International Mathematics Olympiad Problems - 2013 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Tuesday, July 23, 2013
Problem 1. Prove that for any pair of positive integers k and n, there exist k positive integers
m
1
;m
2
;:::;m
k
(not necessarily dierent) such that
1 +
2
k
 1
n
=

1 +
1
m
1

1 +
1
m
2






1 +
1
m
k

:
Problem 2. A conguration of 4027 points in the plane is called Colombian if it consists of 2013 red
points and 2014 blue points, and no three of the points of the conguration are collinear. By drawing
some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian
conguration if the following two conditions are satised:
 no line passes through any point of the conguration;
 no region contains points of both colours.
Find the least value of k such that for any Colombian conguration of 4027 points, there is a good
arrangement of k lines.
Problem 3. Let the excircle of triangle ABC opposite the vertex A be tangent to the side BC at the
point A
1
. Dene the points B
1
on CA and C
1
on AB analogously, using the excircles opposite B and
C, respectively. Suppose that the circumcentre of triangle A
1
B
1
C
1
lies on the circumcircle of triangle
ABC. Prove that triangle ABC is right-angled.
The excircle of triangle ABC opposite the vertex A is the circle that is tangent to the line segment
BC, to the rayAB beyondB, and to the rayAC beyondC. The excircles oppositeB andC are similarly
dened.
Page 2


Tuesday, July 23, 2013
Problem 1. Prove that for any pair of positive integers k and n, there exist k positive integers
m
1
;m
2
;:::;m
k
(not necessarily dierent) such that
1 +
2
k
 1
n
=

1 +
1
m
1

1 +
1
m
2






1 +
1
m
k

:
Problem 2. A conguration of 4027 points in the plane is called Colombian if it consists of 2013 red
points and 2014 blue points, and no three of the points of the conguration are collinear. By drawing
some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian
conguration if the following two conditions are satised:
 no line passes through any point of the conguration;
 no region contains points of both colours.
Find the least value of k such that for any Colombian conguration of 4027 points, there is a good
arrangement of k lines.
Problem 3. Let the excircle of triangle ABC opposite the vertex A be tangent to the side BC at the
point A
1
. Dene the points B
1
on CA and C
1
on AB analogously, using the excircles opposite B and
C, respectively. Suppose that the circumcentre of triangle A
1
B
1
C
1
lies on the circumcircle of triangle
ABC. Prove that triangle ABC is right-angled.
The excircle of triangle ABC opposite the vertex A is the circle that is tangent to the line segment
BC, to the rayAB beyondB, and to the rayAC beyondC. The excircles oppositeB andC are similarly
dened.
Wednesday, July 24, 2013
Problem 4. Let ABC be an acute-angled triangle with orthocentre H, and let W be a point on the
side BC, lying strictly between B and C. The points M and N are the feet of the altitudes from B and
C, respectively. Denote by !
1
the circumcircle of BWN, and let X be the point on !
1
such that WX
is a diameter of !
1
. Analogously, denote by !
2
the circumcircle of CWM, and let Y be the point on !
2
such that WY is a diameter of !
2
. Prove that X, Y and H are collinear.
Problem5. LetQ
>0
be the set of positive rational numbers. Letf:Q
>0
!R be a function satisfying
the following three conditions:
(i) for all x;y2Q
>0
, we have f(x)f(y)f(xy);
(ii) for all x;y2Q
>0
, we have f(x+y)f(x)+f(y);
(iii) there exists a rational number a> 1 such that f(a) =a.
Prove that f(x) =x for all x2Q
>0
.
Problem 6. Let n 3 be an integer, and consider a circle with n+1 equally spaced points marked
on it. Consider all labellings of these points with the numbers 0;1;:::;n such that each label is used
exactly once; two such labellings are considered to be the same if one can be obtained from the other
by a rotation of the circle. A labelling is called beautiful if, for any four labels a < b < c < d with
a +d = b +c, the chord joining the points labelled a and d does not intersect the chord joining the
points labelled b and c.
Let M be the number of beautiful labellings, and let N be the number of ordered pairs (x;y) of
positive integers such that x+yn and gcd(x;y) = 1. Prove that
M =N +1: