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The 2012 Sun Life Financial v ] v K v D Z u Z o o v P
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WůĞĂƐĞ ƉƌŝŶƚ ĐůĞĂƌůLJ ĂŶĚ ĐŽŵƉůĞƚĞ Ăůů ŝŶĨŽƌŵĂƟŽŶ ďĞůŽǁ &ĂŝůƵƌĞ ƚŽ ƉƌŝŶƚ ůĞŐŝďůLJ Žƌ ƉƌŽǀŝĚĞ ĐŽŵƉůĞƚĞ ŝŶĨŽƌŵĂƟŽŶ ŵĂLJ ƌĞƐƵůƚ ŝŶ LJŽƵƌ ĞdžĂŵ ďĞŝŶŐ ĚŝƐƋƵĂůŝĮĞĚ dŚŝƐ ĞdžĂŵ ŝƐ ŶŽƚ ĐŽŶƐŝĚĞƌĞĚ ǀĂůŝĚ ƵŶůĞƐƐ ŝƚ ŝƐ ĂĐĐŽŵƉĂŶŝĞĚ ďLJ LJŽƵƌ ƚĞƐƚ ƐƵƉĞƌǀŝƐŽƌƐ ƐŝŐŶĞĚ ĨŽƌŵ First Name: Last Name: ƌĞ LJŽƵ ĐƵƌƌĞŶƚůLJ ƌĞŐŝƐƚĞƌĞĚ ŝŶ ĨƵůůͲƟŵĞ ĂƩĞŶĚĂŶĐĞ Ăƚ ĂŶ ĞůĞŵĞŶƚĂƌLJ ƐĞĐŽŶĚĂƌLJ Žƌ ĠŐĞƉ ƐĐŚŽŽů Žƌ ŚŽŵĞ ƐĐŚŽŽůĞĚ ĂŶĚ ŚĂǀĞ ďĞĞŶ ƐŝŶĐĞ ^ĞƉƚĞŵďĞƌ ϭϱƚŚ ŽĨ ƚŚŝƐ LJĞĂƌ Y N AƌĞ LJŽƵ Ă ĂŶĂĚŝĂŶ ŝƟnjĞŶ Žƌ Ă WĞƌŵĂŶĞŶƚ ZĞƐŝĚĞŶƚ ŽĨ ĂŶĂĚĂ ;ƌĞŐĂƌĚůĞƐƐ ŽĨ ĐƵƌƌĞŶƚ ĂĚĚƌĞƐƐͿ Y N Grade ϴ ϵ ϭϬ ϭϭ ϭϮ ĠŐĞƉ Other: dͲ^Śŝƌƚ ^ŝnjĞ ;zŽƵƚŚͿ y^ ^ D L XL XXL Date of Birth: y y y y m m d d Gender: ;KƉƟŽŶĂůͿ D & E-mail Address: ^ŝŐŶĂƚƵƌĞ ͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺ The 2012 Sun Life Financial ĂŶĂĚŝĂŶ KƉĞŶ DĂƚŚĞŵĂƟĐƐ ŚĂůůĞŶŐĞ INSTRUCTIONS: K EKd KWE d,/^ KK<>d hEd/> /E^dZhd dK K ^K EXAM: dŚĞƌĞ ĂƌĞ ϯ ƉĂƌƚƐ ƚŽ ƚŚĞ KD ƚŽ ďĞ ĐŽŵƉůĞƚĞĚ ŝŶ Ϯ ŚŽƵƌƐ ĂŶĚ ϯϬ ŵŝŶƵƚĞƐ WZd ŽŶƐŝƐƚƐ ŽĨ ϰ ďĂƐŝĐ ƋƵĞƐƟŽŶƐ ǁŽƌƚŚ ϰ ŵĂƌŬƐ ĞĂĐŚ WZd ŽŶƐŝƐƚƐ ŽĨ ϰ ŝŶƚĞƌŵĞĚŝĂƚĞ ƋƵĞƐƟŽŶƐ ǁŽƌƚŚ ϲ ŵĂƌŬƐ ĞĂĐŚ WZd ŽŶƐŝƐƚƐ ŽĨ ϰ ĂĚǀĂŶĐĞĚ ƋƵĞƐƟŽŶƐ ǁŽƌƚŚ ϭϬ ŵĂƌŬƐ ĞĂĐŚ DIAGRAMS ŝĂŐƌĂŵƐ ĂƌĞ ŶŽƚ ĚƌĂǁŶ ƚŽ ƐĐĂůĞ ƚŚĞLJ ĂƌĞ ŝŶƚĞŶĚĞĚ ĂƐ ĂŝĚƐ ŽŶůLJ WORK AND ANSWERS: ůů ƐŽůƵƟŽŶ ǁŽƌŬ ĂŶĚ ĂŶƐǁĞƌƐ ĂƌĞ ƚŽ ďĞ ƉƌĞƐĞŶƚĞĚ ŝŶ ƚŚŝƐ ŬůĞƚ ŝŶ ƚŚĞ ďŽdžĞƐ ƉƌŽǀŝĚĞĚ DĂƌŬƐ ĂƌĞ ĂǁĂƌĚĞĚ ĨŽƌ ĐŽŵƉůĞƚĞŶĞƐƐ ĂŶĚ ĐůĂƌŝƚLJ &Žƌ ƐĞĐƟŽŶƐ ĂŶĚ ŝƚ ŝƐ ŶŽƚ ŶĞĐĞƐƐĂƌLJ ƚŽ ƐŚŽǁ LJŽƵƌ ǁŽƌŬ ŝŶ ŽƌĚĞƌ ƚŽ ƌĞĐĞŝǀĞ ĨƵůů ŵĂƌŬƐ ,ŽǁĞǀĞƌ ŝĨ LJŽƵƌ ĂŶƐǁĞƌ Žƌ ƐŽůƵƟŽŶ ŝƐ ŝŶĐŽƌƌĞĐƚ ĂŶLJ ǁŽƌŬ ƚŚĂƚ LJŽƵ ĚŽ ĂŶĚ ƉƌĞƐĞŶƚ ŝŶ ƚŚŝƐ ŬůĞƚ ǁŝůů ďĞ ĐŽŶƐŝĚĞƌĞĚ ĨŽƌ ƉĂƌƚ ŵĂƌŬƐ &Žƌ ƐĞĐƟŽŶ LJŽƵ ŵƵƐƚ ƐŚŽǁ LJŽƵƌ ǁŽƌŬ ĂŶĚ ƉƌŽǀŝĚĞ ƚŚĞ ĐŽƌƌĞĐƚ ĂŶƐǁĞƌ Žƌ ƐŽůƵƟŽŶ ƚŽ ƌĞĐĞŝǀĞ ĨƵůů ŵĂƌŬƐ /ƚ ŝƐ ĞdžƉĞĐƚĞĚ ƚŚĂƚ Ăůů ĐĂůĐƵůĂƟŽŶƐ ĂŶĚ ĂŶƐǁĞƌƐ ǁŝůů ďĞ ĞdžƉƌĞƐƐĞĚ ĂƐ ĞdžĂĐƚ ŶƵŵďĞƌƐ ƐƵĐŚ ĂƐ ϰʋ Ϯ н яϳ ĞƚĐ ƌĂƚŚĞƌ ƚŚĂŶ ĂƐ ϭϮϱϲϲ ϰϲϰϲ ĞƚĐ dŚĞ ŶĂŵĞƐ ŽĨ Ăůů ĂǁĂƌĚ ǁŝŶŶĞƌƐ ǁŝůů ďĞ ƉƵďůŝƐŚĞĚ ŽŶ ƚŚĞ ĂŶĂĚŝĂŶ DĂƚŚĞŵĂƟĐĂů ^ŽĐŝĞƚLJ ǁĞď ƐŝƚĞ dŚĞ ĐŽŶƚĞŶƚƐ ŽĨ ƚŚĞ KD ϮϬϭϮ ĂŶĚ LJŽƵƌ ĂŶƐǁĞƌƐ ĂŶĚ ƐŽůƵƟŽŶƐ ŵƵƐƚ ŶŽƚ ďĞ ƉƵďůŝĐĂůůLJ ĚŝƐĐƵƐƐĞĚ ŝŶĐůƵĚŝŶŐ ǁĞď ĐŚĂƚƐ ĨŽƌ Ăƚ ůĞĂƐƚ Ϯϰ ŚŽƵƌƐ DŽďŝůĞ ƉŚŽŶĞƐ ĂŶĚ ĐĂůĐƵůĂƚŽƌƐ ĂƌĞ ŶŽƚ ƉĞƌŵŝƩĞĚ ϭ Ϯ A3 A4 A ϭ Ϯ B3 B4 B ϭ Ϯ C3 C4 C ABC &Žƌ ŽĸĐŝĂů ƵƐĞ ŽŶůLJ DĂƌŬĞƌ ŝŶŝƟĂůƐ ĂƚĂ ĞŶƚƌLJ ŝŶŝƟĂůƐ
Transcript of The 2012 Sun Life Financial v ] v K v D Z u Z o o v P
