2007 Indonesia MO Problems
Contents
[hide]Day 1
Problem 1
Let be a triangle with
. Let point
on side
such that
is the altitude, point
on side
such that
, and point
is the intersection of
and
. Prove that
.
Problem 2
For every positive integer ,
denote the number of positive divisors of
and
denote the sum of all positive divisors of
. For example,
and
. Let
be a positive integer greater than
.
(a) Prove that there are infinitely many positive integers which satisfy
.
(b) Prove that there are finitely many positive integers which satisfy
.
Problem 3
Let be positive real numbers which satisfy
. Prove that these three inequalities hold:
,
,
.
Problem 4
A 10-digit arrangement is called beautiful if (i) when read left to right,
form an increasing sequence, and
form a decreasing sequence, and (ii)
is not the leftmost digit. For example,
is a beautiful arrangement. Determine the number of beautiful arrangements.
Day 2
Problem 5
Let ,
be two positive integers and
a 'chessboard' with
rows and
columns. Let
denote the maximum value of rooks placed on
such that no two of them attack each other.
(a) Determine .
(b) How many ways to place rooks on
such that no two of them attack each other?
[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
Problem 6
Find all triples of real numbers which satisfy the simultaneous equations
Problem 7
Points are on circle
, such that
is the diameter of
, but
is not the diameter. Given also that
and
are on different sides of
. The tangents of
at
and
intersect at
. Points
and
are the intersections of line
with line
and line
with line
, respectively.
(a) Prove that ,
, and
are collinear.
(b) Prove that is perpendicular to line
.
Problem 8
Let and
be two positive integers. If there are infinitely many integers
such that
is a perfect square, prove that
.
See Also
2007 Indonesia MO (Problems) | ||
Preceded by 2006 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2008 Indonesia MO |
All Indonesia MO Problems and Solutions |