10uD7?OP3?tES?BP3?478;:F0uD7?B8w:O0324gh24:B576�:F00uD7?�8�?B?F032/5Qk1T

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t T

A

BC

r

R

XY*�03?�0uD1:O0AB = R + r

:+5QIAC = R − r

T _ D-.Q6FJC>B<;0uD7?�p-<�0uD1:�kO*1P3?O:+5_ D7?B*1P3?B8BC2 = AB2 −AC2

= (R+ r)2 − (R− r)2

= 4Rr

_ D-.Q6=JT t = BC = 2

√RrT

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360 × 1111 = 399960T

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y − 1==⇒ (x− 1)(y − 1) = 1 ==⇒ xy = x+ y

T

Page 6: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

� �

XY*+E15 = x2 + y2 = (x2 + 2xy + y2) − 2xy = (x+ y)2 − 2(x+ y)

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p ≤ 7T _ *�6(D7*+E^0uD-2/6FJB6e.Qd7df*16A?@03*t0uD7?Ygh*15-0hP�:+P{<'0uD1:F0

p ≥ 8T_ D7?O5�0uD7?�51.Q8�>7?BP`*-lK6(�-.Q:+P3?O6�gu*O|1?BP3?BI�>B<s0uD7?�P3?Bg303:B5Qk=,U?O6W2/5�0uD7?SI7?+gh*18;dQ*16e2�032U*15246

a1 + a2 + · · · + ap ≥ 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 = 72J

ESD-2UgeDs246`2/8;dQ*16A6u2U>-,U?t6e2/5Qgu?Y0uD7?SgeD7?O6A6A>Q*1:BP3I;D1:B6�*151, <646A�1.Q:BP3?B6OT

¡1*1Pp = 7

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Page 12: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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�B?F0A:B5QI

B>7?'|1?OP�0324gu?O6Y*-lC0uD7?�03?F0hP�:+D7?+I1P3*15KJf,4?F0

O>Q?�2r0�6�gu?O5-0hP3?1JQ,4?F0

O′>Q?t0uD7?�gu?O5-0hP3?S*-lf0uD7?tlU:�gh?S*1d7df*16u2r03?BJ-:+5QI�,4?F0

x = OA = OBT _ D7?O5

O′A = 2 ×√

3

3=

2

3

√3J

BO′ =

22 −(

2

3

√3

)2

=2

3

√6J

OO′ =2

3

√6 − x

TXY*�03?t0uD1:F0 4OO′A

246WP�2UkOD�0u�u:+5Qk=,4?+I 0uD-.Q6FJ+ES?�D1:F|1?(

2

3

√6 − x

)2

+

(2

3

√3

)2

= x2 J :B5QIx =

√6

2

T_ D7?OP3?Olm*1P3?-J10uD7?�P�:�I-2/.Q6�*-lf0uD7?�?uÔB03?BP357:B,C6edfD7?OP3?�2/6 √

6

2+ 1:B5QI�0uD7?�P�:�I-2/.Q6�*-lC0uD7?

245-03?BP357:B,C6edfD7?OP3?'2/6 √6

2− 1T

354 6#7�6#784 9=;=< >-?���;89 G I Q�� M8I�� K��8C��A<B;13K � �8C 4�� � G ���HR �BR K��7=CED ��B7=C J-K L�� � �=I < � � � J� �M K

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AJB:B5QI

CJQ2/l

8 cosA cosB cosC = 10uD7?O5dQP3*O|1?t0uD1:O0 4ABC 246@?B�1.72r,4:F0h?OP�:B,�T�$�����7�#" ��S��� / ��� � ���� �$���+���$�Q�

11��� ��� "#� � / �F� �{� �-���! �" �K���=� � �1�#"��� ���+�$� +�� ���#�

� ��� "#� �$� 2 +�" � �QT¡-P3*18cosA cosB = 1

2[cos(A+B) + cos(A−B)]

ES?SkO?=01

2[cos(A+B) + cos(A−B)] cosC =

1

8

J

Page 13: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

� �

cos(A−B) =1

4 cosC− cos(A+B)

J

=1

4 cosC+ cosC

T'@6e2/5Qk�0uD7? \ � ��G`� 245Q?B�1.Q:O,/2�0�<SES?t0uD7?O5;kO?F0

cos(A−B) =1

4 cosC+ cosC ≥ 2

√(

1

4 cosC

)

(cosC) = 1T

_ D7?OP3?Olm*1P3?-Jcos(A − B) = 1

ESD-24geDw2/8;d1,/24?B6`0uD1:O0A = B

T \ 6u248�2/,/:+P�:BP3k=.Q8�?O5-0E'2r0uDB:+5QI

C<124?O,4I76

B = CT _ D-.Q6=J

A = B = C = 60◦ :+5QI 4ABC 2/6?+�-.72/,/:O03?BP�:O,�T354 6#716#784:98;=< >-? ��;89 G I!Q���M=I �8K��8C��A<B;

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Jax2 = c

T_ D7?O5ax2 + ax+ a = 7

T�¡-:+g30h*1P�245Qkt:B5QIwP3?B:BPhP�:B5Qk=245Qk�J+ES?SkO?=0a(x2 + x+ 1) = 7

Jx2 + x+ 1 =

7

a

Jx2 + x+

a− 7

a= 0

JxU5Q*�032/5Qk�0uD1:O0

a 6= 0yAT )C?+g3:B.Q6A?~?O:�geDi0h?OPh8�2/5�0uD7?~kO?+*18;?F0hP�2Ug�6A?+�-.f?B5Qgh?b2/6�:B5245-03?+kO?OP(J`*15Q?b6A?+�-.f?B5Qgh?~?eÔB246h036Slm*1PS?B:+geDoP�:O0324*157:B,`P3*7*�0S*-l`0uD7?c:�>7*B|1?b?+�-.Q:O0324*15KJE'2r0uDs2/5-0h?BkO?BP

aT _ *�8w:�j+?Y0uD7?S?uÔ�d7P3?B6e6e24*15�lU:�g303*1P�:�>-,U?-J

a8;:=<;>7?

1*1P

7T

Úmla = 1

Jx2 + x− 6 = 0 =⇒ x = −3

*1Px = 2

JB<124?O,4I12/5Qk�0uD7?Y6(?B�1.f?O5Qgu?O61J −3J9:+5QI

1J2J4J1P3?B6edf?Bg3032�|1?F,�<}T

Úmla = 7

Jx2 + x = 0 =⇒ x = 0

*1Px = −1

Jf<-2U?F,UI-245Qk�0uD7?�6(?B�1.f?O5Qgu?O67J0J0:B5QI

7J −7J7J1P3?B6edf?Bg3032�|1?F,�<}T

Page 14: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

� �

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9:+5QI

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a03*

9:B5QI

4T

Úmla = 9

Jx2 + x + 2

9= 0 =⇒ x = −2

3

*1Px = −1

3

J9<124?O,4I12/5Qk;0uD7?6(?B�1.f?O5Qgu?O69J −6J4:B5QI

9J −3J1J1P3?B6edf?Bg3032�|1?F,�<}T

Úmla = 4

Jx2 +x− 3

4= 0 =⇒ x = 1

2

*1Px = −3

2

JF<-2U?F,UI-245Qk`0uD7?�6(?B�1.f?O5Qgu?O64J2J1:B5QI

4J −6J9J1P3?B6edf?Bg3032�|1?F,�<}T

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�ON

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Jy1 = 847

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limn→∞

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XY*�03?t0uD1:F0xn+1yn+1 =

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xn+1 =xn + yn

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xn +20022

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=

(1365

3367

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limn→∞

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n→∞

(1365

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BCT_ D7?OP3?Olm*1P3?-J

BM = MCT��B?F0

C′ >Q?t0uD7?'df*-2/5-0`*15�0uD7?t,/2/5Q? AM ?uÔB03?B5QI7?+I�E'2�0uDAM = MC′ T _ D7?O5 4AMC

246�gu*15QkFP�.f?B5-0`0h* 4C′MBxU6u245Qgh?

BM = MCJ

AM = MC′ JQ:B5QI∠C′MB = ∠AMC

yAT _ D7?BP3?Flm*1P3?1JAC = BC′ T \ ghgu*1P3I-245Qk0h*�0uD7? _ P32/:+5Qk=,4?sÚ{5Q?+�-.Q:B,r2r0{<KJ

AB +BC′ > AC′ TH)O2/5Qgu? AM = MC′ JAC′

2=

AM +MC′

2=

AM +AM

2= AM

T��?O5Qgu?-J'ES?ÀD1:F|1?

(AB + BC′)/2 > AMT )O245Qgh?

BC′ = ACJtES?^kO?=0

(AB +AC)/2 > AMT

XY*�03?b0uD1:F0(AB + AC)/2 = AM

*151,�<$2/l 4ABC′ 246w5Q*�0s:+g303.Q:B,r,�<Ã:0hP32/:+5Qk=,4?�:F0�:B,r,Cx/0uD1:F0�246=JB2/6W:SdQ*-245-0`*15

AC′ J+ESD-24geD�P3?B6u.7,r0�6W2/5 B = M = CyAT

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Page 17: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

��M

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aJbJcJ

d:BP3?s5Q*15Q�e5Q?BkF:O032 |1?s2/5-0h?BkO?BP�6 ≤ 9

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2T

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T•

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J+ESD-2UgeDwgh*15-0hP�:�I-2Ug30�610c+ d > 0

T•

Úmla = 1

J�0uD7?B5;?O2�0uD7?BPc = 1

:B5QId = 8

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:+5QId = 0

T _ D7?'�QP�6u0`*-l0uD7?B6A?�0�ES*�gh:B6(?O6�lm*1P3gu?O6b = 9

J7ESD-24geD�gu*15-0hP�:+I124g3036abcd = 10c + d

T _ D7?6(?Bgu*15QI�gh:B6(?'k=2�|1?O6�0uD7?Y0�ES*s51.Q8;>Q?OP�6@:B69:B5QI

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Page 18: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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(b2 + c2 − a2

) (c2 + a2 − b2

) (a2 + b2 − c2

)≤ a2b2c2

T x{L�y'@6e2/5Qk�0uD7?Sj+5Q*+E�5;P3?F,4:F032U*1576

b2 + c2 − a2 = 2bc cosAJ

c2 + a2 − b2 = 2ca cosBJ :B5QI

a2 + b2 − c2 = 2ab cosC

245;xmL=yhJ�ES?�?B6h03:+>1,r246AD�0uD7?'P324kOD�0u�AD1:B5QI;6u2UI7?�*-l}x{M�yAT� ÎeÉ1É6��Ü�J�B?F0

OIQ?O5Q*�0h?�0uD7?Ygh24P3gh.Q8;gu?O5-0hP3?Y:+5QI�,U?=0

HIQ?O5Q*�0h?@0uD7?t*1P�0uD7*7gu?O5-0hP3?t*-l10uD7?0hP32/:+5Qk=,4?

ABCT _ D7?O5~ES?cD1:F|1?;0uD7?�lm*-,/,4*+E'245Qkwlm*1P38w.7,/:�lm*1PY0uD7?�6(�-.Q:+P3?c*-l�0uD7?I12/6u0�:+5Qgh?S>Q?=0�ES?+?O5

O:+5QI

HT

OH2 = R2 (1 − 8 cosA cosB cosC)J x ] y

[email protected]�:�I-2/.Q6z*-l+0uD7?�0hP�24:B5Qk=,U?7T _ D7?WP324kOD�0u�AD1:B5QI'6u2UI7?@*-l-x{M�yQlm*-,r,U*+E�6248w8�?BI12/:O03?O, <w6u245Qgh?

OH2 ≥ 0T

�$*16u0`gh,4:B6A6u2Ug3:B,703?eÔO0u>Q*7*7j+6�*15 _ P324kO*15Q*18�?=0hP{<KJ76u.fgeDw:B6 3M �t:+5QI��L��+JQD1:=|1?S:dQP3*7*-l}*-l}x ] yeT "!$#&%('*) +-,-.

c© /10$0$2 4315(3&6$) 31587�39.9,$:-;<39.=) >?31@4A1!(>=) :=.*%

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� ]

� ÎeÉ1É6���JÚml�0uD7?;0hP�24:B5Qk=,U?;246SP�2UkOD�0u�u:+5Qk=,4?+ICJ90uD7?B5~0uD7?�d7P3*7I1.fg30�*-l�0uD7?�gu*16u245Q?O6�*-l�0uD7?0uD1P3?+?�:+5Qk=,4?B6�2/6@�e?OP3*}T'Úml}0uD7?�0hP�24:B5Qk=,U?S2/6Y*7>�03.Q6(?-J70uD7?B5;0uD-2/6�d7P3*7I1.fg30W246�5Q?+kF:F032�|1?7T��?O5Qgu?-J�2/5S0uD7?O6(?�0�ES*�g3:+6A?B6H0uD7?YP�2UkOD�0u�eD1:+5QI�6e24IQ?t*-lfx{M�y}2/6`?O6u0�:�>-,/2/6(D7?BI T�Úml70uD7?�0hP�2U�:+5Qk=,4?W2469:+gh.10h?-JO:B,r,O0uD7?�gu*16u245Q?O69:+P3?Wdf*16u2r032 |1?W:+5QI�0uD7? \ � ��G`� 245Q?B�1.Q:O,/2�0�<'k=2�|1?O69.Q6

(cosA cosB cosC)13 ≤ 1

3(cosA+ cosB + cosC)

T x [ y�A?B576A?B5K� 6YÚ{5Q?+�-.Q:B,r2r0{<;k=2�|1?O6`.Q6

13(cosA+ cosB + cosC) ≤ cos

(13(A+B + C)

)= 1

2

T x{��y_ D7?O5�x [ yz:B5QI;x{��yH?B6h03:+>1,r246AD�0uD7?'P324kOD�0u�AD1:B5QI;6u2UI7?�*-l}x{M�yAT

� ÎeÉ1É6���sJ�x/ab?S>7?O,r2U?(|1?Y0uD-246WdQP3*7*-lC2/6W5Q?=E;T y�B?F090uD7?t6(?O8w24�edQ?BP�248;?F03?BPAJ�2/57P�:�I-2/.Q6`:B5QIwgh2/P3gh.Q8;P�:+I12r.Q6@*-l70hP�24:B5Qk=,U?

