International Mathematics Olympiad Problems - 2014 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


T uesday, July 8, 2014 Problem 1. Let a
0
<a
1
<a
2
<


 b e an innite sequence of p ositiv e in tegers. Pro v e that there exists a unique in teger n 1 suc h that a
n
<
a
0
+a
1
+


+a
n
n
a
n+1
:
Problem 2. Let n 2 b e an in teger. Consider an nn c hessb oard c on si sting of n
2
unit squares. A conguration of n ro oks on this b oard is p e ac eful if ev ery ro w and ev ery column con tains exactly one ro ok. Find the greatest p ositiv e in te ger k suc h that, f or eac h p eacef u l conguration of n ro oks, there is a kk square whic h do es not con tain a ro ok on an y of its k
2
unit squares. Problem 3. Con v ex quadril at e ral ABCD has \ABC =\CDA = 90

. P oin t H is the fo ot of the p erp endicular from A to BD . P oin ts S and T lie on sides AB and AD , resp ectiv ely , suc h that H
lies inside triangle SCT and \CHS\CSB = 90

; \THC\DTC = 90

:
Pro v e that line BD is tangen t to the circumcircle of triangle TSH .
Page 2


T uesday, July 8, 2014 Problem 1. Let a
0
<a
1
<a
2
<


 b e an innite sequence of p ositiv e in tegers. Pro v e that there exists a unique in teger n 1 suc h that a
n
<
a
0
+a
1
+


+a
n
n
a
n+1
:
Problem 2. Let n 2 b e an in teger. Consider an nn c hessb oard c on si sting of n
2
unit squares. A conguration of n ro oks on this b oard is p e ac eful if ev ery ro w and ev ery column con tains exactly one ro ok. Find the greatest p ositiv e in te ger k suc h that, f or eac h p eacef u l conguration of n ro oks, there is a kk square whic h do es not con tain a ro ok on an y of its k
2
unit squares. Problem 3. Con v ex quadril at e ral ABCD has \ABC =\CDA = 90

. P oin t H is the fo ot of the p erp endicular from A to BD . P oin ts S and T lie on sides AB and AD , resp ectiv ely , suc h that H
lies inside triangle SCT and \CHS\CSB = 90

; \THC\DTC = 90

:
Pro v e that line BD is tangen t to the circumcircle of triangle TSH . W e dnesday, July 9, 2014 Problem 4. P oin ts P and Q lie on side BC of acute-angled triangle ABC so that \PAB =\BCA
and \CAQ = \ABC . P oi n ts M and N lie on li nes AP and AQ , resp ectiv ely , suc h that P is the midp oin t of AM , and Q is the midp oin t of AN . Pro v e that lines BM and CN in tersect on the circumcircle of triangle ABC . Problem 5. F or eac h p ositiv e in t e ger n , the Bank of Cap e T o wn issues coins of denomi n a t i on 1
n
. Giv en a nite collection of suc h coins (of not necessarily dieren t denominations) with total v alue at most 99+
1
2
, pr o v e that it is p ossible to split this collec tion in to 100 or few er groups, suc h that eac h group has total v al ue at most 1 . Problem 6. A set of lines in the plane is in gener al p osition if no t w o are parallel and no three pass through the same p oin t. A set of lines in ge neral p osition cuts the plane in to regions, some of whic h ha v e nite area; w e call these its nite r e gions . Pro v e that for all sucien tly large n , in an y set of n lines in general p osition it is p ossible to col our at least p
n of the lines blue in suc h a w a y that none of its nite regions has a completely blue b oundary . Note: Results with p
n replaced b y c
p
n will b e a w arded p oin ts dep ending on the v alue of the constan t c .