AoPS Community 2024 ISL

IMO Shortlist 2024

www.artofproblemsolving.com/community/c3376654
by quantam13, EthanWYX2009, EpicBird08, sqing-inequality-BUST, MarkBcc168, EeEeRUT, IndoMathXdZ,
YaoAOPS, Tintarn, KevinYang2.71, RaymondZhu, LLL2019, Twoisaprime, thdnder, vincentwant, OronSH,
sarjinius, brainfertilzer, peppapig , Scilyse

– Algebra

A1 Determine all real numbers α such that, for every positive integer n, the integer

⌊α⌋ + ⌊2α⌋ + · · · + ⌊nα⌋

is a multiple of n. (Note that ⌊z⌋ denotes the greatest integer less than or equal to z. For example,
⌊−π⌋ = −4 and ⌊2⌋ = ⌊2.9⌋ = 2.)
Proposed by Santiago Rodrı́guez, Colombia

A2 Let n be a positive integer. Find the minimum possible value of

S = 20 x20 + 21 x21 + · · · + 2n x2n ,

where x0 , x1 , . . . , xn are nonnegative integers such that x0 + x1 + · · · + xn = n.

A3 Decide whether for every sequence (an ) of positive real numbers,

3a1 +3a2 +···+3an 1
(2a1 +2a2 +···+2an )2
< 2024

is true for at least one positive integer n.

A4 Let Z>0 be the set of all positive integers. Determine all subsets S of {20 , 21 , 22 , . . .} for which
there exists a function f : Z>0 → Z>0 such that

S = {f (a + b) − f (a) − f (b) | a, b ∈ Z>0 }.

Pitchayut Saengrungkongka, Thailand

A5 Find all periodic sequence a1 , a2 , . . . of real numbers such that the following conditions hold for
all n ⩾ 1:
an+2 + a2n = an + a2n+1 and |an+1 − an | ⩽ 1.

Proposed by Dorlir Ahmeti, Kosovo

© 2025 AoPS Incorporated 1

AoPS Community 2024 ISL

A6 Let a0 , a1 , a2 , . . . be an infinite strictly increasing sequence of positive integers such that for
each n ⩾ 1 we have
 
an−1 + an+1 √
an ∈ , an−1 · an+1 .
2
Let b1 , b2 , . . . be an infinite sequence of letters defined as
(
A if an = 21 (an−1 + an+1 )
bn =
G otherwise.
Prove that there exist positive integers n0 and d such that for all n ⩾ n0 we have bn+d = bn .

A7 Let Q be the set of rational numbers. A function f : Q → Q is called aquaesulian if the following
property holds: for every x, y ∈ Q,

f (x + f (y)) = f (x) + y or f (f (x) + y) = x + f (y).

Show that there exists an integer c such that for any aquaesulian function f there are at most
c different rational numbers of the form f (r) + f (−r) for some rational number r, and find the
smallest possible value of c.

A8 Let p ̸= q be coprime positive integers. Determine all infinite sequences a1 , a2 , . . . of positive

integers such that the following conditions hold for all n ≥ 1:

max(an , an+1 , . . . , an+p ) − min(an , an+1 , . . . , an+p ) = p

max(an , an+1 , . . . , an+q ) − min(an , an+1 , . . . , an+q ) = q

– Combinatorics

C1 Let n be a positive integer. A class of n students run n races, in each of which they are ranked
with no draws. A student is eligible for a rating (a, b) for positive integers a and b if they come
in the top b places in at least a of the races. Their final score is the maximum possible value of
a − b across all ratings for which they are eligible.
Find the maximum possible sum of all the scores of the n students.
Proposed by William Steinberg, Australia

C2 Let n be a positive integer. The integers 1, 2, 3, . . . , n2 are to be written in the cells of an n × n

board such that each integer is written in exactly one cell and each cell contains exactly one
integer. For every integer d with d | n, the d-division of the board is the division of the board into

© 2025 AoPS Incorporated 2

AoPS Community 2024 ISL

(n/d)2 nonoverlapping sub-boards, each of size d × d, such that each cell is contained in exactly
one d × d sub-board.
We say that n is a cool number if the integers can be written on the n × n board such that, for
each integer d with d | n and 1 < d < n, in the d-division of the board, the sum of the integers
written in each d × d sub-board is not a multiple of d.
Determine all even cool numbers.
Proposed by Melek Güngör, Türkiye

C3 Let n be a positive integer. There are 2n knights sitting at a round table. They consist of n pairs
of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to
each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number
of exchanges of adjacent knights such that, regardless of the initial arrangement, every knight
can meet her partner and shake hands at some time.