ABC>Q?IQ?O5Q*�0h?BI�>B<

sJrJRJ7P3?O6AdQ?+g3032 |1?O, < T �B?F0

y = s/R:B5QI

x = r/RT _ D-246@5Q*�03:F032U*15246WlU:B8w2r,/2/:+PH03*wP3?B:+IQ?OP�6�*-l! � ���� 790:*;+ '���������'�T _ D7?B5KJ-0uD7?�lm*-,r,U*+E'2/5Qk�?B�1.Q:O,/2�0�<246@j+5Q*+E�5;:B5QI;g3:+5;>Q?�?B:B6e2r,�<sdQP3*O|1?+IK�

y2 − (x+ 2)2 = 4 cosA cosB cosCT xU�Fy

\ ,46A*y2 ≤ −x2 + 8x+ 3

T x��(yab?�5Q*+EºdQP3*O|1?t0uD1:F0H0uD7?',4?Ol/0u�eD1:+5QI�6e24IQ?�*-lKxm�=y9246�,U?O6A6�0uD1:B5;*1P�?+�-.Q:B,70h* 1

2

J+ESD-24geD248w8�?BI12/:O03?O, <s?B6h03:+>1,r246AD7?B6z0uD7?YP324kOD�0u�AD1:B5QIs6u2UI7?t*-lfx{M�yAT�¡-*1P 0uD-2/6FJFES?t*7>76A?BP{|1?�0uD1:F0y2 − (x+ 2)2 ≤ 1

2⇐=⇒ 2y2 ≤ 2x2 + 8x+ 9

TÚ{5�|124?=E°*-l}x ��y 2r09E'2r,/,Q6e.7v�gu?Y0h*sdQP3*O|1?

2x2 + 8x+ 9 ≥ 2(−x2 + 8x+ 3)T _ D-246,4:B6u0�2/5Q?+�-.Q:B,r2r0{<w246�?+�-.72�|-:O,U?O5-0�0h*

(2x − 1)(2x − 3) ≥ 0J-ESD-24geDw246�gh,4?B:BP3, <�0hP3.f?7T_ D-2/6�gh*18;d1,U?=0h?O6�*-.QP�dQP3*7*-l3T

¡�2/57:B,r,�<CJ�0uD7?t,U?Fl/0u�AD1:B5QIw6e24IQ?�*-l}xmM=y92/6W6(?B?B5;>B<w*7>76A?BP{|-245Qk�0uD1:O0sin2A+ sin2B + sin2C = 2 (1 + cosA cosB cosC) > 0

T� Ë5��ËFÎeËF× � ËOÝ��

3M �~)QT��7T �B*15Q?(<CJ � ���K� �F��" ���� ���c�+�m�8�-JYRC:+8;>7P�2UI7kO? \ 0 _ D7? '@512�|1?OP�6e2�0�< pQP3?B6e6FJM+N����QT3L ���}T a$T��'*7>76A*15KJH �F�e���7�#" �F� $ �( �#� �C� �F��" �$������c���m� �1J+RC:+8;>7P�2UI7kO? \ 0 _ D7?+'@512U�|1?BP�6u2r0{<bpQP3?O6A6=JQM ��N�MQT

�H?+I-.7,4:wXSTK��.QP�0{<] �+N��SR1<7d7P3?B6e6�RK*-.QP�0SW*O|1?BPAJKp \ J '�) \ M� ] M��

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xU:=yzP3?B:O,CP3*7*�0uÓxm>+yH5Q?BkF:O032 |1?�P3*7*�0uÓ

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J�:B5QI�:+5�<i6(?B�1.f?O5Qgu?e1Je2J. . .JeNJESD7?BP3?

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a:B5QI

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b6u.fgeD�0uD1:O0

(aa)5 = bbT

� T _ *�:+5�<w0hP32/:+5Qk=,4?ABC

E'2r0uDAB = c

JBC = a

JCA = b

J∠A = α

J∠B = β

J :B5QI∠C = γ

JCES?w:+6e6e24kF5c0uD7?w6(?uÔB03.Qd1,U?(a, b, c, α, β, γ)

JCESD7?OP3?�0uD7?:+5Qk=,4?B6 :BP3?`8�?O:+6u.QP3?+I�2/5tP�:+I12/:+576OTH¡+245QI�0uD7?`8�24512/8;:O,=|-:O,/.f?W*-lnlm*1P7ESD-2UgeD@0uD7?OP3?`2/6:'5Q*15Q�h246A*16(gh?O,4?B6H0hP�24:B5Qk=,U?

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SE'2r0uD |M | = 50

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JM − {x} 246�5Q*�0Y,4:+>Q?F,/,4?+IE'2r0uD

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n ≤ k6u.fgeD�0uD1:F0

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m:+5QI

nT@pQP3*O|1?t0uD1:F0

gcd(m,n) > 1T

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kI12/6u032/5Qg306e.f>16(?=036W6e.fgeD�0uD1:F0H0uD7?'245-03?BP�6A?+g30324*15;*-lK:B5�<s0�ES*�*-lf0uD7?B8�246@gh*1575Q?+g303?+I}T

� T �B?=0f>7?;:wl4.Q5Qg3032U*15blUP3*18 0uD7?;6A?F0t*-lHdf*16u2r032 |1?w2/5-0h?BkO?BP�6Y0h*�0uD7?;6A?F0Y*-l5Q*15Q�e5Q?BkF:O032 |1?'245-03?+kO?OP�[email protected]:F0

f(1) = 0:+5QI

f(n) = max{f(j) + f(n− j) + j}lm*1P�:O,/,

n ≥ 2T SW?F03?BP38w2/5Q?

f(2000)T

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XY?uÔB0(J�ES? k=2�|1? 6(*-,r.1032U*1576~03*�0uD7? '�j+P�:B2/5124:B5 �o:O0uD7?O8;:F032Ug3:B,�V`, <78wd72/:�ICJ)+?O,4?+g303?+IbpQP3*7>-,U?O8;6�MBN�N�� 3L ����Mw�1� �����1TÒ TYxUNO0uD�GzP�:�I7?=y�R}?F,/,/6�*-l96A*18�?SP3?+g30�:+5Qk=.7,/:+P@>Q*1:BP3I�:+P3?sgh*-,U*-.QP3?BI�:+6YgeD7?O6A6A�>Q*1:BP3I~gu?F,/,/6BTwÚ{5i?O:�geDbgh?O,r,z:B5c245-03?+kO?OP�246�E�P�2r0�03?B5 T;ÚU0�2/6'j�5Q*+E�5�0uD1:O0@0uD7?�6e.Q8 *-l0uD7?�51.Q8�>7?BP�6�245�?O:�geD;P3*+E¼246�?(|1?O5�:+5QIw0uD7?S6e.Q8�*-l 51.Q8;>Q?OP�6�2/5�?B:+geD�gh*-,/.Q8w5�2/6?(|1?B5 T@pQP3*O|1?Y0uD1:O0H0uD7?�6u.Q8 *-lC:B,r,f51.Q8�>7?BP�6W245s0uD7?�>1,/:�gej�gu?F,/,/6`2/6@?(|1?B5 T�$����AB� ����� �C� ���u� 27���� �� �Q� � � ��" AB�=����"�� � 2���� ��� �1� �� � +C� 2 �+��"�� � � � ���+�(� � �O��� �

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XY*+E gu*1576u2UI7?BPC0uD7?�6u.Q8 *O|1?BP}:B,r,B0uD7?�>1,/:�gejYgu?F,/,/6H*15t0uD7?�>Q*1:BP3I T ab?W2/8;:+k=245Q?0uD-246S6u.Q8 03*�>7?�gh:O,Ugh.7,/:O03?+I�>B<ogh*-,/.Q8w576BTi)O2/5Qgu?�ES?�:+P3?;2/5-0h?OP3?B6h0h?BIo*151, <~245~0uD7?dQ:BP32�0�<�*-l}0uD7?s6u.Q8bJ1ES?s:BP3?�lUP3?B?S03*�6u.Q8 *O|1?BPW:B,r,C0uD7?�ESD-2�0h?sgu?F,/,/6�2/576u03?B:+Ic*-l}0uD7?>1,/:�gejYgu?F,/,/692/5�:+5�<�k=2�|1?O5Sgu*-,r.Q8;5 T _ [email protected] *O|1?BPC0uD7?�>1,/:�gejYgu?F,/,/692/5t0uD7?�*7I7IQ�51.Q8�>7?BP3?BI�gh*-,/.Q8w576@:B5QI;*B|1?OPH0uD7?�ESD-2r03?Sgh?O,r,46�245s0uD7?�?(|1?B5Q�u51.Q8�>7?BP3?BI�gu*-,r.Q8;576OT_ D-2/5Qj+2/5QkW5Q*+EÀ2/5Y0h?OPh8w6H*-l-P3*+E�6=JB*-.QPK6e.Q8�246z? � ?Bg3032�|1?F,�<'*O|1?BP���� �-gh?O,r,46zxU>Q*�0uD'>1,/:�gej:+5QI�ESD-2r03?=y}2/5�?(|1?BP{<s6(?Bgu*15QI�P3*+E;T _ D-246`6u.Q8�2/6Wgh,4?B:BP3, <�?(|1?O5KJ-6e2/5Qgu?�0uD7?t6e.Q8 *O|1?BP:B,r,70uD7?Sgu?F,/,/6`2/5w:+5�<wP3*+E�246@?(|1?O5 TÙ THx{M �O0uDwGHP�:+IQ?�yH)+*-,�|1?Y0uD7?�6�<16u03?B8�245�P3?B:O,C51.Q8;>Q?OP�6

{x1 + x2 + · · · + x1997 = 1997x4

1 + x42 + · · · + x4

1997 = x31 + x3

2 + · · · + x31997

T�$����AB� �¼��� / " � �1��� `�c��� � ����� 2���A���� � 2 ���#� ,�" � �1�=� ��� / ��� �����������c�F� ��" ���

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�$*1P3?ikO?O5Q?BP�:O,/, <KJ�,U?=0n>7?~:~dQ*16e2�032�|1?c2/5-0h?BkO?BP�T ab?�E'2r,/,Wd7P3*B|1?�0uD1:O0�0uD7?.Q512U�-.f?S6A*-,/.10324*15s245wP3?O:B,f51.Q8�>7?BP�6�*-lf0uD7?'6�<76h0h?O8

{x1 + x2 + · · · + xn = nx4

1 + x42 + · · · + x4

n = x31 + x3

2 + · · · + x3n246

(1, 1, . . . , 1)T�ÚU0z2/6`?O:+6�<�03*S6(?B?�0uD1:O0

(1, 1, . . . , 1)2/6�:t6A*-,/.10324*15s*-l70uD7?Y6�<76h0h?O8iTR}*15�|1?OP�6(?F,�<CJ-2/l

x1Jx2J. . .Jxn:+P3?'P3?B:O,C51.Q8;>Q?OP�6@6A:F03246ulr<12/5Qk�0uD7?'6�<76h0h?O8bJ�0uD7?O5

0 =

n∑

i=1

x4i −

n∑

i=1

x3i −

n∑

i=1

xi + n =

n∑

i=1

(x4i − x3

i − xi + 1)

=n∑

i=1

(xi − 1)2(x2i + xi + 1) =

n∑

i=1

(xi − 1)2

((

xi +1

2

)2

+3

4

) T)O245Qgh?1J+lm*1PH?B:+geD

iJOES?tD1:=|1?

(xi − 1)2((xi + 1

2)2 + 3

4

)≥ 0J+0uD7?t?B�1.Q:O,/2�0�<s*7gugh.QP�6*151,�<s2/l

xi = 1lm*1P

i = 1J2J. . .JnT

��� "Q� �Y����"�AB� � ���������=� � �1��� �F�Q�K����$� +c��� / �+���!��� �$��� "#�OT)1<;RKD7?+>B<16(D7?(|C� 6W2/5Q?+�-.Q:B,r2r0{<KJ-2rl

a1 ≤ · · · ≤ an:+5QI

b1 ≤ · · · ≤ bnJ+0uD7?B5

n(a1b1 + a2b2 + · · · + anbn) ≥ (a1 + a2 + · · · + an)(b1 + b2 + · · · + bn)T

�7�-.Q:B,r2r0{<;*7ghgh.QP�6W2rlK:+5QI;*151, <s2/lK:F0�,U?O:+6h0�*15Q?�6A?+�-.f?B5Qgh?'246@gh*1576u0�:+5-0 �L����-Mw�-��M [ �-Ta�2r0uD7*-.10�,4*16A6�*-l�kO?O5Q?BP�:O,/2�0�<CJKES?;:B6A6u.Q8�?s0uD1:O0

x1 ≤ x2 ≤ · · · ≤ x1997T_ D7?O5

x31 ≤ x3

2 ≤ · · · ≤ x31997

T _ D7?BP3?Flm*1P3?1J1997(x4

1+x42+· · ·+x4

1997) ≥ (x1+x2+· · ·+x1997)(x31+x3

2+· · ·+x31997)

T)O245Qgh?

x1 + x2 + · · · + x1997 = 1997:+5QI

x41 + x4

2 + · · · + x41997 = x3

1 + x32 + · · · + x3

1997

J0uD7?�:�>7*B|1?t245Q?B�1.Q:O,/2�0�<w>Q?Bgu*18;?B6@?+�-.Q:B,r2r0{< T _ D-.Q6FJ

x1 = x2 = · · · = x1997*1P

x31 = x3

2 = · · · = x31997

TÚ{5�?F2r0uD7?OP`g3:+6A?1J

x1 = x2 = · · · = x1997 = 1T

Û Tzx{M �O0uD�GzP�:�I7?=y �B?F0d(n)

I7?B5Q*�03?@0uD7?YkFP3?B:F0h?O6u0z*7IQI�I-2�|-246A*1Pz*-l-0uD7?�57:O03.QP�:O,51.Q8�>7?BPnTWab?sI7?O�Q5Q?�0uD7?�l4.Q5Qg3032U*15

f : � → �:B6@lm*-,/,4*+E�6O�

f(2n − 1) = 2nJ

f(2n) = n+2n

d(n)

lm*1P�:O,/,n ∈ �

T¡�2/5QIw:B,r,

k6u.fgeD�0uD1:O0

f(f(. . . f(1) . . .)) = 1997J�ESD7?OP3?

f246`2�0h?OP�:O03?+I

k032/8�?O6BT

�$�����7�#" ��b���� "{�F�h�u� � ��� � � �=����" �$� ��}� ��"#�=����,���� ���u��T�B?F0

(xk)>7?�0uD7?S6A?+�-.f?B5Qgh?sIQ?F�f5Q?BI�>B<

x1 = 1:+5QI

xk+1 = f(xk)lm*1P`:B,r,

k ≥ 1T9ab?�E�:+5-0H03*s�f5QI�0uD7?'245-03?+kO?OP�6

k6e.fgeD�0uD1:F0

xk+1 = 1997T

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_ D7?��QP�6u0@03?BP38;6'*-l90uD7?w6A?+�-.f?B5Qgh?(xk)

:BP3?1J2J3J4J6J5J8J12J10J7J

16J24J. . .T9ab?YdQP3?O6(?O5-090uD7?B8�2/5s6e.fghgu?O6A6u2�|1?YP3*+E�6=J

R1JR2JR3J. . .JOESD7?BP3?

Rjgu*15-0�:B2/576�?uÔB:�g303, <j03?BP38;6F�

R1 : 1R2 : 2

J3

R3 : 4J6J5

R4 : 8J12J10J7

R5 : 16J24J20J14J9TTT

ab?'E'2r,/,Kd7P3*B|1?�0uD1:F0(JQlm*1P`:B,r,}dQ*16e2�032�|1?�2/5-0h?BkO?BP�6iJjJ-E'2r0uD

j ≤ iJ70uD7?

j ��� 51.Q8�>7?BP245Ri2/6

(2j − 1)2i−jT

_ D-2/6�2/6�gh,4?B:BP3, <c0hP3.f?;lm*1Pi = 1

T �B?=0i ≥ 1

>7?;��Ô+?+I}T~)O.QdQdQ*16(?w0uD1:F0Y0uD7?P3?B6u.7,r0z2/6H0hP�.f?Ylm*1PRiT _ D7?B5S0uD7?�,4:B6u0 0h?OPh8�*-l

Rixr0uD7?'*15Q?Y:O0 0uD7?YP324kOD�0�y}2/6

2i− 1TÚU0Hlm*-,r,U*+E�6H0uD1:F090uD7?Y�fP�6h090h?OPh8 *-l

Ri+12/6f(2i− 1) = 2i = (2 × 1 − 1)2i+1−1 T_ D-.Q6=J-0uD7?�IQ?O6e2/P3?+Islm*1Ph8�.7,4:'246�0hP�.f?'lm*1PH0uD-2/6`�fP�6h0H03?BP38iT

)O.QdQdQ*16(?�0uD1:F0�0uD7?SP3?B6u.7,r0`2/6W0hP�.f?Slm*1P�0uD7?j ��� 51.Q8�>7?BP`245;P3*+E Ri+1

J-ESD7?OP3?1 ≤ j < i+ 1

T _ D7?O5�0uD7?tlm*-,/,4*+E'245Qk�0h?OPh8�245Ri+1

246F�f((2j− 1)2i+1−j

)= (2j− 1)2i−j +2i−j+1 =

(2(j+1) − 1

)2i+1−(j+1) T

_ D7?OP3?Olm*1P3?-J+0uD7?Ylm*1Ph8�.7,4:Y246H0hP3.f?Ylm*1P 0uD7?�|-:O,/.f?j+ 1

T )1<S2/5QI1.fg30324*15KJ�2r0z2/6H0hP3.f?Ylm*1P:B,r,j ∈ {1

J2J. . .Ji+ 1} T_ D-.Q6=J�0uD7?tlm*1Ph8�.7,4:'2/6�0hP�.f?tlm*1P�:B,r,C*-l

Ri+1T )1<�245QI-.fg3032U*15KJ-2�0�2/6�0hP�.f?tlm*1P�:B,r,0uD7?�P3*+E�6BT

�B?F0a = m2n

J}ESD7?BP3?mJn:BP3?�5Q*15Q�e5Q?BkF:O032 |1?�2/5-0h?BkO?BP�6S:+5QI

m2/6�*7IQI}T_ D7?O5

m2n = (2j − 1)2i−j2rlK:+5QI;*151, <s2/l

{m = 2j − 1n = i− j

0uD1:O0�2/6FJ

k =m+ 1

2

i = n+m+ 1

2

T

ÚU0`lm*-,/,4*+E�6`0uD1:O0a:BdQdQ?B:BP�6Y?uÔB:�g303, <�*15Qgu?�2/5w0uD7?S6(?B�1.f?O5Qgu?-JQ2/5�df*16u2r0324*15 m+ 1

2

2/5Rn+m+1

2

TÚmla = 1997

JW0uD7?B5m = 1997

:+5QIn = 0

T _ D-.Q6FJk = i = 999

T_ D7?OP3?Olm*1P3?-Jxk = 1997

2/lQ:+5QI�*151, <�2rlxk2/6H0uD7?Y,4:B6u0 0h?OPh8�*-l

R999J+245�ESD-24geDsgh:B6(?

k = 1 + 2 + 3 + · · · + 999 = 999 · 500 = 499500T

� THxmM�M=0uD�GHP�:+IQ?�y�ÚU0H2/6Wj+5Q*+E�5�0uD1:O090uD7?'?+�-.Q:O0324*15ax3 + bx2 + cx+ d = 0E'2r0uDiP3?B6edf?Bg30t03*

xD1:+6'0uD1P3?+?cI-246h03245Qg30'P3?B:O,�P3*7*�0�6BTÃ�'*+E¦8;:B5�<�P3*7*�036�I7*7?B6�0uD7??+�-.Q:O0324*15

4(ax3 + bx2 + cx+ d)(3ax+ b) = (3ax2 + 2bx+ c)2D1:F|1?BÓ

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4P3*7*�036OT`p1?OPuD1:BdQ6H0uD7?'*1P�2Uk=2/57:B,+|1?BP�6u2U*15�:+6Aj+?+ISlm*1P 0uD7?Y51.Q8�>7?BP*-lKP3?B:O,CP3*7*�0�6(Ó

�$�����7�#" �� ��� �C� �$�h� 27� �� �� �Q� � ���$"�AB�F��� "�� � 2���� ��� �1� �� � +C� 2 �+��"�� � � �� �B�(��� �O��� ���2�%

�B?F0p(x) = ax3 + bx2 + cx+ d

J1:+5QI�,4?F0f(x) = 4(ax3 + bx2 + cx+ d)(3ax+ b) − (3ax2 + 2bx+ c)2

Tab?':+P3?Y03*-,UIS0uD1:O0z0uD7?�?+�-.Q:O0324*15

p(x) = 0D1:+6�0uD1P3?+?�I12/6u032/5Qg30�P3?B:O,fP3*7*�036=J-:+5QISES?:+P3?�:+6Aj+?+Ic:�>7*-.10@0uD7?wP3*7*�0�6'*-l90uD7?w?+�-.Q:O0324*15f(x) = 0

TwRf,U?O:+P�,�<CJC0uD7?�dQP3*7>-,U?O8246�245-03?BP3?O6u03?+I�245cP3?B:O,9P3*7*�036t*151,�<}T;X�*�0h?S0uD1:O0f(x) = 2p(x)p ′′(x) − [p ′(x)]2

T��?O5Qgu?-Jf ′(x) = 2p(x)p ′′′(x) = 12ap(x)

:+5QIf ′′(x) = 12ap ′(x)

T�B?F0�0uD7?�P3*7*�0�6�*-l

p(x)>7?r1Jr2Jz:+5QI

r3JKESD7?BP3?

r1 < r2 < r3T~)O245Qgh?0uD7?B6A?�P3*7*�0�6t:BP3?wI-246h03245Qg30(J

p ′(ri) 6= 0x/lm*1P

i = 1J2J3yATS)O245Qgh?�0uD7?BP3?�:BP3?�0uD1P3?B?P3*7*�036=Jf0uD7?�IQ?BkFP3?+?�*-l

p(x)g3:+575Q*�0�>7?s,4?B6e6�0uD1:B5�0uD1P3?+?7T _ D7?OP3?Olm*1P3?-J

a 6= 0T'ab?gh:B5o:+6e6e.Q8;?

a > 0ToxmV�0uD7?OP�E'2/6(?wES?�P3?Bd1,4:+gu?

p(x)>B< −p(x) J9E'2�0uD�5Q*�? � ?Bg30*15

f(x)T y _ D7?B5

limx→∞ p(x) = ∞ :+5QIlimx→−∞ p(x) = −∞ Tº�'?B5Qgh?

p ′(r1) > 0Jp ′(r2) < 0

:+5QIp ′(r3) > 0

T¡1*1PK?B:+geD

iJf ′(ri) = 12ap(ri) = 0

JO:B5QIY0uD7?OP3?W:+P3?W5Q*t*�0uD7?BPKdQ*-245-0�6KESD7?OP3?f ′(x) = 0

T')O245Qgh?f ′′(x)

D1:+6@0uD7?s6A:B8�?�6u2UkF5�:B6p ′(x)

J7ES?sD1:F|1?f ′′(r1) > 0

Jf ′′(r2) < 0

:B5QIf ′′(r3) > 0

T _ D-.Q6=JfD1:+6�,4*7gh:O,`8�24512/8;:b:O0

r1:+5QI

r3J�:,U*7g3:B,`8;:�Ô+2/8w.Q8�:O0

r2J`:+5QI$5Q*$*�0uD7?OP�,U*7g3:B,`8w:=ÔB248w:~*1PS8w2/51248w:fTq¡1*1PS?B:+geD

iJ

f(ri) = −[p ′(ri)]2 < 0

T _ D7?BP3?Flm*1P3?1Jf(x) < 0

*15o:B5o2/5-0h?OP{|-:B,`gu*15-0�:B2/51245Qkr1Jr2:B5QI

r3TY¡+.QP�0uD7?OPh8;*1P3?1J

limx→±∞ f(x) = ∞ x46e2/5Qgu?'0uD7?sD-24kOD7?B6h0u�AI7?+kFP3?B?0h?OPh8�2/5f(x)

2463a2x4 yeT \ ,r,9*-l 0uD-246Y2/8;d1,/24?B6�0uD1:F0@0uD7?w?B�1.Q:F032U*15 f(x) = 0

D1:+6?eÔB:+g303,�<'0�ES*SP3?B:O,7P3*7*�0�6FJ�*15Q?t*-l-ESD-24geDS2/6H,U?O6A6z0uD1:+5r1x�ESD7?BP3?

f(x)geD1:B5QkO?B6�lUP3*18df*16u2r032 |1?W0h*'5Q?+kF:F032�|1?�yK:+5QIt0uD7?Y*�0uD7?BP9kFP3?O:O03?BP}0uD1:B5

r3x�ESD7?BP3?

f(x)geD1:B5QkO?B6HlUP3*185Q?+kF:F032�|1?t03*�dQ*16e2�032�|1?�yAT

� Tzx{M+MF0uD�GzP�:�I7?=y �B?F0�� + IQ?O5Q*�0h?@0uD7?�6(?=0z*-lQ:O,/,-df*16u2r032 |1?�P�:O0324*157:B,151.Q8;>Q?OP�6BT¡�2/5QIw:B,r,Ql4.Q5Qg3032U*1576f :

� + → � + 6e.fgeD�0uD1:O0�lm*1P�:B,r, x ∈ � + �xU:=yf(x+ 1) = f(x) + 1

Jxm>+y

f(x2) = (f(x))2T

�$����AB� �w��� / "�� +C� � ���7�#��" � �{����� � �1��� ��,+� ��� �h��� "{�F�h�u� � ��� �$���=�{� "#� � ��K����" �F� �,���� ���u��� � ��� / �+�h������� % ��� �B� " �$� ���$"�AB�F��� "�� � �� � ���7�=�(� ��� ( �-� ��K����$�� � %-Cs���"�AB��� +K���@��"��{�� �F� �� ����7�#��" � �{�-T

�B?F0f>Q?�:+5�<�6e.fgeD�l4.Q5Qg30324*15 T )1<�d7P3*1df?OP�0{<txU:=y7:B5QIY:B5�248w8�?BI12/:O03?�245QI-.fg3032U*15KJES?SkO?=0f(x+ n) = f(x) + n

lm*1P�:B,r,x ∈ � + :B5QI n ∈ �

TVW5�0uD7?�*�0uD7?BP�D1:+5QICJ

(f(x+ n)

)2= f

((x+ n)2

)= f(x2 + 2nx+ n2)

= f(x2 + 2nx) + n2 T

Page 29: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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R}*18wdQ:BP32/5Qk�J�ES?�*7>-0�:B2/5f(x2 + 2nx) = f(x2) + 2nf(x)

x{M�ylm*1P�:O,/,

x ∈ � + :+5QI n ∈ �T

XY*+E�J�,U?=0r =

p

q

>7?b:+5�<Ã?F,U?O8�?O5-0�*-l � + JHESD7?BP3? p ∈ �:B5QI

q ∈ �T

¡-P3*18�xmM=yhJBE'2�0uDx = r

:+5QIn = q

JOES?YkO?F0f(r2 +2p) = f(r2)+2qf(r)

T _ D7?B5f(r2) + 2p = f(r2) + 2qf(r)

JESD-2UgeDS<124?O,4I76

f(r) =2p

2q= rT _ D7?OP3?Olm*1P3?-J

f246�0uD7?t2UI7?B5-032�0�<w*15 �

+T

R}*15�|1?OP�6(?F,�<CJK0uD7?�24IQ?O5-032r0{<~*-l � + gh,U?O:+P�,�<c6A:F03246u�C?O6'gh*15QI12�032U*1576'xU:=yW:+5QI~xm>+yuJESD7?B5Qgh?'2r0H246�0uD7?'.Q5124�1.f?�6A*-,/.10324*15 T� TcxmM�M=0uD�GzP�:�I7?=yY¡�2/5QIc0uD7?;6A8w:B,r,U?O6u0

n6e.fgeD�0uD1:O0Y:+8;*15Qkw:+5�<

n245-03?+kO?OP�60uD7?BP3?�:BP3?

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kxU6A?+? �M��AyATHRK*1576e24IQ?OP�:B5�<2k− 2

2/5-0h?BkO?BP�6W6u.fgeD�0uD1:F0k− 1

*-lf0uD7?O8 :+P3?'?+�-.Q:B,703*08;*7I1.7,4*

kJf:+5QIw0uD7?�*�0uD7?OP

k − 1:+P3?s?B�1.Q:O,K03*

18;*7I1.7,4*

kT'ÚU0W246�?B:B6�<w0h*;6A?+?0uD1:O0(JC:+8;*15Qk�0uD7?B6A?

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kT _ D-.Q6FJ�0uD7?Y|-:O,/.f?

2k − 1246`2/5QIQ?B?+I�8w2/51248w:B,�T

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(x, y)6e:O032/6elr<-245Qk�0uD7?S?B�1.Q:F032U*15

1 + 1996x+ 1998y = xy .

