International Mathematics Olympiad Problems - 2012 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Tuesday, July 10, 2012
Problem 1. Given triangle ABC the point J is the centre of the excircle opposite the vertex A.
This excircle is tangent to the side BC at M, and to the lines AB and AC at K and L, respectively.
The lines LM and BJ meet at F, and the lines KM and CJ meet at G. Let S be the point of
intersection of the lines AF and BC, and let T be the point of intersection of the lines AG and BC.
Prove that M is the midpoint of ST.
(The excircle of ABC opposite the vertex A is the circle that is tangent to the line segment BC,
to the ray AB beyond B, and to the ray AC beyond C.)
Problem 2. Let n = 3 be an integer, and let a
2
,a
3
,...,a
n
be positive real numbers such that
a
2
a
3
···a
n
= 1. Prove that
(1+a
2
)
2
(1+a
3
)
3
···(1+a
n
)
n
> n
n
.
Problem 3. The liar’s guessing game is a game played between two players A and B. The rules
of the game depend on two positive integers k and n which are known to both players.
At the start of the game A chooses integers x and N with 1= x= N. Player A keeps x secret,
and truthfully tellsN to playerB. PlayerB now tries to obtain information aboutx by asking player
A questions as follows: each question consists of B specifying an arbitrary set S of positive integers
(possibly one speci?ed in some previous question), and asking A whether x belongs to S. Player
B may ask as many such questions as he wishes. After each question, player A must immediately
answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is
that, among any k +1 consecutive answers, at least one answer must be truthful.
After B has asked as many questions as he wants, he must specify a set X of at most n positive
integers. If x belongs to X, then B wins; otherwise, he loses. Prove that:
1. If n= 2
k
, then B can guarantee a win.
2. For all su?ciently large k, there exists an integer n= 1.99
k
such that B cannot guarantee a
win.
Page 2


Tuesday, July 10, 2012
Problem 1. Given triangle ABC the point J is the centre of the excircle opposite the vertex A.
This excircle is tangent to the side BC at M, and to the lines AB and AC at K and L, respectively.
The lines LM and BJ meet at F, and the lines KM and CJ meet at G. Let S be the point of
intersection of the lines AF and BC, and let T be the point of intersection of the lines AG and BC.
Prove that M is the midpoint of ST.
(The excircle of ABC opposite the vertex A is the circle that is tangent to the line segment BC,
to the ray AB beyond B, and to the ray AC beyond C.)
Problem 2. Let n = 3 be an integer, and let a
2
,a
3
,...,a
n
be positive real numbers such that
a
2
a
3
···a
n
= 1. Prove that
(1+a
2
)
2
(1+a
3
)
3
···(1+a
n
)
n
> n
n
.
Problem 3. The liar’s guessing game is a game played between two players A and B. The rules
of the game depend on two positive integers k and n which are known to both players.
At the start of the game A chooses integers x and N with 1= x= N. Player A keeps x secret,
and truthfully tellsN to playerB. PlayerB now tries to obtain information aboutx by asking player
A questions as follows: each question consists of B specifying an arbitrary set S of positive integers
(possibly one speci?ed in some previous question), and asking A whether x belongs to S. Player
B may ask as many such questions as he wishes. After each question, player A must immediately
answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is
that, among any k +1 consecutive answers, at least one answer must be truthful.
After B has asked as many questions as he wants, he must specify a set X of at most n positive
integers. If x belongs to X, then B wins; otherwise, he loses. Prove that:
1. If n= 2
k
, then B can guarantee a win.
2. For all su?ciently large k, there exists an integer n= 1.99
k
such that B cannot guarantee a
win.
W e dnesday, July 11, 2012 Problem 4. Find all functions f: Z?Z suc h that , for all in tegers a , b , c that satisfy a+b+c = 0 , the follo wing equalit y holds: f(a)
2
+f(b)
2
+f(c)
2
= 2f(a)f(b)+2f(b)f(c)+2f(c)f(a).
(Here Z denotes the set of in tegers.) Problem 5. Let ABC b e a tr i angle with ?BCA = 90
?
, and let D b e the fo ot of the altitude from C . Let X b e a p oin t in the in terior of the segmen t CD . Let K b e the p oin t on the segmen t AX
suc h that BK =BC . Similarly , let L b e the p oin t on the segmen t BX suc h that AL =AC . Let M
b e the p oin t of in tersection of AL and BK . Sho w that MK =ML . Problem 6. Find all p ositiv e in tegers n for whic h there exist non-negativ e in tegers a
1
,a
2
,...,a
n
suc h that 1
2
a1
+
1
2
a2
+··· +
1
2
an
=
1
3
a1
+
2
3
a2
+··· +
n
3
an
= 1.