International Mathematics Olympiad Problems - 2003

``` Page 1

Problem 1. S is the set {1,2,3,...,1000000}. Show that for any subset A
of S with 101 elements we can ?nd 100 distinct elements x
i
of S, such that
the sets {a+x
i
|a?A} are all pairwise disjoint.
Problem 2. Find all pairs (m,n) of positive integers such that
m
2
2mn
2
-n
3
+1
is a positive integer.
Problem3. Aconvexhexagonhasthepropertythatforanypairofopposite
sides the distance between their midpoints is
v
3/2 times the sum of their
lengths Show that all the hexagon’s angles are equal.
Problem 4. ABCD is cyclic. The feet of the perpendicular from D to the
lines AB,BC,CA are P,Q,R respectively. Show that the angle bisectors of
ABC and CDA meet on the line AC i? RP =RQ.
Problem 5. Given n > 2 and reals x
1
= x
2
= ··· = x
n
, show that
(
P
i,j
|x
i
-x
j
|)
2
=
2
3
(n
2
-1)
P
i,j
(x
i
-x
j
)
2
. Show that we have equality i?
the sequence is an arithmetic progression.
Problem 6. Show that for each prime p, there exists a prime q such that
n
p
-p is not divisible by q for any positive integer n.
1
```

## Olympiad Preparation for Class 10

11 videos|36 docs|201 tests

## Olympiad Preparation for Class 10

11 videos|36 docs|201 tests

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