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ab?��f5QI�:B,r,7dQ:O24P�6`*-lQ2/5-0h?BkO?BP�6(x, y)

6A:F03246ulr<12/5Qk@0uD7?Y8�*1P3?tkO?O5Q?BP�:O,f?+�-.Q:O0324*151 + (p− 1)x+ (p+ 1)y = xy

JESD7?BP3?

p > 12/6W:�dQP�248;?�51.Q8;>Q?OP=T _ D-246@?+�-.Q:O0324*15�246@?B�1.72 |-:B,4?B5-0H0h*

px+ py = xy + x− y − 1J

ESD-2UgeDwgh:B5;>Q?'P3?=E�P�2r0�03?B5w:+6p((x− 1) + (y + 1)

)= (x− 1)(y + 1)

T x{M�yab?s*7>76A?BP{|1?�0uD1:F0

(x, y) = (1,−1)246�:s6A*-,/.10324*15�:+5QIw0uD1:O0W5Q*�*�0uD7?OP�g3:+5Q�I124I7:F0h?O6Wlm*1P

(x, y)E'2�0uD

x = 1*1Py = −1

gh:B5w6A:F03246ulr<wx{M�yATXY*+E 6u.QdQdQ*16(?s0uD1:F0

x 6= 1:+5QI

y 6= −1J :+5QI~I7?B5Q*�03?;>B<

d0uD7?;kFP3?O:O03?B6h0gu*18w8�*15;I-2�|-246A*1P�*-l

x− 1:+5QI

y + 1T9ab?�D1:F|1?

x− 1 = du, y + 1 = dvJ x{L�y

ESD7?BP3?u:B5QI

v:+P3?�P3?F,4:F032�|1?F,�<;d7P32/8�?S2/5-0h?BkO?BP�6OT�)O.f>76h032r03.1032/5Qk'0uD7?B6A?s?uÔ�d7P3?B6e6e24*1576lm*1P

x− 1:B5QI

y + 12/5-0h*�x{M�yz:+5QI;I-2�|-2UI-245Qk'>Q*�0uD�6u2UI7?B6@>B<

dk=2 |1?B6

p(u+ v) = duvT x ] y

��?O5Qgu?-JuvI-2�|-2UI7?B6�0uD7?'d7P3*7I1.fg30

p(u+ v):+5QIs2/6`P3?O,/:O032 |1?O, <sdQP�248;?Y0h*

u+ vT )1<0uD7?s¡�.Q5QI1:+8;?B5-0�:B, _ D7?+*1P3?O8 *-l \ P�2r0uD18;?F0324g�J

uv246W:SI12 |12/6(*1P�*-l

pT _ D-.Q6FJ

uv ∈ {1J −1JpJ −p} T)O245Qgh?

p > 0:B5QI

d > 0JQ2�0`lm*-,/,4*+E�6@lUP3*18¦x ] yH0uD1:F0

u + v:B5QI

uv:+kFP3?+?S2/56e24kF5 T _ D-246`,4?B:+I76�0h*S0uD7?'lm*-,r,U*+E'2/5QktdQ*16A6u2U>-2/,r2r0324?B6F�

• u = v = 1T _ D7?B5KJ}>B<bx ] yuJ

d = 2pT�)O.f>16u032�03.103245Qk�0uD7?B6A?S|-:O,/.f?O6�245-03*cx{L�yuJES?t�f5QIs0uD1:F0

x = 2p+ 1Jy = 2p− 1

T• u = 1

Jv = p

T _ D-246�<124?O,4I76d = p+ 1

:B5QIx = p+ 2

Jy = p(p+ 1) − 1

T• u = 1

Jv = −p T _ D-2/6�<-2U?F,UI16 d = p− 1

:+5QIx = p

Jy = −p(p− 1) − 1

T• u = p

Jv = 1

T _ D-246�<124?O,4I76d = p+ 1

:B5QIx = p(p+ 1) + 1

Jy = p

T• u = −p J v = 1

T _ D-2/6�<-2U?F,UI16d = p− 1

:+5QIx = 1 − p(p− 1)

Jy = p− 2

T

Page 31: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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ab?Sgh*15Qgh,/.fI7?t0uD1:F0H0uD7?�6A?F0`*-lC6(*-,r.1032U*1576`lm*1P(x, y)

246{(1,−1)

J(2p+ 1, 2p− 1)

J(p+ 2, p(p+ 1) − 1)

J(p,−p(p− 1) − 1)

J(p(p+ 1) + 1, p)

J(1 − p(p− 1), p− 2)} T

_ D7?Sk=2 |1?B5wd7P3*7>1,4?B8 2/6�0uD7?�6edf?Bgh24:O,Cgh:B6(?�ESD7?O5p = 1997

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p(x+y) = xyJB6(?B?�d7P3*7>1,4?B8 [ *-l�0uD7?

31� � )Bd7:+512/6(D��o:F0uD7?B8w:O0u�2Ug3:B,QV`, <78wd72/:�Isk=2 |1?B5�2/5S0uD7? 2���� �C�F�c2/5S0uD7?S�o:=<wL �����'51.Q8;>Q?OP 3L �������+L ������:B5QIpQP3*7>1,4?B8 �K� M�� ] L+L 2/5 0uD7?dPK*18;:B5124:B5 Gz:O�e?=03: �o:F0h?O8;:F032Ug3: X�P ���-MBN�N�L1JdQd TCM ��� �-M �$�1T

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6e:O032/6elr<-245Qk�0uD7?S?B�1.Q:F032U*15(x− 16)p(2x) = 16(x− 1)p(x)

lm*1P�:O,/,xT�$����AB� � ��� )F������� 2�� � �$�Q� � �-��"{���A"��� ��� � �C��� ,+� ��� �h��� / "�� +C� � ���1� ��" � ��� �

� � �-���$��,���� ���u��� � �Q�7�F�(� � " � " �$�� " ��$ �7�m�u�F� ��K� � 0�2�� "��=�h�e� � ��� � � �=����" �$� ��K����" �F� �,���� ���u��� � ��� ( � � �+��� � %�? % C � � ��� C " � U��"������ �+��"{�F�����$"�AB�F��� "�� ��� C �1���=����� � � $��!%Cs� ���=��� +C� �������1�#"���c��� ���-��"���B" %p1*16(*1576

p(x) = anxn + an−1x

n−1 + · · · + a1x+ a0J

:F|1?Bgan 6= 0

T S`:B576(x− 16)p(2x) = 16(x− 1)p(x)

J�,{�"�?BkF:B,r2r0��?tI7?B6`gu*7?Fv�gh24?B5-0�6IQ?O6H03?BP38�?O6`IQ?�d7,r.Q6`D1:O.10�I7?+kFP��?tIQ*1575Q?2nan = 16an

T-VW5�?O5sI��?BI1.72�0�I7*15Qg��-.f?n = 4

T S�� *�.p(x) = a4x

4 + a3x3 + a2x

2 + a1x+ a0T

VW5;:(x− 16)p(2x) = 16a4x

5 + (8a3 − 256a4)x4 + (4a2 − 128a3)x

3

+(2a1 − 64a2)x2 + (a0 − 32a1)x− 16a0?F0

16(x − 1)p(x) = 16a4x5 + (16a3 − 16a2)x

4 + (16a2 − 16a3)x3

+(16a1 − 16a2)x2 + (16a0 − 16a1)x− 16a0

TVW5�*7>�032U?O5-0�d7:+P�24IQ?O5-032/�fgh:F032U*15wIQ?O6�gh*7?Ov�gh2U?O5-036

a3 = 30a4Ja2 = 280a4

Ja1 = −960a4

Ja0 = 1024a4

JIK� * �.

p(x) = a4(x4 − 30x3 + 280x2 − 960x + 1024)

= a4(x− 2)(x− 4)(x− 8)(x− 16)T

� T �B?F0aJbJc>7?Y5Q*15Q�e5Q?BkF:O032 |1?YP3?B:O,751.Q8�>7?BP�6�6e.fgeD�0uD1:F0

a+ b+ c ≥ abcTpQP3*B|1?Y0uD1:O0

a2 + b2 + c2 ≥ abcT

�$����AB� �w��� / "�� +C� � ���7�#��" � �{����� � �1��� ��,+� ��� �h��� "{�F�h�u� � ��� �$���=�{� "#� � ��K����" �F� �,���� ���u��� / �+�h����� � % ���#�+� "#� � � ��" AB�=����"�� � �� ���7�=�(� ��� ( �-� ��K����$� ���$��A ��� � / �B� ���$� ����� " � � "U�u����� � �}�u�+���u��� ?S��" � ��� )���+� �1��� �-� " �z�=�(� � � �F��� " �$��z�F�h� ��� � �+$��� � � ( % ������� ��� ��������� ���� �� +C���$� � �}�e�B� �h��� � ��� ( � � �+��� � %�? %.C � � ���C " � m��" ����� �+��"{�F� ���$"�AB�F��� "�� ���.C �1���=����� � � $��!%3Cs����"�AB�'�1��" �9�F�(� � � �������7�#" �� %

Page 33: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

N��

�$*1P3?�kO?O5Q?BP�:O,/, <KJ�,4?F0n ≥ 2

>7?':B5s245-03?+kO?OP�:B5QI1 ≤ p ≤ n

T9ab?'gh,4:O248 0uD1:F02/la1, a2, . . . , an

:BP3?t5Q*15Q�u5Q?+kF:F032�|1?YP3?B:O,751.Q8�>7?BP�6`6u.fgeD�0uD1:O0 ∑ai ≥ ∏

aiJB0uD7?B5

api ≥ n(p−1)/(n−1)∏

aiT

x��'?BP3?�:B5QI;>Q?F,U*+E�J-:B,r,f6e.Q8w6W:+5QIwd7P3*7I1.fg30�6@:+P3?�?eÔO0h?O5QIQ?BI;*B|1?OPi = 1

J2J. . .JnT y

f� � �� �T9ab?Sgh*1576e24IQ?OPH0�ES*�gh:B6(?O6O�B � ÝFË'ÏCT ∏

ai ≤ nn/(n−1) T¡-P3*18�0uD7? \ �$�eGW� Ú{5Q?B�1.Q:O,/2�0�<CJ�ES?�D1:F|1?∑

api ≥ n(∏

api

)1/n

= n(∏

ai

)(p−n)/n∏

ai

≥ n(p−1)/(n−1)∏

aiJ

>Q?Bgh:O.Q6(?p ≤ n

TB � ÝFËSÜQT ∏

ai > nn/(n−1) T'@6e2/5Qkt0uD7?;p1*+ES?BPh�=�$?O:+5cÚ{5Q?+�-.Q:B,r2r0{<�:B5QIw0uD7?�gh*15QI12�032U*15 ∑

ai ≥∏aiJ-ES?*7>-0�:B2/5

(1

n

api

)1/p

≥ 1

n

ai ≥ 1

n

aiJ

ESD-2UgeDs248wd7,r2U?O6∑

api ≥ n1−p(∏

ai

)p

= n1−p(∏

ai

)p−1∏

ai

≥ n(p−1)/(n−1)∏

aiT

_ D-2/6�gh*18;d1,U?=0h?O6`0uD7?'dQP3*7*-l}*-lQ0uD7?Sgh,4:O248~T_ :+j+2/5Qk

n = 3:B5QI

p = 2JB:+5QI'P3?O57:+8�245Qk

a1Ja2Ja3>B<aJbJcJ�ES?@6A?+?`0uD1:F0.Q5QIQ?OPz0uD7?'gh*15QI12�032U*1576`k=2�|1?O5s2/5S0uD7?tdQP3*1dQ*16A:O,�JB0uD7?BP3?'D7*-,4I76H0uD7?'>Q?=0�0h?OPz245Q?B�1.Q:O,/2�0�<

a2 + b2 + c2 ≥√

3 abcT _ D-246`2/5Q?+�-.Q:B,r2r0{<�246W6h0hP3*15QkO?BPz0uD1:+5s0uD7?S*15Q?'d7P3*1df*16A?+I}T

Cs� ��� ��� ��" AB� ���#�+� "#� � � �������1�#"������� �s�u�F� �B� ��%ab?�5Q?B?+Iw*151,�<wgu*1576u2UI7?BPz0�ES*wg3:+6A?B6OTx{M�y�Úml

aJbJc:BP3?�:O,/, ≥ 1

J�0uD7?B5;gh,U?O:+P�,�<a2 + b2 + c2 ≥ a+ b+ c ≥ abc

Tx{L�y�Úml}:F0�,4?B:B6u0�*15Q?S*-l

aJbJc ≤ 1

x46A:=<c ≤ 1

yuJ�0uD7?O5a2 + b2 ≥ ab ≥ abc

T2����;�c���K�3T _ D7?b6e:+8;?cdQP3*7>-,U?O8 E'2�0uD7*-.10'0uD7?bgh*15QI12�032U*15�0uD1:F0

aJbJc>Q?5Q*15Q�e5Q?BkF:O032 |1?Y246`k=2�|1?O5�:B6�:'d7P3*7>1,4?B8¼E'2�0uD7*-.10H6(*-,r.1032U*15S245 StTQ¡-*18w2/5KJ1)QT1GH?B5QjB245KJÚuTCÚU0h?O5Q>Q?OP3k�J / �1� +K�=� �7�#" ����� 2�"4� ���{���=J \ 8�?OP=TK�o:F0uD}TQ)+*7gFTUJQMBN�N+�7J1d9TfM �-�7T

ab?�5Q*+E�gh*15Qgh,/.fI7?S0uD7?�d7P3*7*-l9>B<�:B,r,U*+E'2/5Qk�5Q?+kF:F032�|1?�51.Q8;>Q?OP�6BTtÚml &U.Q6h0�*15Q?*-laJbJc2/6@5Q?BkF:O032 |1?S*1PH2/lK:O,/,70uD1P3?B?�:+P3?�5Q?BkF:O032 |1?1J�0uD7?O5

abc ≤ 0J-245�ESD-24geDwgh:B6(?0uD7?wP3?B6u.7,r0�2/6t2/8;8;?+I-24:F0h?7T _ D-.Q6=JCES?�8;:=<b:+6e6e.Q8;?s0uD1:F0Y*151,�<�0�ES*b*-l90uD7?B8cJ 6e:F<

b:+5QI

cJC:BP3?s5Q?BkF:O032 |1?QT _ D7?B5KJf,4?F0�032/5Qk

x = −b :+5QI y = −c J7ES?sD1:=|1?S03*�6(D7*+E0uD1:O0@ESD7?O5a ≥ x + y + axy

JCES?;gh:B5igu*15Qgh,r.fIQ?s0uD1:O0a2 + x2 + y2 ≥ axy

T

Page 34: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

N��

_ D7?�:B6A6u.Q8�?BIi2/5Q?+�-.Q:B,r2r0{<i248wd7,r2U?O6'0uD1:O0xy ≤ 1

T$Úmla ≥ 1

J90uD7?B5a2 ≥ axy

JESD-2UgeD�248wd7,r2U?O6S0uD7?~IQ?O6e2/P3?+I$P3?O6e.7,�0=T Úmla ≤ 1

J�0uD7?B5x ≤ 1

:B5QIy ≤ 1

T Ú{50uD-246Sgh:B6(?a > x + y ≥ xy

Jz6u245Qgh?�0uD7?;,4:F0�0h?OP�245Q?B�1.Q:O,/2�0�<�g3:+5�>7?�P3?�E�P32�0�0h?O5�:B61 ≥ (1 −x)(1 − y)

T _ D-.Q6=JBES?Y6u032r,/,7D1:F|1?a2 ≥ axy

JBE'2�0uD�0uD7?t6e:+8;?tgu*15Qgh,r.Q6e24*15:+6@>Q?Flm*1P3?QT� T �B?=0

S>7?�0uD7?W6A?F0}*-l�:O,/,+*7I7IY245-03?+kO?OP�6zkFP3?B:F0h?OPf0uD1:+5'*15Q?QTH¡-*1P}?O:�geD

x ∈ SJIQ?O5Q*�0h?S>B<

δ(x)0uD7?t.Q512U�-.f?�2/5-0h?BkO?BP�6A:F03246ulr<12/5Qk�0uD7?'2/5Q?+�-.Q:B,r2r0{<

2δ(x) < x < 2δ(x)+1 T¡1*1P

aJb ∈ S

J1IQ?F�f5Q?a ∗ b = 2δ(a)−1(b− 3) + a

Th¡-*1PH?uÔB:+8wd7,4?1JO0h*Sgh:O,Ugh.7,/:O03?

