International Mathematics Olympiad Problems - 2010 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Wednesday, July 7, 2010
Problem 1. Determine all functions f :R?R such that the equality
f
?
bxcy
?
=f(x)
?
f(y)

holds for all x,y?R. (Herebzc denotes the greatest integer less than or equal to z.)
Problem 2. Let I be the incentre of triangle ABC and let G be its circumcircle. Let the line AI
intersect G again at D. Let E be a point on the arc
ú
BDC and F a point on the side BC such that
?BAF =?CAE <
1
2
?BAC.
Finally, let G be the midpoint of the segment IF. Prove that the lines DG and EI intersect on G.
Problem 3. LetN be the set of positive integers. Determine all functions g :N?N such that
?
g(m) +n
??
m +g(n)
?
is a perfect square for all m,n?N.
Page 2


Wednesday, July 7, 2010
Problem 1. Determine all functions f :R?R such that the equality
f
?
bxcy
?
=f(x)
?
f(y)

holds for all x,y?R. (Herebzc denotes the greatest integer less than or equal to z.)
Problem 2. Let I be the incentre of triangle ABC and let G be its circumcircle. Let the line AI
intersect G again at D. Let E be a point on the arc
ú
BDC and F a point on the side BC such that
?BAF =?CAE <
1
2
?BAC.
Finally, let G be the midpoint of the segment IF. Prove that the lines DG and EI intersect on G.
Problem 3. LetN be the set of positive integers. Determine all functions g :N?N such that
?
g(m) +n
??
m +g(n)
?
is a perfect square for all m,n?N.
Thursday, July 8, 2010
Problem 4. Let P be a point inside the triangle ABC. The lines AP, BP and CP intersect the
circumcircle G of triangle ABC again at the points K, L and M respectively. The tangent to G atC
intersects the line AB at S. Suppose that SC =SP. Prove that MK =ML.
Problem 5. In each of six boxes B
1
,B
2
,B
3
,B
4
,B
5
,B
6
there is initially one coin. There are two
types of operation allowed:
Type 1: Choose a nonempty box B
j
with 1=j= 5. Remove one coin from B
j
and add two
coins to B
j+1
.
Type 2: Choose a nonempty boxB
k
with 1=k= 4. Remove one coin fromB
k
and exchange
the contents of (possibly empty) boxes B
k+1
and B
k+2
.
Determinewhetherthereisa?nitesequenceofsuchoperationsthatresultsinboxes B
1
,B
2
,B
3
,B
4
,B
5
being empty and box B
6
containing exactly 2010
2010
2010
coins. (Note that a
b
c
=a
(b
c
)
.)
Problem 6. Let a
1
,a
2
,a
3
,... be a sequence of positive real numbers. Suppose that for some
positive integer s, we have
a
n
= max{a
k
+a
n-k
| 1=k=n- 1}
for alln>s. Prove that there exist positive integers` andN, with`=s and such thata
n
=a
`
+a
n-`
for all n=N.