International Mathematics Olympiad Problems - 2021

# International Mathematics Olympiad Problems - 2021 - Olympiad Preparation for Class 10

``` Page 1

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
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.
```

## Olympiad Preparation for Class 10

11 videos|36 docs|201 tests

## FAQs on International Mathematics Olympiad Problems - 2021 - Olympiad Preparation for Class 10

 1. What is the International Mathematics Olympiad (IMO)?
Ans. The International Mathematics Olympiad (IMO) is an annual mathematics competition for high school students from around the world. It is considered one of the most prestigious mathematical competitions and aims to stimulate interest in mathematics and develop problem-solving skills.
 2. Who can participate in the International Mathematics Olympiad (IMO)?
Ans. The International Mathematics Olympiad (IMO) is open to high school students up to the age of 19. Each country can send a team of up to six students to represent their nation in the competition. Participants are selected through a series of national and regional mathematics contests.
 3. How difficult are the problems in the International Mathematics Olympiad (IMO)?
Ans. The problems in the International Mathematics Olympiad (IMO) are known for their high level of difficulty. They require advanced problem-solving skills, mathematical creativity, and a deep understanding of various mathematical concepts. The problems are designed to challenge the brightest young mathematicians from around the world.
 4. What is the format of the International Mathematics Olympiad (IMO)?
Ans. The International Mathematics Olympiad (IMO) consists of six problems, each to be solved within a time limit of four and a half hours. The problems are essay-type questions, and participants are required to provide rigorous mathematical proofs for their solutions. The competition is conducted over two consecutive days.
 5. How can I prepare for the International Mathematics Olympiad (IMO)?
Ans. To prepare for the International Mathematics Olympiad (IMO), it is essential to have a strong foundation in mathematics and problem-solving techniques. Practice solving challenging mathematical problems from various sources, such as past IMO papers and other Olympiad-style books. Participating in local and national mathematics contests can also help improve your problem-solving skills. Additionally, seek guidance from experienced mathematics teachers or coaches who can provide valuable insights and strategies for tackling advanced mathematical problems.

## Olympiad Preparation for Class 10

11 videos|36 docs|201 tests

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