5∗75Q*�03?�0uD1:F0

22 < 5 < 23J+6(*t0uD1:O0

δ(5) = 2J�:B5QID7?B5Qgh?1J

5 ∗ 7 = 22−1(7 − 3) + 5 = 13T \ ,46A*

22 < 7 < 23 JQ6(*�0uD1:F0 δ(7) = 2:+5QI7 ∗ 5 = 22−1(5 − 3) + 7 = 11

�-TpQP3*B|1?Y0uD1:O0�2rlaJbJc ∈ S

J�0uD7?O5xU:=ya ∗ b ∈ S

:B5QIxm>+y

(a ∗ b) ∗ c = a ∗ (b ∗ c) T�$����AB� �w��� / "�� +C� � ���7�#��" � �{����� � �1��� ��,+� ��� �h��� "{�F�h�u� � ��� �$���=�{� "#� � ��K����" �F� �,���� ���u��� $��A ��� � / �+���!��� �$��� "#� � "4�e����� � �}�e�B� �h��� � ��� ( � � �B� � ��%�? % C � � ���C " � m��" ����� �+��"{�F� ���$"�AB�F��� "�� ���.C �1���=����� � � $��!%3Cs����"�AB� C � � � � � �������7�#" �� %

xU:=yC)O2/5Qgu?b−32469?(|1?B5KJ

a∗b 2/6zgh,4?B:BP3, <'*7I7I T \ ,46A*CJF6u245Qgh? 2δ(a)−1(b−3) ≥ 0JES?SD1:=|1?

a ∗ b ≥ a > 1T��'?B5Qgh?1J

a ∗ b ∈ ST

xm>+y`XY*�03?'�QP�6u0H0uD1:F0(a ∗ b) ∗ c = 2δ(a∗b)−1(c− 3) + (a ∗ b)

= 2δ(a∗b)−1(c− 3) + 2δ(a)−1(b− 3) + ax{M�y

:+5QIa ∗ (b ∗ c) = a ∗ (2δ(b)−1(c− 3) + b)

= 2δ(a)+δ(b)−2(c− 3) + 2δ(a)−1(b− 3) + aT x{L�y

)1<;xmM=yz:B5QI;x{L�y92r0�gh,U?O:+P�,�<�6u.7v�gh?B6�0h*�6AD7*+E�0uD1:F0δ(a ∗ b) = δ(a) + δ(b) − 1

T x ] y)1<;I7?O�Q512r0324*15KJ

2δ(a) < a < 2δ(a)+1 :+5QI 2δ(b) < b < 2δ(b)+1 T'@6e2/5Qk�0uD7?'2/5Q?+�-.Q:B,r2r0324?B6

a < 2δ(a)+1 :B5QI b < 2δ(b)+1 JBES?SD1:=|1?2δ(a)−1(b− 3) + a < 2δ(a)−1(b− 3) + 2δ(a)+1

= 2δ(a)−1(b− 3 + 4)

= 2δ(a)−1(b+ 1) ≤ 2δ(a)−1 · 2δ(b)+1

Page 35: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

N+N

0uD1:O0�2/6FJa ∗ b < 2δ(a)+δ(b)

T x [ y)O248�2/,/:+P�,�<CJ-.Q6u245Qk�0uD7?'2/5Q?+�-.Q:B,r2r0324?B6

2δ(a) < a:B5QI

2δ(b) < bJ�ES?�D1:=|1?

2δ(a)−1(b− 3) + a > 2δ(a)−1(b− 3) + 2δ(a)

= 2δ(a)−1(b− 3 + 2)

= 2δ(a)−1(b− 1) ≥ 2δ(a)−1 · 2δ(b) 0uD1:O0�2/6FJ

a ∗ b > 2δ(a)+δ(b)−1 T x{��y¡-P3*18 x [ yz:+5QI;xm�=y}ES?�D1:=|1?

2δ(a)+δ(b)−1 < a ∗ b < 2δ(a)+δ(b)T _ D7?OP3?Olm*1P3?-J

δ(a ∗ b) = δ(a) + δ(b) − 1J

ESD-2UgeDs246@x ] yeT _ D-246@gu*18wd7,4?F03?B6�0uD7?'dQP3*7*-l3T� T9G92�|1?O5t:WdQ*16e2�032�|1?�245-03?+kO?OP

nJBI7?B5Q*�03?W>B<

σ(n)0uD7?`6u.Q8º*-l�:O,/,=0uD7?`df*16u2r032 |1?245-03?+kO?OP�6WESD-24geD�I12 |124IQ?

nT%h¡-*1PW?uÔB:+8wd7,4?1J

σ(3) = 1 + 3 = 4Jσ(6) = 1 + 2 +

3 + 6 = 12Jσ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28

T �ab?�6A:=<~0uD1:O0

n246S:�>-.Q5QI7:B5-0�2rl

σ(n) > 2nTÃx _ D-.Q6=J�lm*1P�?uÔB:+8wd7,4?1J

122/6:�>-.Q5QI7:B5-0�yAT �B?=0

aJb>Q?�df*16u2r032 |1?�2/5-0h?BkO?BP�6�:+5QI�6e.Qd7df*16A?;0uD1:O0

a246S:�>-.Q5QI7:B5-0=TpQP3*B|1?Y0uD1:O0

ab2/6W:�>-.Q5QI7:B5-0=T

�$����AB� �w��� / "�� +C� � ���7�#��" � �{����� � �1��� ��,+� ��� �h��� "{�F�h�u� � ��� �$���=�{� "#� � ��K����" �F� �,���� ���u��� $��A ��� � / �+���!��� �$��� "#� � "4�e����� � �}�e�B� �h��� � ��� ( � � �B� � ��%�? % C � � ���C " � m��" ����� �+��"{�F� ���$"�AB�F��� "�� ���.C �1���=����� � � $��!%3Cs����"�AB� / �B� ���$� ����� " � ���������1�#"����%

�B?F0d1Jd2J. . .Jdk>7?�:B,r,90uD7?�df*16u2r032 |1?;2/5-0h?BkO?BP�6tESD-24geD�I-2�|-2UI7?

aT _ D7?B5

d1 + d2 + · · · + dk > 2aT \ ,46A*CJ

d1bJd2bJ. . .Jdkb:+P3?�:O,/,}I12�� ?OP3?B5-0Wdf*16u2r032 |1?245-03?+kO?OP�6`0uD1:O0`I-2�|-2UI7?

abT _ D-.Q6FJ

σ(ab) ≥ d1b+ d2b+ · · · + dkb > 2abJ

ESD7?B5Qgh?ab2/6@:+>1.Q5QI1:+5-0=T

� TABCD

2/6�:c�-.Q:�I1P32r,4:F0h?OP�:B,9ESD-24geD~2/6�gh2/P3gh.Q8;6AguP324>Q?BIi:�>7*-.10':�gh24P3gh,4?Γx/0uD1:F0H246=J-?B:+geDs6u2UI7?'*-l70uD7?'�-.Q:�I1P32r,4:F0h?OP�:B,72/6�0�:+5QkO?O5-0z0h*

ΓyAT�Úml

∠A = ∠B = 120◦ J∠D = 90◦ :B5QI BC D1:B6W,4?B5Qk�0uD 1

J-�f5QICJ�E'2�0uD�dQP3*7*-l�J�0uD7?t,U?O5Qk�0uDw*-lADT

�$����AB� �w��� / " � +K��� ���1� ��" � ��� ��� � �1��� ��,���� ���u��� / �+�h�����'� % ���#�+� "#� � ���$"�AB�F���� "�� � �� 1 ���7�F�(�#��� ( �-� ��K����$� ��� � ��� $� A�� � � / �+���!��� �$��� "#� � "4�e����� � �}�u�+���u�%Cs� ��"�AB��� +K� �������7�#" ��b�������1� ��" � ���%

Page 36: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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D K C

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�B?F0O>7?�0uD7?'gu?O5-0hP3?�*-l

ΓJ-:B5QIs,4?F0

IJJJKJ-:B5QI

L>7?�0uD7?tdf*-2/5-036`:F09ESD-24geD

ABJBCJCDJO:+5QI

DA03*-.fgeD

ΓJBP3?B6edf?Bg3032�|1?F,�<}T _ P�24:B5Qk=,U?

IBJ24692/6(*16Agu?F,U?O6}E'2�0uD

∠B = 120◦ T _ D7?BP3?Flm*1P3?1J∠BIJ = ∠BJI = 30◦ ESD7?B5Qgh?1J ∠OIJ = 60◦ T�ÚU0lm*-,/,4*+E�6W0uD1:F0 4IOJ 2/6�?+�-.72/,/:O03?BP�:O,K:+5QICJCgu*1576A?+�-.f?B5-03, <KJ IJ = OI = OJ = R

J0uD7?�P�:�I-2/.Q6@*-l

ΓTH)O245Qgh? √

3

2= cos 30◦ =

IJ/2

BJ

J+ES?�kO?F0BJ =

R√3

TXY*+E�J9*7>76A?BP{|-245Qk�0uD1:F0

OKDL2/6':�6A�1.Q:BP3?;:B5QIc0uD1:O0 4IOJ :+5QI 4IOL:+P3?�?+�-.72/,/:O03?BP�:O,z0hP32/:+5Qk=,4?B6=J}ES?�*7>-0�:B2/5∠KOJ = 150◦ T _ D7?B5

∠OCJ = 15◦ ESD7?B5Qgh?1J2 −

√3 = tan 15◦ =

OJ

CJ

T _ D-246�2/8;d1,/24?B6�0uD1:F0CJ = R(2 +

√3)T _ D7?

P3?O,/:O0324*151 = BC = BJ + CJ

5Q*+E¼<124?O,4I76R =

√3

4 + 2√

3

T('@6u245QkDL = R

:+5QIAL = BJ =

R√3

J�ES?�gh:B5�gh*18;d1.10h?AD = AL+DL =

R√3

+RJ�ESD-24geD

P3?B:+I12r,�<wk=2�|1?O6AD =

√3 − 1

2

T� T �B?=0

A>7?�:w6u.f>76A?F0�*-l {0

J1J2J3J. . .J1997} gh*15-03:O24512/5Qks8�*1P3?S0uD1:+5

1000?F,U?O8�?O5-036OTWpQP3*O|1?�0uD1:O0�?O2�0uD7?BP

Agu*15-0�:B2/576`:'dQ*+ES?BPH*-l

2xr0uD1:O0z246=J-:t51.Q8�>7?BP*-l�0uD7?~lm*1P38

2kE'2�0uD

k:�5Q*15Q�e5Q?BkF:O032 |1?~245-03?+kO?OP�yS*1PS0uD7?OP3?o?uÔ+2/6u0�0�ES*�I12/6u032/5Qg30?O,4?B8;?B5-0�6

aJb ∈ A

6e.fgeD�0uD1:F0a+ b

2/6W:SdQ*+ES?BP�*-l2T

�$����AB� �����! "{�F�h�u� � ��� �$���=�{� "#� � ��K����" �F� ��,���� ���u��� � ��� ( � � �B� � ��%�? % C � � ���C " � m��" ����� �+��"{�F� ���$"�AB�F��� "�� ���.C �1���=����� � � $��!%3Cs����"�AB� C � � � � � �������7�#" �� %

�B?F0S = {0

J1J2J. . .J1997} T�ab?�d7P3*B|1?�0uD7?Slm*-,/,4*+E'245Qk�6h0hP3*15QkO?BPWP3?O6e.7,�0��2/l

A2/6S:c6u.f>76A?F0'*-l

Sgh*15-03:O24512/5Qk�8;*1P3?;0uD1:B5

997?F,U?O8�?O5-036=J90uD7?O5

Agh*15-03:O24576:wdQ*+ES?BP�*-l

2J}*1P

Agh*15-03:O24576�0�ES*cI-246h03245Qg30�?F,U?O8�?O5-036

aJb6e.fgeD;0uD1:F0

a + b246Y:df*+ES?OP`*-l

2T�¡�.QP�0uD7?BP38�*1P3?-J-0uD7?S>7*-.Q5QI

997246�0uD7?�>Q?O6u0�dQ*16A6u2U>-,U?7T

)O.QdQdQ*16(?A ⊆ S

IQ*7?O6 ���Q��D1:=|1?�0uD7?wI7?B6AguP324>Q?BIcdQP3*1dQ?BP�0�<}TSab?�E'2/,r, 6(D7*+E0uD1:O0 |A| ≤ 997T

XY*�03?@0uD1:O0210 = 1024 < 1997 < 2048 = 211

T ab?�:BPhP�:B5QkO?�0uD7?�51.Q8�>7?BP�6

Page 37: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M���M

245S − {0

J1J2J8J32J1024} 2/5-0h*�:B5w:+P3P�:F<�:+6`lm*-,r,U*+E�6F�3 13 14 50 51 19974 12 15 49 52 19965 11 16 48 53 19956 10 17 47 54 19947 9 18 46 55 1993

· · · ·· · · ·

31 33 · ·· ·· ·

1023 1025

�B?F0CiI7?B5Q*�03?�0uD7?

i ��� gh*-,/.Q8w5�*-lz0uD7?;:+P3P�:F<ix i = 1J2J. . .J6yeTiX�*�0h?�0uD1:F0Y2/5

C1 ∪C2JO0uD7?@0�ES*S51.Q8;>Q?OP�6�245'0uD7?Y6e:+8;?�P3*+Eq:�I7IS.QdS03*

16 245 C3 ∪C4JO0uD7?�0�ES*51.Q8�>7?BP�6W2/5s0uD7?'6e:+8;?'P3*+Eº:�I7Is.Qd�03*

64 :+5QIs245 C5 ∪C6J+0uD7?Y0�ES*s51.Q8�>7?BP�6W2/50uD7?s6e:+8;?�P3*+E�:+IQI;.Qd�0h*

2048Tt)O245Qgh?