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WůĞĂƐĞƉƌŝŶƚĐůĞĂƌůLJĂŶĚĐŽŵƉůĞƚĞĂůůŝŶĨŽƌŵĂƟŽŶďĞůŽǁ&ĂŝůƵƌĞƚŽƉƌŝŶƚůĞŐŝďůLJŽƌƉƌŽǀŝĚĞĐŽŵƉůĞƚĞŝŶĨŽƌŵĂƟŽŶŵĂLJƌĞƐƵůƚŝŶLJŽƵƌĞdžĂŵďĞŝŶŐĚŝƐƋƵĂůŝĮĞĚdŚŝƐĞdžĂŵŝƐŶŽƚĐŽŶƐŝĚĞƌĞĚǀĂůŝĚƵŶůĞƐƐŝƚŝƐĂĐĐŽŵƉĂŶŝĞĚďLJLJŽƵƌƚĞƐƚƐƵƉĞƌǀŝƐŽƌƐƐŝŐŶĞĚĨŽƌŵ
First Name:
Last Name:
ƌĞLJŽƵĐƵƌƌĞŶƚůLJƌĞŐŝƐƚĞƌĞĚŝŶĨƵůůͲƟŵĞĂƩĞŶĚĂŶĐĞĂƚĂŶĞůĞŵĞŶƚĂƌLJ ƐĞĐŽŶĚĂƌLJŽƌĠŐĞƉƐĐŚŽŽůŽƌŚŽŵĞƐĐŚŽŽůĞĚĂŶĚŚĂǀĞďĞĞŶƐŝŶĐĞ^ĞƉƚĞŵďĞƌϭϱƚŚŽĨƚŚŝƐLJĞĂƌ
Y N AƌĞLJŽƵĂĂŶĂĚŝĂŶŝƟnjĞŶŽƌĂWĞƌŵĂŶĞŶƚZĞƐŝĚĞŶƚŽĨĂŶĂĚĂ;ƌĞŐĂƌĚůĞƐƐŽĨĐƵƌƌĞŶƚĂĚĚƌĞƐƐͿ Y N
Grade ϴϵϭϬϭϭϭϮĠŐĞƉ Other:
dͲ^Śŝƌƚ^ŝnjĞ;zŽƵƚŚͿy^^D
L XL XXL
Date of Birth: y y y y m m d d
Gender: ;KƉƟŽŶĂůͿD&
E-mail Address:
^ŝŐŶĂƚƵƌĞͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺ
The 2012 Sun Life Financial ĂŶĂĚŝĂŶKƉĞŶDĂƚŚĞŵĂƟĐƐŚĂůůĞŶŐĞ
INSTRUCTIONS: KEKdKWEd,/^KK<>dhEd/>/E^dZhddKK^K
EXAM: dŚĞƌĞĂƌĞϯƉĂƌƚƐƚŽƚŚĞKDƚŽďĞĐŽŵƉůĞƚĞĚŝŶϮŚŽƵƌƐĂŶĚϯϬŵŝŶƵƚĞƐWZdŽŶƐŝƐƚƐŽĨϰďĂƐŝĐƋƵĞƐƟŽŶƐǁŽƌƚŚϰŵĂƌŬƐĞĂĐŚ WZdŽŶƐŝƐƚƐŽĨϰŝŶƚĞƌŵĞĚŝĂƚĞƋƵĞƐƟŽŶƐǁŽƌƚŚϲŵĂƌŬƐĞĂĐŚWZdŽŶƐŝƐƚƐŽĨϰĂĚǀĂŶĐĞĚƋƵĞƐƟŽŶƐǁŽƌƚŚϭϬŵĂƌŬƐĞĂĐŚ
DIAGRAMSŝĂŐƌĂŵƐĂƌĞŶŽƚĚƌĂǁŶƚŽƐĐĂůĞƚŚĞLJĂƌĞŝŶƚĞŶĚĞĚĂƐĂŝĚƐŽŶůLJ
