International Mathematics Olympiad Problems - 2018 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Monday, July 9, 2018
Problem 1. Let  be the circumcircle of acute-angled triangle ABC. Points D and E lie on
segments AB and AC, respectively, such that AD = AE. The perpendicular bisectors of BD and
CE intersect the minor arcs AB and AC of  at points F and G, respectively. Prove that the lines
DE and FG are parallel (or are the same line).
Problem 2. Find all integers n 3 for which there exist real numbers a
1
;a
2
;:::;a
n+2
, such that
a
n+1
=a
1
and a
n+2
=a
2
, and
a
i
a
i+1
+ 1 =a
i+2
for i = 1; 2;:::;n.
Problem3. Ananti-Pascal triangle is an equilateral triangular array of numbers such that, except
for the numbers in the bottom row, each number is the absolute value of the di?erence of the two
numbers immediately below it. For example, the following array is an anti-Pascal triangle with four
rows which contains every integer from 1 to 10.
4
2 6
5 7 1
8 3 10 9
Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to
1 + 2 +


 + 2018?
Page 2


Monday, July 9, 2018
Problem 1. Let  be the circumcircle of acute-angled triangle ABC. Points D and E lie on
segments AB and AC, respectively, such that AD = AE. The perpendicular bisectors of BD and
CE intersect the minor arcs AB and AC of  at points F and G, respectively. Prove that the lines
DE and FG are parallel (or are the same line).
Problem 2. Find all integers n 3 for which there exist real numbers a
1
;a
2
;:::;a
n+2
, such that
a
n+1
=a
1
and a
n+2
=a
2
, and
a
i
a
i+1
+ 1 =a
i+2
for i = 1; 2;:::;n.
Problem3. Ananti-Pascal triangle is an equilateral triangular array of numbers such that, except
for the numbers in the bottom row, each number is the absolute value of the di?erence of the two
numbers immediately below it. For example, the following array is an anti-Pascal triangle with four
rows which contains every integer from 1 to 10.
4
2 6
5 7 1
8 3 10 9
Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to
1 + 2 +


 + 2018?
Tuesday, July 10, 2018
Problem 4. A site is any point (x;y) in the plane such that x andy are both positive integers less
than or equal to 20.
Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy
going ?rst. On her turn, Amy places a new red stone on an unoccupied site such that the distance
between any two sites occupied by red stones is not equal to
p
5. On his turn, Ben places a new blue
stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from
any other occupied site.) They stop as soon as a player cannot place a stone.
Find the greatest K such that Amy can ensure that she places at least K red stones, no matter
how Ben places his blue stones.
Problem 5. Let a
1
;a
2
;::: be an in?nite sequence of positive integers. Suppose that there is an
integer N > 1 such that, for each nN, the number
a
1
a
2
+
a
2
a
3
+


 +
a
n1
a
n
+
a
n
a
1
is an integer. Prove that there is a positive integer M such that a
m
=a
m+1
for all mM.
Problem 6. A convex quadrilateral ABCD satis?es AB
CD = BC
DA. Point X lies inside
ABCD so that
\XAB =\XCD and \XBC =\XDA:
Prove that\BXA +\DXC = 180

.