16J64JC:B5QI

2048:+P3?�df*+ES?OP�6Y*-l

2JC:B5QI6e2/5Qgu? |C1| = |C2| = 5

J |C3| = |C4| = 18Jf:+5QI |C5| = |C6| = 973

J7ES?�D1:=|1?|A ∩ (C1 ∪ C2)| ≤ 5

J |A ∩ (C3 ∪ C4)| ≤ 18JC:+5QI |A ∩ (C5 ∪ C6)| ≤ 973

T\ ,/,4*+E'245Qk0 ∈ A

J+ES?t�Q5QIs0uD1:O0 |A| ≤ 1 + 5 + 18 + 973 = 997J7:B6�gh,/:B2/8�?BI T

¡�.QP�0uD7?BP38�*1P3?-JS2/l'ES?$03:+j+?A = {0} ∪ C2 ∪ C4 ∪ C6

J�0uD7?O5°gh,4?B:BP3, <|A| = 997

J�:+5QI�2r0'246s?B:B6�<i0h*ogeD7?+gejc0uD1:F0AI7*7?B6s5Q*�0Sgu*15-0�:B2/5$:bdf*+ES?OPS*-l

2*1P}0�ES*S?F,U?O8�?O5-0369ESD-24geDS:+IQI�.Qd�03*S:�df*+ES?OPz*-l2T _ D-.Q6FJO0uD7?�.QdQdQ?BPz>Q*-.Q5QI�*-lQN�N��246�0uD7?S>7?B6h0�dQ*16A6u2U>-,U?7T

�z�=� �+� f� \ |1?OP{<S245-03?BP3?O6u032/5Qk�dQP3*7>-,U?O8 � IQ?O6(?OP{|12/5Qk��@:t5124gu?Y6A*-,/.10324*15KJOESD-24geDD7*1df?Fl4.7,/, <�246`lUP3*18 � _ D7?()C*7*7j ��xm>B<��-P3I �*16F�-I7?O�Q512r0324*15�yAT� T �B?F0

S>Q?s0uD7?w6(?=0t*-lH:B,r,z57:F03.QP�:B,z51.Q8�>7?BP�6

n6e:O032/6elr<-245QkS0uD7?�lm*-,/,4*+E'245Qkgu*15QI-2r0324*1576O�

xU:=ynD1:+6

1000I124k=2r0�6FJ

xm>+yH:O,/,70uD7?�I124k=2r0�6@*-ln:+P3?�*7IQICJ1:+5QI

xmg�y 0uD7?':�>16(*-,r.10h?�|-:B,r.f?S*-lQ0uD7?SI12�� ?OP3?B5Qgh?S>7?F0�ES?B?B5w:�I &m:�gh?B5-0`I124k=2r0�6@*-ln246

2T

SW?F03?BP38w2/5Q?t0uD7?�51.Q8;>Q?OP`*-l}I12/6u032/5Qg30`?F,U?O8�?O5-036@*-lST

�$����AB� � ��� / " � +K�������1� ��" � ��� � � � �1��� ��,+� ��� �h��� ��� � "{�F�h�u� � ��� �$���=�{� "#� � ��}� ��"#�=���1,+� ��� �h��% Cs����"�AB� � ��� � � �=����" � � � �������7�#" �� %

�B?F0k>Q?�:�dQ*16e2�032�|1?'2/5-0h?BkO?BP�T SW?O5Q*�0h?S>B<

Sk0uD7?�6(?=0W*-l}:O,/,

k�AI-2Uk=2�0`57:O03.QP�:O,51.Q8�>7?BP�6

n6A:F03246ulr<12/5Qktxm>+yz:B5QI;xUg�yAT��B?F0

Uk>Q?Y0uD7?'6A?F0�*-lK?O,4?B8;?B5-0�6@*-l

Sk?B5QI-245Qk245

1*1P

9JVk0uD7?b6A?F0S*-l�?F,U?O8�?O5-036�*-l

Sk?O5QI12/5Qk�2/5

3*1P

7J�:+5QI

Wk0uD7?c6(?=0*-l�?O,4?B8;?B5-0�6S*-l

Sk?B5QI-245Qk�2/5

5T �B?=0

sk = |Sk|Juk = |Uk|

Jvk = |Vk|

J9:B5QIwk = |Wk|

TXY*�03?�0uD1:F0@?B:+geD�?O,4?B8;?B5-0W*-l

Sk+12/6�*7>-0�:B2/5Q?+IwlUP3*18¦:B5�?O,4?B8;?B5-0@*-l

Sk>B<:�I7I12/5Qk;:+5�*7IQIi51.Q8;>Q?OP':O0�0uD7?�P�2UkOD�0�*-l�2�036SIQ?Bgh248w:B,�?eÔ+dQ:B576e24*15KJzlm*-,r,U*+E'2/5Qk;xUg�yAT\ 5�<

n ∈ Uk8;:=<�>Q?S.Q6(?BI;0h*�gu*1576h0hP3.fg30�?uÔB:�g303, <�*15Q?s51.Q8;>Q?OP�*-l

Sk+1>7?+g3:B.Q6A?

Page 38: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M���L

0uD7?�*151,�<�dQ*16A6u2U>-,U?sI-2Uk=2�0`0uD1:F0@8;:=<c>7?s:+IQI7?+I;03*n246

32/l}0uD7?�P324kOD�0h8�*16h0�I-2Uk=2�0�*-l

n2/6

1J9:B5QI

72rlz0uD7?;P324kOD�0h8�*16h0tI-2Uk=2�0t*-l

n2/6

9T _ D7?O5cES?�*7>�03:O245

uk?F,U?O8�?O5-036>Q?F,U*15Qk=2/5Qk�03*

Vk+1Tt)O2/8w2r,4:BP3, <KJf:+5�<

n ∈ Vk8;:=<c>7?�.Q6A?+I;03*�gh*1576u0hP�.fg30�?eÔB:+g303,�<*15Q?�51.Q8�>7?BP'*-l

Uk+1:+5QI�*15Q?;51.Q8�>7?BP'*-l

Wk+1 :+5QIi:B5�< n ∈ Wk8;:=<�>Q?.Q6(?BIs0h*�gu*1576h0hP3.fg30`?uÔB:�g303, <s0�ES*s51.Q8�>7?BP�6�*-l

Vk+1T

)O245Qgh?ng3:+575Q*�0t>7?O,4*15Qk�0h*c8�*1P3?s0uD1:B5�*15Q?;*-l90uD7?�g3:+6A?B6':+>Q*O|1?1JK0uD7?;gu*15Q�6u0hP�.fg3032U*15�,4?B:+I76�03*wI12�� ?OP3?B5-0�51.Q8;>Q?OP�6@2/5

Sk+1T _ D-.Q6=J

uk+1 = vkvk+1 = uk + 2wk

Jwk+1 = vkE'2r0uD

u1 = v1 = 2:+5QI

w1 = 1T�¡-*1P

k ≥ 2J}ES?�D1:=|1?

uk = wk = vk−1JESD-2UgeDo,4?B:+I76�03*

vk+1 = 3vk−1T _ D7?BP3?Flm*1P3?1J

v2k+1 = 3kv1 = 2 × 3k:B5QI

v2k+2 = 3kv2 = 4 × 3klm*1PH:B,r,

k ≥ 0T _ D7?O5

u2k+1 = w2k+1 = 4 × 3k−1:B5QI

u2k = w2k = 2 × 3k−1 lm*1P�:B,r, k ≥ 1T

ab?SI7?+I-.fgu?t0uD1:F0(J-lm*1P�:O,/,k ≥ 1

Js2k = 2 × 3k−1 + 4 × 3k−1 + 2 × 3k−1 = 8 × 3k−1 J

s2k+1 = 4 × 3k−1 + 2 × 3k + 4 × 3k−1 = 14 × 3k−1 TÚ{5;d7:+P�032Ugh.7,/:+PAJ

s1000 = 8 × 3499T

Ò � T �B?=0p>7?^: dQP�248;?^51.Q8�>7?BP~:B5QI

n: 57:O03.QP�:O,�51.Q8�>7?BPAJw:B5QI�,U?=0

T = {1J2J3J. . .Jn} T _ D7?O5 n 2/6'gh:O,/,4?+I p �h� �B�(�#"��#" �� �1�����S2rl90uD7?BP3?;?uÔ+2/6u0�5Q*15Q�?B8wd10{<w6e.f>16(?=036

T1JT2J. . .JTp*-lT6u.fgeD�0uD1:O0

x42 yT = T1 ∪ T2 ∪ · · · ∪ Tp

Jx42r2�y

T1JT2J. . .JTp:BP3?sI-246 &{*-245-0Wxr0uD1:O0`2/6FJ

Ti ∩ Tj2/6W0uD7?�?B8wd10{<�6A?F0�lm*1P`:B,r,

iJjE'2r0uD

i 6= jyuJ1:+5QI

x42r2/2 y 0uD7?'6e.Q8 *-lf0uD7?S?F,U?O8�?O5-036`2/5Ti2/6�0uD7?�6e:+8;?tlm*1P

i = 1J2J. . .JpT

h¡-*1P�?eÔB:B8;d1,U?-J5246

3�ed7:+P�032r0324*157:�>-,U?Ã6u245Qgh?1Jt2/lYES?o03:+j+?

T1 = {1J

4} JT2 = {2

J3} J T3 = {5} JY0uD7?B5 x/2�yhJ�x/2/2 ys:+5QIqx42r2/2 ys:BP3?$6A:F03246u�C?BI T \ ,/6(* 6

2/63�udQ:BP�032�032U*157:+>1,4?�6u245Qgh?1J 2rlzES?�0�:�j+?

T1 = {1J6} J T2 = {2

J5} J T3 = {3

J4} J0uD7?B5;x42 yuJ7x/2/2 yz:B5QI;x42r2/2 yz:BP3?�6e:O032/6e�f?+I}T �

xU:=yH)O.QdQdQ*16(?t0uD1:F0n246p�udQ:BP�032�032U*157:+>1,4?QT`pQP3*B|1?Y0uD1:O0

pI-2�|-2UI7?B6

n*1Pn+ 1

Txm>+y�)O.Qd7df*16A?t0uD1:F0

n246@I-2�|-246u2U>-,U?�>B<

2pTWpQP3*B|1?Y0uD1:O0

n2/6p�ed7:+P�032r0324*157:�>-,U?7T

�$����AB� �w��� / "�� +C� � ���7�#��" � �{����� � �1��� ��,+� ��� �h��� "{�F�h�u� � ��� �$���=�{� "#� � ��K����" �F� �,���� ���u��� � ��� ( � � �+��� � %�? % C � � ��� C " � U��"������ �+��"{�F�����$"�AB�F��� "�� ��� C �1���=����� � � $��!%Cs� ��"�AB� ���7�#��" � �{� � � �������7�#" �� %xU:=y SW?O5Q*�0h?Y>B<

s(A)0uD7?Y6u.Q8�*-l10uD7?Y?O,4?B8;?B5-0�6�*-l10uD7?��Q512r03?�6e.f>16(?=0

A*-l

�T_ D7?O5

n(n+ 1)

2= s(T ) =

p∑

i=1

s(Ti) = pmJ

Page 39: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�� ]

ESD7?BP3?m = s(Ti)

Ji = 1

J2J. . .JpT _ D-.Q6FJ

2pm = n(n + 1)T _ D7?�dQP�248;?51.Q8�>7?BP

pJ+ESD-2UgeDwI-2�|-2UI7?B6�0uD7?'dQP3*7I-.fg30

n(n+ 1)J18�.Q6u0`I-2�|-2UI7?

n*1Pn+ 1

Txm>+y��B?F0

k>Q?Y0uD7?'2/5-0h?BkO?BP�6u.fgeD�0uD1:O0

n = 2pkJ7:B5QIw6A?F0

T1 ={1J2J. . .Jk}

∪{(2p− 1)k + 1

J(2p− 1)k + 2

J. . .J2pk

} JT2 =

{k + 1

Jk + 2

J. . .J2k}

∪{(2p− 2)k + 1

J(2p− 2)k + 2

J. . .J(2p− 1)k

} JTTT

Tj+1 ={jk + 1

J. . .J(j + 1)k

}∪{(2p− j)k + 1

J. . .J (

2p− (j − 1))k} J

TTTTp =

{(p− 1)k + 1

J. . .JpkJpk + 1

Jpk + 2

J. . .J(p+ 1)k

} TXY*+E ES?�D1:=|1?

s(Ti) = k(2pk + 1)lm*1P

i = 1J2J. . .JpJ1:B5QI;gh,U?O:+P�,�<�x42 yz:B5QI;x42r2�y:+P3?':B,/6(*s6A:F03246u�C?BI T�ÚU0�lm*-,/,4*+E�6�0uD1:F0

n246p�udQ:BP�032�032U*157:+>1,4?QT

_ D1:F0Wgu*18wd7,4?F03?B6`0uD-2/[email protected]�>7?BPW*-lf0uD7? 2���� �C�F��T _ D-2/6W246�V`, <78wd72/:�I;)+?O:+6A*15 �6(?O5QI�8�?�V`,�<18;d124:+IoRK*15-0h?O6u0�6�:B6tES?O,r,�:+6t<1*-.QP'5124gu?�6(*-,r.1032U*1576S:B5QIokO?O5Q?BP�:O,/2��u:��032U*1576OT

Page 40: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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_ :�>-,U?SMQTWp7.10h57:+8 )+gu*1P3?O6@:B5QIQPC:+5Qj+6_ D7?�03:+>1,4?tIQ*7?O6�5Q*�0z245Qgh,r.fIQ?�0uD7?�lUP3?+�-.f?B5Qgh24?B6W*-l10uD7?t6Agu*1P3?O6BT _ D-246`*18�246e6e24*15246�245-03?BP3?O6u032/5Qk�>Q?Bgh:O.Q6(?;0uD7?�lUP3?B�1.f?O5Qgh2U?O6s:BP3?�.Q6(?BI~0h*�I7?F03?BP38w2/5Q?�0uD7?�P�:B5Qj+2/5QkF6BT)O.QdQdQ*16(?�:�6h03.fIQ?O5-0(JHp7.10h57:+8"p-:O0(JC6(gh*1P3?+I

20T \ ghgu*1P3I-245Qk�03*;0uD7?S03:+>1,4?1Jzp-:O0(� 6Y2/60uD7?

