imo 2014 problems with selected solutionssms.math.nus.edu.sg/smsmedley/vol-40-2/imo 2014...
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Mathematical Medley • Volume 40 No. 2 December 201428
Language: English
Day: 1
a0 < a1 < a2 < · · ·n ≥ 1
an <a0 + a1 + · · ·+ an
n≤ an+1.
n ≥ 2 n× n n2n
k nk × k k2
ABCD ∠ABC = ∠CDA = 90◦ HA BD S T AB AD H
SCT
∠CHS − ∠CSB = 90◦, ∠THC − ∠DTC = 90◦ .
BD TSH
Language: English
Day: 1
a0 < a1 < a2 < · · ·n ≥ 1
an <a0 + a1 + · · ·+ an
n≤ an+1.
n ≥ 2 n× n n2n
k nk × k k2
ABCD ∠ABC = ∠CDA = 90◦ HA BD S T AB AD H
SCT
∠CHS − ∠CSB = 90◦, ∠THC − ∠DTC = 90◦ .
BD TSH
Language: EnglishDay: 1
Handwritten solutions for Problems 1 and 2 by SIMO team members can be found on page 30 - 33.
IMO 2014 problems with selected solutions
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Mathematical Medley • Volume 40 No. 2 December 2014 29
Language: English
Day: 1
a0 < a1 < a2 < · · ·n ≥ 1
an <a0 + a1 + · · ·+ an
n≤ an+1.
n ≥ 2 n× n n2n
k nk × k k2
ABCD ∠ABC = ∠CDA = 90◦ HA BD S T AB AD H
SCT
∠CHS − ∠CSB = 90◦, ∠THC − ∠DTC = 90◦ .
BD TSH
Language: English
Day: 2
P Q BC ABC ∠PAB = ∠BCA∠CAQ = ∠ABC M N AP AQ P
AM Q AN BM CNABC
n 1n
99 + 12
1001
nn
√n
√n c
√n
c
Language: English
Day: 1
a0 < a1 < a2 < · · ·n ≥ 1
an <a0 + a1 + · · ·+ an
n≤ an+1.
n ≥ 2 n× n n2n
k nk × k k2
ABCD ∠ABC = ∠CDA = 90◦ HA BD S T AB AD H
SCT
∠CHS − ∠CSB = 90◦, ∠THC − ∠DTC = 90◦ .
BD TSH
Language: EnglishDay: 2
Language: English
Day: 2
P Q BC ABC ∠PAB = ∠BCA∠CAQ = ∠ABC M N AP AQ P
AM Q AN BM CNABC
n 1n
99 + 12
1001
nn
√n
√n c
√n
c
IMO 2014 problems with selected solutions
Handwritten solutions for Problems 4 and 5 by SIMO team members can be found on page 34 - 35.
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Mathematical Medley • Volume 40 No. 2 December 201430
Contestant : Tan Siah Yong
Problem : 1
Page : 1
Reprinted from original manuscripts of the solutions written by SIMO team members during the competition.
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Mathematical Medley • Volume 40 No. 2 December 2014 31
Contestant : Tan Siah Yong
Problem : 1 (cont.)
Page : 2
Reprinted from original manuscripts of the solutions written by SIMO team members during the competition.
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Mathematical Medley • Volume 40 No. 2 December 201432
Contestant : Sheldon Kieran Tan
Problem : 2
Page : 1
Reprinted from original manuscripts of the solutions written by SIMO team members during the competition.
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Mathematical Medley • Volume 40 No. 2 December 2014 33
Contestant : Sheldon Kieran Tan
Problem : 2 (cont.)
Page : 2
Reprinted from original manuscripts of the solutions written by SIMO team members during the competition.
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Mathematical Medley • Volume 40 No. 2 December 201434
Contestant : Kewei David Lin
Problem : 4
Page : 1
Reprinted from original manuscripts of the solutions written by SIMO team members during the competition.
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Mathematical Medley • Volume 40 No. 2 December 2014 35
Contestant : Sheldon Kieran Tan
Problem : 5
Page : 1
Reprinted from original manuscripts of the solutions written by SIMO team members during the competition.