AoPS Community 2019 ISL

IMO Shortlist 2019

www.artofproblemsolving.com/community/c1308102
by naman12, lminsl, y-is-the-best- , tapir1729, tastymath75025, ilovemath04, TelMarin, parmenides51, Kas-
suno, InternetPerson10, nukelauncher

– Algebra

A1 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)).
Proposed by Liam Baker, South Africa

A2 Let u1 , u2 , . . . , u2019 be real numbers satisfying

u1 + u2 + · · · + u2019 = 0 and u21 + u22 + · · · + u22019 = 1.
Let a = min (u1 , u2 , . . . , u2019 ) and b = max (u1 , u2 , . . . , u2019 ). Prove that
1
ab ⩽ − .
2019

A3 Let n ⩾ 3 be a positive integer and let (a1 , a2 , . . . , an ) be a strictly increasing sequence of n

positive real numbers with sum equal to 2. Let X be a subset of {1, 2, . . . , n} such that the value
of
X
1− ai
i∈X
is minimised. Prove that there exists a strictly increasing sequence of n positive real numbers
(b1 , b2 , . . . , bn ) with sum equal to 2 such that
X
bi = 1.
i∈X

A4 Let n ⩾ 2 be a positive integer and a1 , a2 , . . . , an be real numbers such that

a1 + a2 + · · · + an = 0.
Define the set A by
A = {(i, j) | 1 ⩽ i < j ⩽ n, |ai − aj | ⩾ 1}
Prove that, if A is not empty, then X
ai aj < 0.
(i,j)∈A

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A5 Let x1 , x2 , . . . , xn be different real numbers. Prove that

0, if n is even;
X Y 1 − xi xj 
=
xi − xj 1, if n is odd.
1⩽i⩽n j̸=i

A6 A polynomial P (x, y, z) in three variables with real coefficients satisfies the identities

P (x, y, z) = P (x, y, xy − z) = P (x, zx − y, z) = P (yz − x, y, z).

Prove that there exists a polynomial F (t) in one variable such that

P (x, y, z) = F (x2 + y 2 + z 2 − xyz).

A7 Let Z be the set of integers. We consider functions f : Z → Z satisfying

f (f (x + y) + y) = f (f (x) + y)

for all integers x and y. For such a function, we say that an integer v is f-rare if the set

Xv = {x ∈ Z : f (x) = v}

is finite and nonempty.

(a) Prove that there exists such a function f for which there is an f -rare integer.
(b) Prove that no such function f can have more than one f -rare integer.
Netherlands

– Combinatorics

C1 The infinite sequence a0 , a1 , a2 , . . . of (not necessarily distinct) integers has the following prop-
erties: 0 ≤ ai ≤ i for all integers i ≥ 0, and
     
k k k
+ + ··· + = 2k
a0 a1 ak

for all integers k ≥ 0. Prove that all integers N ≥ 0 occur in the sequence (that is, for all N ≥ 0,
there exists i ≥ 0 with ai = N ).

C2 You are given a set of n blocks, each weighing at least 1; their total weight is 2n. Prove that for
every real number r with 0 ≤ r ≤ 2n − 2 you can choose a subset of the blocks whose total
weight is at least r but at most r + 2.

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C3 The Bank of Bath issues coins with an H on one side and a T 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 kth coin from the left; otherwise, all
coins show T and he stops. For example, if n = 3 the process starting with the configuration
T HT would be T HT → HHT → HT T → T T T , which stops after three operations.
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration C, let L(C) be the number of operations before Harry stops.
For example, L(T HT ) = 3 and L(T T T ) = 0. Determine the average value of L(C) over all 2n
possible initial configurations C.
Proposed by David Altizio, USA

C4 On a flat plane in Camelot, King Arthur builds a labyrinth L consisting of n walls, each of which
is an infinite straight line. No two walls are parallel, and no three walls have a common point.
Merlin then paints one side of each wall entirely red and the other side entirely blue.
At the intersection of two walls there are four corners: two diagonally opposite corners where
a red side and a blue side meet, one corner where two red sides meet, and one corner where
two blue sides meet. At each such intersection, there is a two-way door connecting the two
diagonally opposite corners at which sides of different colours meet.
After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights
can walk through doors, but cannot walk through walls.
Let k(L) be the largest number k such that, no matter how Merlin paints the labyrinth L, Morgana
can always place at least k knights such that no two of them can ever meet. For each n, what
are all possible values for k(L), where L is a labyrinth with n walls?