59 ��� D-24kOD7?B6h0H6(gh*1P3?1J-k=2 |12/5Qk�:tP�:+5Qj�*-l 494 THRC:+5s:B5�<-0uD-2/5QkY8;*1P3?t>Q?Y6A:O2UI7Ó�V`.QPkO*1:B,Q245s0uD-2/6W5Q*�0h?t246�03*�P3?+gh*B|1?OPH0uD7?tlUP3?+�-.f?B5Qgh24?B6WlUP3*18�0uD7?�P�:+5QjB245QkF6OTab?��fP�6h0SgeD7*7*16A?�6A*18�?�5Q*�03:F032U*15�:+5QIo?uÔ�d1,4:O245oD7*+E 0uD7?�P�:+5QjB245QkF6�:+P3?cI7?+�0h?OPh8�245Q?BI T�¡-*1P

i = 1J2J. . .J76x76>7?O2/5Qk�0uD7?w51.Q8�>7?BPY*-l�I-2 �}?BP3?O5-0Y6Agu*1P3?O6hyhJ,U?=0

siIQ?O5Q*�0h?'0uD7?

i ��� 6(gh*1P3?�x�ESD7?BP3? s1 246`0uD7?SD-2UkOD7?O6u0�yhJ7,4?F0 ni >7?'0uD7?S51.Q8;>Q?OPW*-l6u03.fI7?B5-0�6�E'2r0uDs6(gh*1P3?siJ-:+5QIs,U?=0

ci>Q?Y0uD7?'51.Q8;>Q?OP`*-lC6h03.fIQ?O5-036�E'2�0uD�6Agu*1P3?

si*1PD-2UkOD7?OP=T _ D-.Q6FJ

ni246W0uD7?

i��� lUP3?B�1.f?O5Qg�<�:B5QI ci 246W0uD7? i��� gh.Q8w.7,/:O032 |1?�lUP3?B�1.f?O5Qg�<}T¡�2/57:B,r,�<CJ-,4?F0ri>7?t0uD7?'P�:+5Qjs*-lf0uD7?'6(gh*1P3?

siT

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)O245Qgh?w0uD7?;P�:B5Qj+2/5QkF6�lm*1P�0uD7?�03*1db0uD1P3?+?;6(gh*1P3?B6':+P3?1J2Jz:B5QI

3J92�0�246�gh,U?O:+P0uD1:O0H0uD7?B6A?�6Agu*1P3?O6�ES?BP3?�*7>-0�:B2/5Q?+I;>B<w*15Q?�6h03.fIQ?O5-0W?O:�geD}T _ D7?�P�:B5Qj

4.5:+6e6e24kF5Q?+I0h*Y0uD7?@lm*-.QP�0uD'6(gh*1P3?�*-l

802/5QI124gh:F0h?O690uD1:O0K0uD1:O0K0�ES*'6h03.fIQ?O5-036�D1:+I'0uD-2/6H6Agu*1P3?-JB6e2/5Qgu?

4.524690uD7?Y:=|1?BP�:+kO?t*-l-0uD7?Yd1,4:+gu?O6

4:B5QI

5T��'?BP3?�24690uD7?�lm*1P38w.7,/:�lm*1P 0uD7?�P�:B5Qj+2/5QkF6O�

ri =

i−1∑

k=1

nk +ni + 1

2= ci−1 +

ni + 1

2(1 ≤ i ≤ 76)

T x{M�y¡1*1Ps?uÔB:+8wd7,4?1J

r6 = 7.58�?O:+576s0uD1:O0s?eÔB:+g303,�<o0�ES*$6u03.fI7?B5-0�6;6Agu*1P3?BI ���J@6e2/5Qgu?

7.5 = 6 +2 + 1

2

Tab?sE'2/,r,�IQ?OP32 |1?�:�6u248wd7,4?wlm*1Ph8�.7,4:;lm*1P

ni245c0h?OPh8w6�*-lz0uD7?

riTi¡�2/P�6u0Y5Q*�0h?0uD1:O0

ri−ri−1 = 12(ni + ni−1)

T _ D7?BP3?Flm*1P3?1Jni+ni−1 = 2 (ri − ri−1)

Tz)O245Qgh?ni − ni−2 = (ni + ni−1) − (ni−1 + ni−2)

JBES?SD1:=|1?ni − ni−2 = 2 (ri − 2ri−1 + ri−2)

T)O248�2/,/:+P�,�<CJ16u245Qgh?

ni + ni−3 = (ni − ni−2) + (ni−2 + ni−3)J+ES?�D1:F|1?

ni + ni−3 = 2 (ri − 2ri−1 + 2ri−2 − ri−3)T

R}*15-032/51.7245Qkt2/5s0uD-246`lU:B6(D-24*15KJ+ES?�*7>-0�:B2/5ni+(−1)

in1 = 2

(

ri − 2ri−1 + 2ri−2 − · · · + (−1)i2r2 + (−1)

i−1r1

) T¡�2/57:B,r,�<CJ16u245Qgh?

n1 = 2r1 − 1J+ES?�D1:F|1?

ni = 2(

ri − 2ri−1 + 2ri−2 − · · · + (−1)i 2r2 + (−1)i−1 2r1

)

+ (−1)i

= 2ri + 4

i−1∑

k=1

(−1)k ri−k + (−1)i (1 ≤ i ≤ 76)T x{L�y

a�2r0uDs0uD-2/6@lm*1P38w.7,/:sx4:+5QI;:�,/2�0�03,U?SD7?F,4d;lUP3*18 :sgh*18;d1.10h?OP�yhJ-ES?�*7>�03:O245�0uD7?SI7:F03:S2/5_ :+>1,4?�LS*15s0uD7?'5Q?eÔO0�dQ:+kO?QTXY*+E ES?�g3:+5�?O:+6u2/, <w6e:F<w8�*1P3?�:+>Q*-.10�0uD7?�P3?B6u.7,r0�6�*-lf0uD7?�gh*18;dQ?F032�032U*15 TW¡1*1P?eÔB:B8;d1,U?-J1D7?OP3?Y2/6H0uD7?�2451lm*1P38;:F032U*15sP3?F,U?(|-:B5-090h*;p7.10h57:+8¦p-:O0��Q0uD7?BP3?@ES?BP3?

996u03.f�IQ?O5-036fESD7*YD1:+IY:`6(gh*1P3?`*-l

20J=6e2/5Qgu?

n59 = 99 0uD7?BP3?zES?BP3? 444 6h03.fIQ?O5-036fESD7*YD1:+I:�6(gh*1P3?S:+>Q*O|1?20Jf6u245Qgh?

c58 = 444 0uD7?BP3?tES?BP3? 2954 − 543 = 24116u03.fI7?B5-0�6ESD7*;6Agu*1P3?BI�>7?O,4*+E

20Jf6u245Qgh?

c59 = 543*-.10W*-l

29546u03.fI7?B5-0�6�d7:+P�032Ugh2/dQ:F03245Qk'2/50uD7?Sgu*18wdf?=032r0324*15 T

¡�.QP�0uD7?BP38�*1P3?-J 0uD7?;6(gh*1P3?B6�P3?O6e.7,�03245Qks245id7P32��e?;8�*15Q?(<~:BP3?�8;*1P3?;:+d7dQ:BP3?B5-0=T_ D7?�p7.10h57:B8 lm?F,/,4*+E�6;xr0uD7?b0h*1dÃ�1|1?�ySD1:+I�6(gh*1P3?B6101J100J86J@:+5QI

80T _ D7?

��M �����;E'2/575Q?BP�6�xr0uD7?�5Q?eÔO0�0h?O5�yYD1:+Ii6(gh*1P3?B679J77J73J72Jz:+5QI

71T _ D7? �+L�� �E'24575Q?OP�6�x/0uD7?'5Q?eÔO0`?O,4?(|1?B5�yHD1:+Iw6(gh*1P3?B6

70J69J1:+5QI

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_ :+>1,4?�LQT�¡�P3?+�-.f?B5Qgh24?B6@:B5QI;Rf.Q8w.7,/:O032 |1?s¡-P3?B�1.f?O5Qgh2U?O6�*-lf0uD7?�p7.10h57:+8 )+gh*1P3?B6

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M��+N

_ D7?Y245�|1?OP�6e24*15slm*1P38w.7,/:�x{L�yzg3:+5w>7?'IQ?OP32 |1?+I�.Q6e2/5Qk�,/2/5Q?B:BPH:B,4kO?+>1P�:fT9ab?Y6u0�:+P�0E'2r0uD�:�8;:F0hP32 Ô�lm*1Ph8 *-l}?B�1.Q:F032U*15;x{M�ye� −→r = A−→n +

−→1/2J+ESD7?BP3?

−→r =

r1r2TTTr76

J −→n =

n1

n2TTTn76

J −→1/2 =

1/21/2TTT1/2

J

:+5QI

A =

1/2 0 0 · · · 01 1/2 0 · · · 01 1 1/2 · · · 0TTT TTT TTT T T T TTT1 1 1 · · · 1/2

T

)+*-,�|-245Qks0uD7?w,r245Q?O:+PY6�<16u03?B8 lm*1P −→n <-2U?F,UI16 −→n = A−1(−→r − −→

1/2) T )C?+g3:B.Q6A?�*-l

0uD7?�6e2/8;d1,U?tlm*1P38 *-lQ0uD7?�8w:O0hP�2 ÔAJ+ES?�gh:B5�?O:+6u2/, <wg3:B,4gh.7,4:F0h?

A−1 =

2 0 0 0 · · · 0−4 2 0 0 · · · 0

4 −4 2 0 · · · 0−4 4 −4 2 · · · 0TTT TTT TTT TTT T T T TTT−4 4 −4 4 · · · 2

T

_ D7?S?B�1.Q:F032U*15�lm*1P −→n >7?+gh*18�?O6

−→n =

2 0 0 0 · · · 0−4 2 0 0 · · · 0

4 −4 2 0 · · · 0−4 4 −4 2 · · · 0TTT TTT TTT TTT T T T TTT−4 4 −4 4 · · · 2

r1 − 1/2r2 − 1/2TTTr76 − 1/2

J

ESD-2UgeDs246W:�8w:O0hP�2 Ô�lm*1P38 lm*1Pz0uD7?'2/5�|1?BP�6u2U*15�lm*1P38w.7,/:�xmL=yeTZ ?O2�0uD )fP�:B5QI-0 SW*157:O,UI �7T��H?O6u0�:B,{J��eP�TSW?Bd7:+P�0h8�?O5-0W*-lz�o:F0uD7?B8w:O0324gh6=J SW?Bd7:+P�0h8�?O5-0W*-lKR}*18wd7.103?BP`)+gh24?B5Qgh?1JR}*18wd7.103?BP`)+gh24?B5Qgh?1J1:+5QIbpfD+<16e24gh6 �o:O0uD7?O8;:F032Ug36FJ1:B5QIbpfD+<76u2Ug36PK*7gej-D-.QP�6h0 '@512�|1?OP�6e2�0�< ��2/6A6A*-.QP327ab?B6h0h?OPh5;)F0�:O03?SR}*-,r,U?BkO?Z :+576e:+6@RC2�0�<CJC�$V^� [ M�M � )B:B2/5-0�A*16(?OdfDCJC�$V^� [ � �$�[email protected] [email protected]

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r,U?'P�:=<1*15;I1.�gu?OP3gh,U?t24576AguP32�0=TK�$*15-0hP3?OP`�1.f?

1

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?B6h0 �?+�-.72/,/:O0��?OP�:B,�TT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

Ú{5�0hP32/:+5Qk=,4? 4ABC J-,4?F0 wa J wb J wc >7?t0uD7?',U?O5Qk�0uD16�*-lf0uD7?'245-03?BP�2U*1P�:B5Qk=,U?>12/6(?Bg30h*1P�6=J1:B5QIr0uD7?t2457P�:+I12r.Q6BTWpQP3*B|1?Y0uD1:O0

1

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T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T)O.QdQdQ*16(?t0uD1:O0

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JF:+5QICLM

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_ P3*-.-|1?BP[BCP ] : [CAP ] : [ABP ]

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ABCT

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T)O.QdQdQ*16(?'0uD1:O0

LJMJN:BP3?�dQ*-245-0�6�*15

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..

.................................................................................................................................................................................................................................................................................................................................................

A

B C

D

E

O

α.............

............................

β

2β............................

.

..

..

..

..

..

..

..

.

..

.

..

.

.

γ

2γ...................

..

................

α+β

..............................α+γ

...................

.............

�B?F0α = ∠A

Jβ = ∠ABD (= ∠B/3)

JH:B5QIγ = ∠ACE (= ∠C/3)

T_ D7?O5∠BEC = 180◦ − ∠AEC = α+ γ

T)O248�2/,/:+P�,�<CJ

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BOE:B5QI

CODJ+ES?�*7>�03:O245OE

sin β=

BE

sin(α+ β + γ)

:B5QI OD

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CD

sin(α+ β + γ)

T��?O5Qgu?-J

OE

OD=

sinβ

sin γ· BECD

T x{M�y)O248�2/,/:+P�,�<CJ-lUP3*18�0hP�24:B5Qk=,U?O6

BDC:+5QI

BECJ

CD

sin(2β)=

BC

sin(α+ β)

:B5QI BE

sin(2γ)=

BC

sin(α+ γ)

T��?O5Qgu?-J

BE

CD=

sin(2γ)

sin(2β)· sin(α+ β)

sin(α+ γ)

T)O.f>76h032r03.1032/5Qkt2/5;x{M�yuJ+ES?S*7>�03:O245

OE

OD=

cos γ

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sinα+ tan γ cosα

T

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Page 54: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�M �

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Page 55: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�MBN

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T_ D-.Q6=J

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2

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Page 56: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�L �

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4ABCK�D ��;#I

2r ≤ RR � � ;

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C

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R

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cyclic

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*

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c = 0�-*

Page 57: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�L+M

�B?F0F (n, k) =

∞∑

j=0

cj(k + j)n

j!

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:+5QIF (2, k) = ec(k2 + 2ck + c2 + c)

xUgu*18wdQ:BP3?c03*pQP3*7>1,4?B8 L��BL��=yAT`�'?B5Qgh?1JF (1, k) = −F (1,−k − 2c)

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J:+5QIs0uD7?'P3?B6u.7,r0�lm*-,r,U*+E�6OT

_ *�6(D7*+E�0uD1:O0H0uD7?�P3?B6u.7,r0`IQ*7?O6@5Q*�0`D7*-,UI�lm*1Pn > 2

JQgh*1576e24IQ?OPF (n, k)

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A@������O���Q>�����D35�.��)3 ��}�������)���x3−19x+30 = 0

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0 ���Q��ρ

13 =

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2�Q���

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Page 60: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

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cos(3x) = 4 cos3 x − 3 cos x�NG:D �

x = θ/3D:7 6 � 7 ��D ��� D1D � ; ;��HM�� D G 7=I

4u3 −3u = cos θ���86ND � ; C�7A7AD 6

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CD"��Q���

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Page 61: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�L+�

��?O5Qgu?-JSB =

BC ·BEBC +DE

=BC · (AE −AB)

BC +DE

T x{M�y)O248�2/,/:+P�,�<CJ-lUP3*18

SE/SB = DE/BCES?�kO?F0

SE

BE=

SE

SB + SE=

DE

BC +DE

T��?O5Qgu?-J

SE =DE ·BEBC +DE

=DE · (AE −AB)

BC +DE

T x{L�y¡-P3*18 x{M�yz:+5QI;xmL=y}ES?Y0uD7?O5�D1:=|1?

