International Mathematics Olympiad Problems - 2021 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

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Monday, 19. July 2021
Problem 1. Letn>100 b e an in teger. Iv an writes the n um b ersn,n+1,...,2n eac h on di fferen t
cards. He then sh uffles these n+1 cards, and divides them in to t w o piles. Pro v e that at least one of
the piles con tains t w o cards suc h that t he sum of their n um b ers is a p erfect square.
Problem 2. Sho w that the inequalit y
n
?
i=1
n
?
j=1
v
|x
i
-x
j
|6
n
?
i=1
n
?
j=1
v
|x
i
+x
j
|
holds for all real n um b ers x
1
,...,x
n
.
Problem 3. LetD b e an in terior p oin t of the acute triangleABC withAB >AC so that\DAB =
\CAD . The p oin t E on the segmen t AC satisfies \ADE =\BCD , the p oin t F on the segmen t
AB satisfies \FDA=\DBC , and the p oin tX on the lineAC satisfies CX =BX . LetO
1
andO
2
b e the circumcen tres of the triangles ADC and EXD , resp ectiv ely . Pro v e that the lines BC , EF ,
and O
1
O
2
are concurren t.
Page 2


Monday, 19. July 2021
Problem 1. Letn>100 b e an in teger. Iv an writes the n um b ersn,n+1,...,2n eac h on di fferen t
cards. He then sh uffles these n+1 cards, and divides them in to t w o piles. Pro v e that at least one of
the piles con tains t w o cards suc h that t he sum of their n um b ers is a p erfect square.
Problem 2. Sho w that the inequalit y
n
?
i=1
n
?
j=1
v
|x
i
-x
j
|6
n
?
i=1
n
?
j=1
v
|x
i
+x
j
|
holds for all real n um b ers x
1
,...,x
n
.
Problem 3. LetD b e an in terior p oin t of the acute triangleABC withAB >AC so that\DAB =
\CAD . The p oin t E on the segmen t AC satisfies \ADE =\BCD , the p oin t F on the segmen t
AB satisfies \FDA=\DBC , and the p oin tX on the lineAC satisfies CX =BX . LetO
1
andO
2
b e the circumcen tres of the triangles ADC and EXD , resp ectiv ely . Pro v e that the lines BC , EF ,
and O
1
O
2
are concurren t.
T uesday, 20. July 2021
Problem 4. Let G b e a circle with cen tre I , and ABCD a con v ex quadrilateral suc h that eac h of
the segmen ts AB , BC , CD and DA is tangen t to G . Let ? b e the circumcircle of the triangle AIC .
The extension of BA b ey ond A meets ? at X , and the extension of BC b ey ond C meets ? at Z .
The extensions of AD and CD b ey ond D meet ? at Y and T , resp ectiv ely . Pro v e that
AD+DT +TX+XA=CD+DY +YZ+ZC.
Problem 5. T w o squirrels, Bush y and Jump y , ha v e collected 2021 w aln uts for the win ter. Jump y
n um b ers the w aln uts from 1 through 2021 , and digs 2021 little holes in a circular pattern in the
ground around their fa v ourite tree. The next morning Jump y notices that Bush y had placed one
w aln ut in to eac h hole, but had paid no atten tion to the n um b ering. Unhapp y , Jump y decides to
reorder the w aln uts b y p erforming a sequence of 2021 mo v es. In the k -th mo v e, Jump y sw aps the
p ositions of the t w o w aln uts adjacen t to w aln ut k .
Pro v e that there exists a v alue of k suc h that, on the k -th mo v e, Jump y sw aps some w aln uts a
and b suc h that a<k <b .
Problem 6. Let m> 2 b e an in teger, A b e a finite set of (not necessarily p ositiv e) in tegers, and
B
1
,B
2
,B
3
,...,B
m
b e subsets of A . Assume that for eac h k =1,2,...,m the sum of the elemen ts of
B
k
is m
k
. Pro v e that A con tains at least m/2 elemen ts.