C5 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 repeat-
edly, 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 and C 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.
Proposed by Adrian Beker, Croatia

C6 Let n > 1 be an integer. Suppose we are given 2n points in the plane such that no three of them
are collinear. The points are to be labelled A1 , A2 , . . . , A2n in some order. We then consider the
2n angles ∠A1 A2 A3 , ∠A2 A3 A4 , . . . , ∠A2n−2 A2n−1 A2n , ∠A2n−1 A2n A1 , ∠A2n A1 A2 . We measure

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each angle in the way that gives the smallest positive value (i.e. between 0◦ and 180◦ ). Prove that
there exists an ordering of the given points such that the resulting 2n angles can be separated
into two groups with the sum of one group of angles equal to the sum of the other group.

C7 There are 60 empty boxes B1 , . . . , B60 in a row on a table and an unlimited supply of pebbles.
Given a positive integer n, Alice and Bob play the following game.
In the first round, Alice takes n pebbles and distributes them into the 60 boxes as she wishes.
Each subsequent round consists of two steps:
(a) Bob chooses an integer k with 1 ≤ k ≤ 59 and splits the boxes into the two groups B1 , . . . , Bk
and Bk+1 , . . . , B60 .
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes
one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest n such
that Alice can prevent Bob from winning.
Czech Republic

C8 Alice has a map of Wonderland, a country consisting of n ≥ 2 towns. For every pair of towns,
there is a narrow road going from one town to the other. One day, all the roads are declared to
be ”one way” only. Alice has no information on the direction of the roads, but the King of Hearts
has offered to help her. She is allowed to ask him a number of questions. For each question
in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road
connecting those two towns.
Alice wants to know whether there is at least one town in Wonderland with at most one outgoing
road. Prove that she can always find out by asking at most 4n questions.

C9 For any two different real numbers x and y, we define D(x, y) to be the unique integer d satisfy-
ing 2d ≤ |x − y| < 2d+1 . Given a set of reals F, and an element x ∈ F, we say that the scales of
x in F are the values of D(x, y) for y ∈ F with x ̸= y. Let k be a given positive integer.

Suppose that each member x of F has at most k different scales in F (note that these scales
may depend on x). What is the maximum possible size of F?

– Geometry

G1 Let ABC be a triangle. Circle Γ passes through A, meets segments AB and AC again at points
D and E respectively, and intersects segment BC at F and G such that F lies between B and
G. The tangent to circle BDF at F and the tangent to circle CEG at G meet at point T . Suppose
that points A and T are distinct. Prove that line AT is parallel to BC.
(Nigeria)

G2 Let ABC be an acute-angled triangle and let D, E, and F be the feet of altitudes from A, B, and

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C to sides BC, CA, and AB, respectively. Denote by ωB and ωC the incircles of triangles BDF
and CDE, and let these circles be tangent to segments DF and DE at M and N , respectively.
Let line M N meet circles ωB and ωC again at P ̸= M and Q ̸= N , respectively. Prove that
M P = N Q.
(Vietnam)

G3 In triangle ABC, point A1 lies on side BC and point B1 lies on side AC. Let P and Q be points
on segments AA1 and BB1 , respectively, such that P Q is parallel to AB. Let P1 be a point on
line P B1 , such that B1 lies strictly between P and P1 , and ∠P P1 C = ∠BAC. Similarly, let Q1 be
the point on line QA1 , such that A1 lies strictly between Q and Q1 , and ∠CQ1 Q = ∠CBA.
Prove that points P, Q, P1 , and Q1 are concyclic.
Proposed by Anton Trygub, Ukraine