CS =

(BC)2 +(BC)2 · (AE −AB)2

(BC +DE)2

=BC

BC +DE

(BC +DE)2 + (AE −AB)2

:+5QI

DS =

(DE)2 +(DE)2(AE −AB)2

(BC +DE)2

=DE

BC +DE

(BC +DE)2 + (AE −AB)2T

XY*�03?`0uD1:O082+152 = 172 :B5QI 402+92 = 412 THÚmlBES?`0�:�j+? AB = 8

JBC = 15

JAC = 17

JAE = 40

JDE = 9

JK:+5QIAD = 41

JC0uD7?O5BC + DE = 24

:+5QIAE −AB = 32

T`�'?B5Qgh?1J(BC +DE)2 + (AE −AB)2 = 242 + 322 = 402 TÚU0�lm*-,r,U*+E�6�0uD1:F0

CS =15

24(40) = 25

JDS =

9

24(40) = 15

J1:+5QICD = 40

T354 6#7 6#784:98;=< >-? Q � 3 @ � � � 3�����FA3�Q � � @�K � G � � �AD ��� K � 3�K ��� 3

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� � ; � 7=I ���BM8C��AD G:7=I G I D ��; � ;(� D M8C�;=< 6#784 M D G 7=I � �=6 �84 6#7 7A>BD �8G I�;=< >-? 356 ��> � � � ;#C��8I�<�BL � � � � � ��� &�� � �BL ��@ � � ��3 � � ����� � � @���� � % @5L F�� � & �BL�� �!� � �@5L � %�R � � ;#C�; � ;#C�;IAG I�; 7AD ��;#C � 7=I � �AM=C�� D G 7=IB6 7A>BD �8G I�;=< R 35O�7=I�� D � ;#O K D � ; 7=I�; �NG:D � D ��;�6�O �84 4:;#6#D 9(� 4 M ; 7��Q �

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20)� �86 � G:98;#I >-? �=O�;=;#IAJ�R �$��;#6#; � 7=I � �AM=C�� D G 7=IB6 � �8I >B; C�;�� C�;#6#;#I�D:;=< >-?M=6 G I�� D ��;

7 D M$�H4 ;

(AC,BC,AD,DE,AB,AE,CD)�=6 � 784 4 7 � 6

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Page 62: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�LB�

Ù � ÙYÛ T �L����-Lw�7M+M [ � f� �O� � �=� �����@C ����� +K�=��)�� ��� ��� ���}� ����� "#�K�������+� � �����"��B��� " � � �F��� �$�$�5 ���F�m��" �7T¡1*1P

1 ≤ k ≤ NJH,4?F0

n1Jn2J. . .Jnk>Q?c5Q*15Q�u5Q?+kF:F032�|1?�2/5-0h?BkO?BP�6�6e.fgeD0uD1:O0

n1 + n2 + · · · + nk = NT SW?F03?BP38w2/5Q?�0uD7?�8w2/51248�.Q8¦|-:O,/.f?w*-l 0uD7?�6e.Q8

k∑

j=1

(njm

) ESD7?O5 x4:Fym = 2 xU>�y ?

m ≥ 3T

�$�����7�#" ��b��� � ��A "������7� ���=� ���O� ���Q���K� � �F��"#�$"�� ��2���� �{� �-���12 �+�b����" ���1� � � ��TxU:=ysÚmlkI12 |124IQ?O6

NJ�0uD7?~:+5763ES?BPS2/6;*7>B|124*-.Q6O�iX�*�03245Qk�0uD1:F0S0uD7?bl4.Q5Qg3032U*15

f(x) =(x2

)

= 12x(x− 1)

2/6@gu*15�|1?uÔQJ+ES?�D1:F|1?k∑

j=1

f(nj) ≥ kf

1

k

k∑

j=1

nj

= kf(N

k

)

=1

2N(N

k− 1

) T

ÚmlkIQ*7?O6W5Q*�0`I12 |124IQ?O6

NJ16A?F0

N = ak + bJ+ESD7?BP3?

0 ≤ b < kTWX�*+E�J-,4?F0

rj =

{a+ 1

lm*1Pj ≤ b

Ja

lm*1Pj > b

Tab?Sgh,/:B2/8�0uD1:O0

k∑

j=1

f(rj) ≤k∑

j=1

f(nj)

lm*1P�:B5�<nj6e.fgeD�0uD1:F0 n∑

j=1

nj = NT

� ÎeÉ1É6� � _ D-246Y8;:=<b>Q?sdQP3*O|1?+I�.Q6e2/5Qk Z :+P�:B8;:F03:1� 6��o:�&{*1P32��u:O0324*15iÚ{5Q?+�-.Q:B,r2r0{<KJESD-2UgeDw6u0�:O03?B6`0uD1:F0@k=2 |1?B5�0�ES*w6A?+�-.f?B5Qgh?B6=J {ci}ni=1

J {di}ni=1

JQ:BPhP�:B5QkO?+I�xrE'2�0uD7*-.10,U*16e6s*-l@kO?O5Q?BP�:O,/2�0�<+y�245o5Q*15Q�h245QguP3?O:+6u245Qk�*1P3I7?BPAJH0uD7?O5$:c5Q?+gh?B6e6A:BP{<o:B5QIo6u.7v�gh24?B5-0gu*15QI-2r0324*15s0uD1:O0 n∑

i=1

f(ci) ≥n∑

i=1

f(di)lm*1P�:B,r,Cgu*15�|1?uÔSl4.Q5Qg3032U*1576

f246�0uD1:F0

n∑

i=1

ci =n∑

i=1

diJ

:+5QI�lm*1P�:+5�<s245-03?+kO?OPtJ-6e.fgeD�0uD1:O0

1 ≤ t < nJ

t∑

i=1

ci ≥t∑

i=1

diT

ab?�5Q?B?+Io*151, <�6AD7*+E 0uD1:F0 {rj}nj=1

246S8w:�&{*1P�2r�e?BIo>B<�:B5�:+Ph>12�0hP�:+P{<�6A?+�-.f?B5Qgh?c*-l245-03?+kO?OP�6 {nj}nj=1

D1:=|12/5Qk'6u.Q8NT

)O.QdQdQ*16(?'2/576u03?B:+Is0uD1:F0H0uD7?BP3?t246W6A*18�?tx1 ≤ t < n

yz6u.fgeD�0uD1:O0t∑

j=1

nj <

t∑

j=1

rjJ

Page 63: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�L�

:+5QI;gh*1576e24IQ?OPH0uD7?t,U?O:+6h0�6u.fgeDtT _ D7?B5

n1 + · · · + nt < r1 + · · · + rtJ

>1.10n1 + · · · + nt−1 ≥ r1 + · · · + tt−1xmgh*1576e24IQ?OP32/5Qk;>7*�0uDb6e24IQ?O6�*-lz0uD7?w,/:+6h0�2/5Q?+�-.Q:B,r2r0{<c03*~>Q?

0lm*1P

t = 1yAT _ D-.Q6=JKES?D1:F|1?

nt < rt ≤ a+ 1T���?O5Qgu?-J

ni ≤ a ≤ rilm*1P

i ≥ tJ+ESD-2UgeDs2/8;d1,/24?B6

N =k∑

j=1

nj ≤t∑

j=1

nj +k∑

j=t+1

rj <k∑

j=1

rj = NJ

:�gh*15-0hP�:�I-2Ug30324*15 TÚU0�lm*-,/,4*+E�6�0uD1:F0H0uD7?

n� 6@8w:�&{*1P�2r�e?Y0uD7?

r� 6=J1:+5QIwD7?B5Qgh?t0uD1:O0

n∑

j=1

(nj

2

)

≥ b

(a+ 1

2

)

+ (k − b)

(a

2

) T

xm>+y9ab?�8w:F<w?eÔO0h?O5QIs0uD-2/6@:B57:B, <76u246�0h*m ≥ 3

.Q6e2/5Qkt:S6u248wd7,4?Y0hP324gejCT��B?=0

fm(x) =

{1m!x(x− 1) · · · (x−m+ 1)

lm*1Px > m− 1

J0

lm*1P0 ≤ x ≤ m− 1

T_ D7?O5�lm*1PW:B,r,K2/5-0h?BkO?BP�6

nJ1ES?sD1:=|1?

fm(n) =(nm

) T�ab?�gh,4:O248 0uD1:O0fm2/6�gu*15�|1?uÔ*15

[0,∞)T

� ÎeÉ1É6� �@Rf,U?O:+P�,�<CJQ0uD7?�df*-, <75Q*18�24:O,g(x) = x(x − 1) · · · (x − m + 1)

246Y*-lIQ?BkFP3?+?m:+5QI�D1:B6

m6u248wd7,4?'�e?OP3*7?B6@:F0�0uD7?�245-03?+kO?OP�6

0J1J. . .Jm − 1

T��'?B5Qgh?1J2r0�6�I7?BP�2�|-:F032�|1?w8�.Q6u0YD1:F|1?m − 1

�e?BP3*7?O6FJ >B<WPK*-,r,U?-� 6 _ D7?B*1P3?B8cJ9*15Q?�245~?O:�geDb*-l0uD7?'245-03?BP{|-:O,46(r, r + 1)

lm*1P0 ≤ r ≤ m− 2

T�1:+geDq*-lY0uD7?B6A?��e?BP3*7?O6c8�.Q6u0�:O,46A* >7?Ã6u248wd7,4?1J'>7?+g3:B.Q6A?�*�0uD7?OP�E'2/6(?

g′(x)ES*-.7,UI~D1:F|1?�03*7*b8w:+5�<c�e?OP3*7?B6tlm*1P�2r0�6SI7?+kFP3?B?QT;ab?w8w:F<bP3?BdQ?B:F0�0uD7?wdQP3*7gh?B6e6t03*6(?B?'0uD1:F0g′′(x)

8w.Q6h0WD1:F|1?m − 2

6e2/8;d1,U?t�e?OP3*7?B6=JQ?B:+geDw2/5;:+5w245-03?BP{|-:O,}>7?F0�ES?B?B5gu*1576A?+gh.1032 |1?t�e?OP3*7?B6@*-lg′(x)

T��?O5Qgu?-JQ245w0uD7?S245-03?BP{|-:O,

[m − 1,∞)0uD7?OP3?�:+P3?�5Q*��e?OP3*7?B6�*-l

g′′(x)T�)O245Qgh?

g(x)D1:B6SdQ*16e2�032�|1?w,U?O:�I-245Qk;gh*7?Ov�gh2U?O5-0(JH6A*iI7*7?B6

g′′(x)T _ D-.Q6=J

g′′(x)0h?O5QI76'03*

+∞ lm*1P�,4:BP3kO? x >-.10W:B6W2r0�246�gh*15-032451.f*-.Q6=J72�0`8�.Q6u0W>7?Sdf*16u2r032 |1?S*15�0uD-2/6@2/5-0h?OP{|-:B,�T_ D7?OP3?Olm*1P3?-Jg(x)J7:+5QI�0uD-.Q6

fm(x)J12/6@gu*15�|1?uÔs*15s0uD7?t245-03?BP{|-:O,�T

��*+ES?(|1?OP(J76e2/5Qgu?g′(x) 6= 0

*15�0uD-246W245-03?BP{|-:O,K:+5QIg′(x)

246�gh*15-032451.f*-.Q6�:B5QI0h?O5QI76z0h* ∞ lm*1P ,/:+P3kO? x JFES?�6(?B?@0uD1:F0 g′(m−1)2/6�dQ*16e2�032�|1?7Tz)O2/5Qgu?

fm(x)2/6z�e?OP3**15s0uD7?'245-03?BP{|-:O,

[0,m− 1]J-2�0�lm*-,/,4*+E�6�0uD1:O0

fm(x)2/6@gu*15�|1?uÔs?(|1?OP{<�ESD7?OP3?QT

)1<s0uD7?s�o:�&{*1P�2r�u:F032U*15�Ú{5Q?+�-.Q:B,r2r0{<KJ+ES?�D1:F|1?n∑

j=1

(nj

m

)

≥ b

(a+ 1

m

)

+ (k − b)

(a

m

)

Page 64: mayhem-editors@cms.math.ca. · 5 _ · 1 · \ _ $ 7 #" ! %$ $ q } '& ( { *) + -, . "# / & # ab _ b b

M�L �

lm*1P�:O,/,f6u.fgeDk�h03.Qd1,U?O6

njT

2����;�c���K��F�ÚU0�2/6�*7>B|-2U*-.Q6�0uD1:O0t0uD7?k�h03.Qd1,U?

[N, 0, . . . , 0]8;:�&{*1P32��e?B6�:B5�<*�0uD7?BP

k�h03.Qd1,U?s*-l 5Q*15Q�u5Q?+kF:F032�|1?S2/5-0h?BkO?BP�6WE'2�0uD�6u.Q8

NJf6(*s0uD7?�8;:�Ô+2/8w.Q8 |-:O,/.f?246@gh,4?B:BP3, < (N

m

) T354 6#7 6#784 9=;=< >-? %!� � @�@ � F�L @�� �, � � � � K % 7=I�D:78G 6#; K !-C��8I � ;

(���8CED � + � 7=IA4 ?

)���N3 ��3���� L

� R� � � @�� � � � 3 � KB� I G:98;#C 6 G:< � < Q=3 � Q � K F�M ;#I�7=6 35G C�;#6 K 35C �A;#I D G I�� �1@ � Q � 3 @ � � R � � � �8K�@ �8I � � 7 %�� 4:7=6���;#C�<B;#6 K Q=3�K.�!�#3 � & ��@�@H3 ��HR � � 3 &���� � K/� I G:98;#C 6 G:D ? 7�� 354 >B;#CED � K � <-O�7=I�D:7=IAK�3 F �� � �3� � 3 �1R�& ��@ � BK ��7 98;#C K %83�K �!�#3

(���8CED � +��N7=IA4 ?

)� �=I < � � ��L � K % 784 J�Q 7=OO'M8IAG D ?Q 784 4 ; �A; K

��G I�D:;#C�� � 9=;#I K�!��8K �!�#3�R �MA;#C 6#;#I$� 9(� G � I 7AD ;#65D ���AD G I1D ��; � �86#;#6

m = 0�=I <

m = 1D ��;5O G I G O'M8O 6 M8O6��8C�;

k�8I�<

NKC�;#6 ��; � D G:98; 4 ?BR &%7=C�;=7 9=;#C K � ;�6#D �AD:;#6�D ��� D G:D � �=I'>B; � C�7 98;=<=K59 G �D C��=IA6 4 � D G 7=IB6 K5D ��� D�� 7=C �=IB?���7=6 G D G 9=;G I D ; � ;#C

kK �8I�<�=IB?1I 7=I I ; ��� D G 9=;�G I D ; � ;#C 6

mKNKcK �8I�<

d ≤ N/kK�D � ;�O G IAG O'M=O 9 �84:MA; K

MK�7��D � ;�6 M=O

k∑

j=1

(c+ nj

m

) �� ��;#C�;�D � ;

nj� 6��8C�; G I�D:; �A;#C 6 6 �AD G 6 � ? G I �

nj ≥ d�8I�<

n1 + · · · + nk = NK G 6�� G:98;#I�>-?

M =k∑

j=1

(c+ n∗j

m

)

= r(c+ d+ q

m− 1

)

+ k(c+ d+ q

m

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