WORK AND ANSWERS:ůůƐŽůƵƟŽŶǁŽƌŬĂŶĚĂŶƐǁĞƌƐĂƌĞƚŽďĞƉƌĞƐĞŶƚĞĚŝŶƚŚŝƐŬůĞƚŝŶƚŚĞďŽdžĞƐƉƌŽǀŝĚĞĚDĂƌŬƐĂƌĞĂǁĂƌĚĞĚĨŽƌĐŽŵƉůĞƚĞŶĞƐƐĂŶĚĐůĂƌŝƚLJ&ŽƌƐĞĐƟŽŶƐĂŶĚŝƚŝƐŶŽƚŶĞĐĞƐƐĂƌLJƚŽƐŚŽǁLJŽƵƌǁŽƌŬŝŶŽƌĚĞƌƚŽƌĞĐĞŝǀĞĨƵůůŵĂƌŬƐ,ŽǁĞǀĞƌ ŝĨLJŽƵƌĂŶƐǁĞƌŽƌƐŽůƵƟŽŶŝƐŝŶĐŽƌƌĞĐƚĂŶLJǁŽƌŬƚŚĂƚLJŽƵĚŽĂŶĚƉƌĞƐĞŶƚŝŶƚŚŝƐŬůĞƚǁŝůůďĞĐŽŶƐŝĚĞƌĞĚĨŽƌƉĂƌƚŵĂƌŬƐ&ŽƌƐĞĐƟŽŶLJŽƵŵƵƐƚƐŚŽǁLJŽƵƌǁŽƌŬĂŶĚƉƌŽǀŝĚĞƚŚĞĐŽƌƌĞĐƚĂŶƐǁĞƌŽƌƐŽůƵƟŽŶƚŽƌĞĐĞŝǀĞĨƵůůŵĂƌŬƐ
/ƚŝƐĞdžƉĞĐƚĞĚƚŚĂƚĂůůĐĂůĐƵůĂƟŽŶƐĂŶĚĂŶƐǁĞƌƐǁŝůůďĞĞdžƉƌĞƐƐĞĚĂƐĞdžĂĐƚŶƵŵďĞƌƐƐƵĐŚĂƐϰʋϮняϳĞƚĐƌĂƚŚĞƌƚŚĂŶĂƐϭϮϱϲϲϰϲϰϲĞƚĐdŚĞŶĂŵĞƐŽĨĂůůĂǁĂƌĚǁŝŶŶĞƌƐǁŝůůďĞƉƵďůŝƐŚĞĚŽŶƚŚĞĂŶĂĚŝĂŶDĂƚŚĞŵĂƟĐĂů^ŽĐŝĞƚLJǁĞďƐŝƚĞ
dŚĞĐŽŶƚĞŶƚƐŽĨƚŚĞKDϮϬϭϮĂŶĚLJŽƵƌĂŶƐǁĞƌƐĂŶĚƐŽůƵƟŽŶƐŵƵƐƚŶŽƚďĞƉƵďůŝĐĂůůLJĚŝƐĐƵƐƐĞĚŝŶĐůƵĚŝŶŐǁĞďĐŚĂƚƐĨŽƌĂƚůĞĂƐƚϮϰŚŽƵƌƐ
DŽďŝůĞƉŚŽŶĞƐĂŶĚĐĂůĐƵůĂƚŽƌƐ
ĂƌĞŶŽƚƉĞƌŵŝƩĞĚ
ϭ Ϯ A3 A4 A ϭ Ϯ B3 B4 B ϭ Ϯ C3 C4 C ABC&ŽƌŽĸĐŝĂůƵƐĞŽŶůLJ
DĂƌŬĞƌŝŶŝƟĂůƐ ĂƚĂĞŶƚƌLJŝŶŝƟĂůƐ
Martin Muñoz
Martin Muñoz
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ǁǁǁĐŵƐŵĂƚŚĐĂ ΞϮϬϭϮE/EDd,Dd/>^K/dz
^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϮŽĨϭϲ
WĂƌƚYƵĞƐƟŽŶϭ;ϰŵĂƌŬƐͿ
WĂƌƚYƵĞƐƟŽŶϮ;ϰŵĂƌŬƐͿ
COMC 2012: Final Version 1
COMC 2012 : Final Version
Part A
A1 Determine the positive integer n such that 84 = 4n.
A2 Let x be the average of the following six numbers: 12, 412, 812, 1212, 1612, 2012. Determinethe value of x.
