International Mathematics Olympiad Problems - 2008 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Wednesday, July 16, 2008
Problem1. An acute-angled triangleABC has orthocentreH. The circle passing throughH with
centre the midpoint of BC intersects the lineBC atA
1
andA
2
. Similarly, the circle passing through
H with centre the midpoint ofCA intersects the lineCA atB
1
andB
2
, and the circle passing through
H with centre the midpoint of AB intersects the line AB at C
1
and C
2
. Show that A
1
, A
2
, B
1
, B
2
,
C
1
, C
2
lie on a circle.
Problem 2. (a) Prove that
x
2
(x-1)
2
+
y
2
(y-1)
2
+
z
2
(z-1)
2
= 1
for all real numbers x, y, z, each di?erent from 1, and satisfying xyz = 1.
(b) Prove that equality holds above for in?nitely many triples of rational numbers x, y, z, each
di?erent from 1, and satisfying xyz = 1.
Problem 3. Prove that there exist in?nitely many positive integers n such thatn
2
+1 has a prime
divisor which is greater than 2n+
v
2n.
Page 2


Wednesday, July 16, 2008
Problem1. An acute-angled triangleABC has orthocentreH. The circle passing throughH with
centre the midpoint of BC intersects the lineBC atA
1
andA
2
. Similarly, the circle passing through
H with centre the midpoint ofCA intersects the lineCA atB
1
andB
2
, and the circle passing through
H with centre the midpoint of AB intersects the line AB at C
1
and C
2
. Show that A
1
, A
2
, B
1
, B
2
,
C
1
, C
2
lie on a circle.
Problem 2. (a) Prove that
x
2
(x-1)
2
+
y
2
(y-1)
2
+
z
2
(z-1)
2
= 1
for all real numbers x, y, z, each di?erent from 1, and satisfying xyz = 1.
(b) Prove that equality holds above for in?nitely many triples of rational numbers x, y, z, each
di?erent from 1, and satisfying xyz = 1.
Problem 3. Prove that there exist in?nitely many positive integers n such thatn
2
+1 has a prime
divisor which is greater than 2n+
v
2n.
Thursday, July 17, 2008
Problem 4. Find all functions f : (0,8)? (0,8) (so, f is a function from the positive real
numbers to the positive real numbers) such that

f(w)

2
+

f(x)

2
f(y
2
)+f(z
2
)
=
w
2
+x
2
y
2
+z
2
for all positive real numbers w, x, y, z, satisfying wx =yz.
Problem 5. Let n and k be positive integers with k=n and k-n an even number. Let 2n lamps
labelled 1, 2, ..., 2n be given, each of which can be either on or o?. Initially all the lamps are o?.
We consider sequences of steps: at each step one of the lamps is switched (from on to o? or from o?
to on).
Let N be the number of such sequences consisting of k steps and resulting in the state where
lamps 1 through n are all on, and lamps n+1 through 2n are all o?.
Let M be the number of such sequences consisting of k steps, resulting in the state where lamps
1 through n are all on, and lamps n + 1 through 2n are all o?, but where none of the lamps n + 1
through 2n is ever switched on.
Determine the ratio N/M.
Problem 6. Let ABCD be a convex quadrilateral with|BA|6=|BC|. Denote the incircles of
triangles ABC and ADC by ?
1
and ?
2
respectively. Suppose that there exists a circle ? tangent to
the ray BA beyond A and to the ray BC beyond C, which is also tangent to the lines AD and CD.
Prove that the common external tangents of ?
1
and ?
2
intersect on ?.