International Mathematics Olympiad Problems - 2019 | International Mathematics Olympiad (IMO) for Class 10 PDF Download

 Page 1


Tuesday, July 16, 2019
Problem 1. Let Z be the set of integers. Determine all functions f: Z! Z such that, for all
integers a and b,
f(2a)+2f(b) =f(f(a+b)):
Problem2. IntriangleABC, pointA
1
liesonsideBC andpointB
1
liesonsideAC. LetP andQ
be points on segmentsAA
1
andBB
1
, respectively, such thatPQ is parallel toAB. LetP
1
be a point
on line PB
1
, such that B
1
lies strictly between P and P
1
, and\PP
1
C =\BAC. Similarly, let Q
1
be a point on line QA
1
, such that A
1
lies strictly between Q and Q
1
, and\CQ
1
Q =\CBA.
Prove that points P, Q, P
1
, and Q
1
are concyclic.
Problem 3. A social network has 2019 users, some pairs of whom are friends. Whenever user A
is friends with user B, user B is also friends with user A. Events of the following kind may happen
repeatedly, one at a time:
Three users A, B, and C such that A is friends with both B and C, but B and C are
not friends, change their friendship statuses such that B andC are now friends, but A is
no longer friends with B, and no longer friends with C. All other friendship statuses are
unchanged.
Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there
exists a sequence of such events after which each user is friends with at most one other user.
Page 2


Tuesday, July 16, 2019
Problem 1. Let Z be the set of integers. Determine all functions f: Z! Z such that, for all
integers a and b,
f(2a)+2f(b) =f(f(a+b)):
Problem2. IntriangleABC, pointA
1
liesonsideBC andpointB
1
liesonsideAC. LetP andQ
be points on segmentsAA
1
andBB
1
, respectively, such thatPQ is parallel toAB. LetP
1
be a point
on line PB
1
, such that B
1
lies strictly between P and P
1
, and\PP
1
C =\BAC. Similarly, let Q
1
be a point on line QA
1
, such that A
1
lies strictly between Q and Q
1
, and\CQ
1
Q =\CBA.
Prove that points P, Q, P
1
, and Q
1
are concyclic.
Problem 3. A social network has 2019 users, some pairs of whom are friends. Whenever user A
is friends with user B, user B is also friends with user A. Events of the following kind may happen
repeatedly, one at a time:
Three users A, B, and C such that A is friends with both B and C, but B and C are
not friends, change their friendship statuses such that B andC are now friends, but A is
no longer friends with B, and no longer friends with C. All other friendship statuses are
unchanged.
Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there
exists a sequence of such events after which each user is friends with at most one other user.
Wednesday, July 17, 2019
Problem 4. Find all pairs (k;n) of positive integers such that
k! = (2
n
1)(2
n
2)(2
n
4)


(2
n
2
n1
):
Problem5. The Bank of Bath issues coins with an H on one side and aT on the other. Harry has
n of these coins arranged in a line from left to right. He repeatedly performs the following operation:
if there are exactly k > 0 coins showing H, then he turns over the k
th
coin from the left; otherwise,
all coins show T and he stops. For example, if n = 3 the process starting with the con?guration
THT would be THT!HHT!HTT!TTT, which stops after three operations.
(a) Show that, for each initial con?guration, Harry stops after a ?nite number of operations.
(b) For each initial con?guration C, let L(C) be the number of operations before Harry stops. For
example, L(THT) = 3 and L(TTT) = 0. Determine the average value of L(C) over all 2
n
possible initial con?gurations C.
Problem6. LetI be the incentre of acute triangleABC withAB6=AC. The incircle! ofABC is
tangent to sides BC, CA, and AB at D, E, and F, respectively. The line through D perpendicular
to EF meets ! again at R. Line AR meets ! again at P. The circumcircles of triangles PCE
and PBF meet again at Q.
Prove that lines DI and PQ meet on the line through A perpendicular to AI.