A3 Let ABCDEF be a hexagon all of whose sides are equal in length and all of whose anglesare equal. The area of hexagon ABCDEF is exactly r times the area of triangle ACD.Determine the value of r.
F
A B
C
DE
A4 Twelve different lines are drawn on the coordinate plane so that each line is parallel to exactlytwo other lines. Furthermore, no three lines intersect at a point. Determine the total numberof intersection points among the twelve lines.
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COMC 2012: Final Version 1
COMC 2012 : Final Version
Part A
A1 Determine the positive integer n such that 84 = 4n.
A2 Let x be the average of the following six numbers: 12, 412, 812, 1212, 1612, 2012. Determinethe value of x.
A3 Let ABCDEF be a hexagon all of whose sides are equal in length and all of whose anglesare equal. The area of hexagon ABCDEF is exactly r times the area of triangle ACD.Determine the value of r.
F
A B
C
DE
A4 Twelve different lines are drawn on the coordinate plane so that each line is parallel to exactlytwo other lines. Furthermore, no three lines intersect at a point. Determine the total numberof intersection points among the twelve lines.
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....
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1
zŽƵƌ^ŽůƵƟŽŶ
zŽƵƌ^ŽůƵƟŽŶ
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ǁǁǁĐŵƐŵĂƚŚĐĂ ΞϮϬϭϮE/EDd,Dd/>^K/dz
^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϯŽĨϭϲ ^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϮŽĨϭϲ
WĂƌƚYƵĞƐƟŽŶϯ;ϰŵĂƌŬƐͿ
WĂƌƚYƵĞƐƟŽŶϰ;ϰŵĂƌŬƐͿ
COMC 2012: Final Version 1
COMC 2012 : Final Version
Part A
A1 Determine the positive integer n such that 84 = 4n.
A2 Let x be the average of the following six numbers: 12, 412, 812, 1212, 1612, 2012. Determinethe value of x.
A3 Let ABCDEF be a hexagon all of whose sides are equal in length and all of whose anglesare equal. The area of hexagon ABCDEF is exactly r times the area of triangle ACD.Determine the value of r.
F
A B
C
DE
A4 Twelve different lines are drawn on the coordinate plane so that each line is parallel to exactlytwo other lines. Furthermore, no three lines intersect at a point. Determine the total numberof intersection points among the twelve lines.
. . . . . .. . . . . .. . . . . .
....
....
....
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....
.......
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...
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1
COMC 2012: Final Version 1
COMC 2012 : Final Version
Part A
A1 Determine the positive integer n such that 84 = 4n.
A2 Let x be the average of the following six numbers: 12, 412, 812, 1212, 1612, 2012. Determinethe value of x.
A3 Let ABCDEF be a hexagon all of whose sides are equal in length and all of whose anglesare equal. The area of hexagon ABCDEF is exactly r times the area of triangle ACD.Determine the value of r.
F
A B
C
DE
A4 Twelve different lines are drawn on the coordinate plane so that each line is parallel to exactlytwo other lines. Furthermore, no three lines intersect at a point. Determine the total numberof intersection points among the twelve lines.
. . . . . .. . . . . .. . . . . .
....
....
....
....
....
.......
...
...
...
...
...
. . . . . .. . . . . .
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1
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COMC 2012: Final Version 2
Part B
B1 Alice and Bob run in the clockwise direction around a circular track, each running at aconstant speed. Alice can complete a lap in t seconds, and Bob can complete a lap in 60seconds. They start at diametrically-opposite points.
Alice Bob
When they meet for the first time, Alice has completed exactly 30 laps. Determine all possiblevalues of t.
B2 For each positive integer n, define ϕ(n) to be the number of positive divisors of n. For exam-ple, ϕ(10) = 4, since 10 has 4 positive divisors, namely 1, 2, 5, 10.
Suppose n is a positive integer such that ϕ(2n) = 6. Determine the minimum possible valueof ϕ(6n).
B3 Given the following 4 by 4 square grid of points, determine the number of ways we can labelten different points A,B,C,D,E, F,G,H, I, J such that the lengths of the nine segments
AB,BC,CD,DE,EF, FG,GH,HI, IJ
are in strictly increasing order.
2
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WĂƌƚYƵĞƐƟŽŶϮ;ϲŵĂƌŬƐͿ
COMC 2012: Final Version 2
Part B
B1 Alice and Bob run in the clockwise direction around a circular track, each running at aconstant speed. Alice can complete a lap in t seconds, and Bob can complete a lap in 60seconds. They start at diametrically-opposite points.