G4 Let P be a point inside triangle ABC. Let AP meet BC at A1 , let BP meet CA at B1 , and let
CP meet AB at C1 . Let A2 be the point such that A1 is the midpoint of P A2 , let B2 be the point
such that B1 is the midpoint of P B2 , and let C2 be the point such that C1 is the midpoint of P C2 .
Prove that points A2 , B2 , and C2 cannot all lie strictly inside the circumcircle of triangle ABC.
(Australia)

G5 Let ABCDE be a convex pentagon with CD = DE and ∠EDC ̸= 2 · ∠ADB.

Suppose that a point P is located in the interior of the pentagon such that AP = AE and
BP = BC.
Prove that P lies on the diagonal CE if and only if area (BCD) + area (ADE) = area (ABD) +
area (ABP ).
(Hungary)

G6 Let I be the incentre of acute-angled triangle ABC. Let the incircle meet BC, CA, and AB at
D, E, and F, respectively. Let line EF intersect the circumcircle of the triangle at P and Q, such
that F lies between E and P . Prove that ∠DP A + ∠AQD = ∠QIP .
(Slovakia)

G7 Let I be the incentre of acute triangle ABC with AB ̸= AC. The incircle ω of ABC is tangent
to sides BC, CA, and AB at D, E, and F , respectively. The line through D perpendicular to EF
meets ω at R. Line AR meets ω again at P . The circumcircles of triangle P CE and P BF meet
again at Q.
Prove that lines DI and P Q meet on the line through A perpendicular to AI.
Proposed by Anant Mudgal, India

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G8 Let L be the set of all lines in the plane and let f be a function that assigns to each line ℓ ∈ L a
point f (ℓ) on ℓ. Suppose that for any point X, and for any three lines ℓ1 , ℓ2 , ℓ3 passing through
X, the points f (ℓ1 ), f (ℓ2 ), f (ℓ3 ), and X lie on a circle.
Prove that there is a unique point P such that f (ℓ) = P for any line ℓ passing through P .
Australia

– Number Theory

N1 Find all pairs (k, n) of positive integers such that

k! = (2n − 1)(2n − 2)(2n − 4) · · · (2n − 2n−1 ).

Proposed by Gabriel Chicas Reyes, El Salvador

N2 Find all triples (a, b, c) of positive integers such that a3 + b3 + c3 = (abc)2 .

N3 We say that a set S of integers is rootiful if, for any positive integer n and any a0 , a1 , · · · , an ∈ S,
all integer roots of the polynomial a0 + a1 x + · · · + an xn are also in S. Find all rootiful sets of
integers that contain all numbers of the form 2a − 2b for positive integers a and b.

N4 Find all functions f : Z>0 → Z>0 such that a + f (b) divides a2 + bf (a) for all positive integers a
and b with a + b > 2019.

Let a be a positive integer. We say that a positive integer b is [i]a-good[/i] if an

b − 1 is divisible


N5
by an + 1 for all positive integers n with an ≥ b. Suppose b is a positive integer such that b is
a-good, but b + 2 is not a-good. Prove that b + 1 is prime.
√
N6 Let H = {⌊i 2⌋ : i ∈ Z>0 } = {1, 2, 4, 5, 7, . . . } and let n be a positive integer. Prove that there
√
exists a constant C such that, if A ⊆ {1, 2, . . . , n} satisfies |A| ≥ C n, then there exist a, b ∈ A
such that a − b ∈ H. (Here Z>0 is the set of positive integers, and ⌊z⌋ denotes the greatest
integer less than or equal to z.)

N7 Prove that there is a constant c > 0 and infinitely many positive integers n with the following
property: there are infinitely many positive integers that cannot be expressed as the sum of
fewer than cn log(n) pairwise coprime nth powers.
Canada

N8 Let a and b be two positive integers. Prove that the integer

 2
4a
a2 +
b

is not a square. (Here ⌈z⌉ denotes the least integer greater than or equal to z.)

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Russia

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