Alice Bob
When they meet for the first time, Alice has completed exactly 30 laps. Determine all possiblevalues of t.
B2 For each positive integer n, define ϕ(n) to be the number of positive divisors of n. For exam-ple, ϕ(10) = 4, since 10 has 4 positive divisors, namely 1, 2, 5, 10.
Suppose n is a positive integer such that ϕ(2n) = 6. Determine the minimum possible valueof ϕ(6n).
B3 Given the following 4 by 4 square grid of points, determine the number of ways we can labelten different points A,B,C,D,E, F,G,H, I, J such that the lengths of the nine segments
AB,BC,CD,DE,EF, FG,GH,HI, IJ
are in strictly increasing order.
2
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ǁǁǁĐŵƐŵĂƚŚĐĂ ΞϮϬϭϮE/EDd,Dd/>^K/dz
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WĂƌƚYƵĞƐƟŽŶϯ;ϲŵĂƌŬƐͿ
COMC 2012: Final Version 2
Part B
B1 Alice and Bob run in the clockwise direction around a circular track, each running at aconstant speed. Alice can complete a lap in t seconds, and Bob can complete a lap in 60seconds. They start at diametrically-opposite points.
Alice Bob
When they meet for the first time, Alice has completed exactly 30 laps. Determine all possiblevalues of t.
B2 For each positive integer n, define ϕ(n) to be the number of positive divisors of n. For exam-ple, ϕ(10) = 4, since 10 has 4 positive divisors, namely 1, 2, 5, 10.
Suppose n is a positive integer such that ϕ(2n) = 6. Determine the minimum possible valueof ϕ(6n).
B3 Given the following 4 by 4 square grid of points, determine the number of ways we can labelten different points A,B,C,D,E, F,G,H, I, J such that the lengths of the nine segments
AB,BC,CD,DE,EF, FG,GH,HI, IJ
are in strictly increasing order.
2
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ǁǁǁĐŵƐŵĂƚŚĐĂ ΞϮϬϭϮE/EDd,Dd/>^K/dz
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COMC 2012: Final Version 3
B4 In the following diagram, two lines that meet at a point A are tangent to a circle at pointsB and C. The line parallel to AC passing through B meets the circle again at D. Join thesegments CD and AD. Suppose AB = 49 and CD = 28. The length of AD is a positiveinteger n. Determine n.
B D
CA
3
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ǁǁǁĐŵƐŵĂƚŚĐĂ ΞϮϬϭϮE/EDd,Dd/>^K/dz
^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϴŽĨϭϲ
WĂƌƚYƵĞƐƟŽŶϭ;ϭϬŵĂƌŬƐͿ
COMC 2012: Final Version 4
Part C
C1 Let f(x) = x2 and g(x) = 3x− 8.
(a) (2 marks) Determine the values of f(2) and g(f(2)).
(b) (4 marks) Determine all values of x such that f(g(x)) = g(f(x)).
(c) (4 marks) Let h(x) = 3x− r. Determine all values of r such that f(h(2)) = h(f(2)).
C2 We fill a 3 × 3 grid with 0s and 1s. We score one point for each row, column, and diagonalwhose sum is odd.
0 1 1
1 0 1
1 1 0
0 1 1
1 0 1
1 1 1
For example, the grid on the left has 0 points and the grid on the right has 3 points.
(a) (2 marks) Fill in the following grid so that the grid has exactly 1 point. No additionalwork is required. Many answers are possible. You only need to provide one.
(b) (4 marks) Determine all grids with exactly 8 points.
(c) (4 marks) Let E be the number of grids with an even number of points, and O be thenumber of grids with an odd number of points. Prove that E = O.
4
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WĂƌƚYƵĞƐƟŽŶϮ;ϭϬŵĂƌŬƐͿ
COMC 2012: Final Version 4
Part C
C1 Let f(x) = x2 and g(x) = 3x− 8.
(a) (2 marks) Determine the values of f(2) and g(f(2)).
(b) (4 marks) Determine all values of x such that f(g(x)) = g(f(x)).
(c) (4 marks) Let h(x) = 3x− r. Determine all values of r such that f(h(2)) = h(f(2)).
C2 We fill a 3 × 3 grid with 0s and 1s. We score one point for each row, column, and diagonalwhose sum is odd.
0 1 1
1 0 1
1 1 0
0 1 1
1 0 1
1 1 1
For example, the grid on the left has 0 points and the grid on the right has 3 points.
(a) (2 marks) Fill in the following grid so that the grid has exactly 1 point. No additionalwork is required. Many answers are possible. You only need to provide one.
(b) (4 marks) Determine all grids with exactly 8 points.
(c) (4 marks) Let E be the number of grids with an even number of points, and O be thenumber of grids with an odd number of points. Prove that E = O.
4
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^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϭϭŽĨϭϲ ^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϭϬŽĨϭϲ
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WĂƌƚYƵĞƐƟŽŶϯ;ϭϬŵĂƌŬƐͿCOMC 2012: Final Version 5
C3 Let ABCD be a parallelogram. We draw in the diagonal AC. A circle is drawn inside ∆ABCtangent to all three sides and touches side AC at a point P .
A B
CD
P
(a) (2 marks) Prove that DA+AP = DC + CP .
(b) (4 marks) Draw in the line DP . A circle of radius r1 is drawn inside ∆DAP tangent toall three sides. A circle of radius r2 is drawn inside ∆DCP tangent to all three sides.Prove that
r1r2
=AP
PC.
(c) (4 marks) Suppose DA+DC = 3AC and DA = DP . Let r1, r2 be the two radii definedin (b). Determine the ratio r1/r2.
C4 For any positive integer n, an n-tuple of positive integers (x1, x2, · · · , xn) is said to be super-
squared if it satisfies both of the following properties:
(1) x1 > x2 > x3 > · · · > xn.
(2) The sum x21 + x22 + · · ·+ x2kis a perfect square for each 1 ≤ k ≤ n.
For example, (12, 9, 8) is super-squared, since 12 > 9 > 8, and each of 122, 122 + 92, and122 + 92 + 82 are perfect squares.
(a) (2 marks) Determine all values of t such that (32, t, 9) is super-squared.
(b) (2 marks) Determine a super-squared 4-tuple (x1, x2, x3, x4) with x1 < 200.
(c) (6 marks) Determine whether there exists a super-squared 2012-tuple.
5
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^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϭϯŽĨϭϲ ^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϭϮŽĨϭϲ
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ǁǁǁĐŵƐŵĂƚŚĐĂ ΞϮϬϭϮE/EDd,Dd/>^K/dz
^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϭϰŽĨϭϲ
WĂƌƚYƵĞƐƟŽŶϰ;ϭϬŵĂƌŬƐͿ
COMC 2012: Final Version 5
C3 Let ABCD be a parallelogram. We draw in the diagonal AC. A circle is drawn inside ∆ABCtangent to all three sides and touches side AC at a point P .
A B
CD
P
(a) (2 marks) Prove that DA+AP = DC + CP .
(b) (4 marks) Draw in the line DP . A circle of radius r1 is drawn inside ∆DAP tangent toall three sides. A circle of radius r2 is drawn inside ∆DCP tangent to all three sides.Prove that
r1r2
=AP
PC.
(c) (4 marks) Suppose DA + DC = 3AC and DA = DP . Let r1, r2 be the two radiusesdefined in (b). Determine the ratio r1/r2.
C4 For any positive integer n, an n-tuple of positive integers (x1, x2, · · · , xn) is said to be super-
squared if it satisfies both of the following properties:
(1) x1 > x2 > x3 > · · · > xn.
(2) The sum x21 + x22 + · · ·+ x2kis a perfect square for each 1 ≤ k ≤ n.
For example, (12, 9, 8) is super-squared, since 12 > 9 > 8, and each of 122, 122 + 92, and122 + 92 + 82 are perfect squares.
(a) (2 marks) Determine all values of t such that (32, t, 9) is super-squared.
(b) (2 marks) Determine a super-squared 4-tuple (x1, x2, x3, x4) with x1 < 200.
(c) (6 marks) Determine whether there exists a super-squared 2012-tuple.
5
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ǁǁǁĐŵƐŵĂƚŚĐĂ ΞϮϬϭϮE/EDd,Dd/>^K/dz
^hE>/&&/EE/>E/EKWEDd,Dd/^,>>E'ϮϬϭϮ WĂŐĞϭϲŽĨϭϲ
ĂŶĂĚŝĂŶKƉĞŶDĂƚŚĞŵĂƟĐƐŚĂůůĞŶŐĞ
Canadian Mathematical SocietySociété mathématique du Canada
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