C4 Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden
monsters in 2022 of the cells. Initially, Turbo does not know where any of the monsters are, but
he knows that there is exactly one monster in each row except the first row and the last row,
and that each column contains at most one monster.
Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he
chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing
a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a
monster, his attempt ends and he is transported back to the first row to start a new attempt. The
monsters do not move, and Turbo remembers whether or not each cell he has visited contains
a monster. If he reaches any cell in the last row, his attempt ends and the game is over.
Determine the minimum value of n for which Turbo has a strategy that guarantees reaching the
last row on the n-th attempt or earlier, regardless of the locations of the monsters.
Proposed by Cheuk Hei Chu, Hong Kong

C5 Let N be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers
1, 2, . . . , N on a board. They then take turns to make a move, starting with Geoff. Each move
consists of choosing a pair of integers (k, n), where k ≥ 0 and n is one of the integers on the
board, and then erasing every integer s on the board such that 2k | n − s. The game continues
until the board is empty. The player who erases the last integer on the board loses.
Determine all values of N for which Geoff can ensure that he wins, no matter how Ceri plays.

C6 Let n and T be positive integers. James has 4n marbles with weights 1, 2, . . . , 4n. He places
them on a balance scale, so that both sides have equal weight. Andrew may move a marble
from one side of the scale to the other, so that the absolute difference in weights of the two
sides remains at most T .

© 2025 AoPS Incorporated 3

AoPS Community 2024 ISL

Find, in terms of n, the minimum positive integer T such that Andrew may make a sequence
of moves such that each marble ends up on the opposite side of the scale, regardless of how
James initially placed the marbles.

C7 Let a1 , a2 , a3 , . . . be an infinite sequence of positive integers, and let N be a positive integer.

Suppose that, for each n > N , an is equal to the number of times an−1 appears in the list
a1 , a2 , . . . , an−1 .
Prove that at least one of the sequence a1 , a3 , a5 , . . . and a2 , a4 , a6 , . . . is eventually periodic.
(An infinite sequence b1 , b2 , b3 , . . . is eventually periodic if there exist positive integers p and M
such that bm+p = bm for all m ≥ M .)

C8 Let n be a positive integer. Given an n × n board, the unit cell in the top left corner is initially
coloured black, and the other cells are coloured white. We then apply a series of colouring oper-
ations to the board. In each operation, we choose a 2 × 2 square with exactly one cell coloured
black and we colour the remaining three cells of that 2 × 2 square black.
Determine all values of n such that we can colour the whole board black.
Proposed by Eduardo Aragon, Peru

– Geometry

G1 Let ABCD be a cyclic quadrilateral such that AC < BD < AD and ∠DBA < 90◦ . Point E lies
on the line through D parallel to AB such that E and C lie on opposite sides of line AD, and
AC = DE. Point F lies on the line through A parallel to CD such that F and C lie on opposite
sides of line AD, and BD = AF .
Prove that the perpendicular bisectors of segments BC and EF intersect on the circumcircle
of ABCD.
Proposed by Mykhailo Shtandenko, Ukraine

G2 Let ABC be a triangle with AB < AC < BC. Let the incenter and incircle of triangle ABC be I
and ω, respectively. Let X be the point on line BC different from C such that the line through X
parallel to AC is tangent to ω. Similarly, let Y be the point on line BC different from B such that
the line through Y parallel to AB is tangent to ω. Let AI intersect the circumcircle of triangle
ABC at P ̸= A. Let K and L be the midpoints of AC and AB, respectively.
Prove that ∠KIL + ∠Y P X = 180◦ .
Proposed by Dominik Burek, Poland

G3 Let ABCDE be a convex pentagon and let M be the midpoint of AB. Suppose that segment
AB is tangent to the circumcircle of triangle CM E at M and that D lies on the circumircles

© 2025 AoPS Incorporated 4

AoPS Community 2024 ISL

of AM E and BM C. Lines AD and M E interesect at K, and lines BD and M C intersect at L.

Points P and Q lie on line EC so that ∠P DC = ∠EDQ = ∠ADB.
Prove that lines KP, LQ, and M D are concurrent.

G4 Let ABCD be a quadrilateral with AB parallel to CD and AB < CD. Lines AD and BC intersect
at a point P . Point X distinct from C lies on the circumcircle of triangle ABC such that P C =
P X. Point Y distinct from D lies on the circumcircle of triangle ABD such that P D = P Y .
Lines AX and BY intersect at Q.
Prove that P Q is parallel to AB.
Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine

G5 Let ABC be a triangle with incentre I, and let Ω be the circumcircle of triangle BIC. Let K be a
point in the interior of segment BC such that ∠BAK < ∠KAC. The angle bisector of ∠BKA
intersects Ω at points W and X such that A and W lie on the same side of BC, and the angle
bisector of ∠CKA intersects Ω at points Y and Z such that A and Y lie on the same side of
BC.
Prove that ∠W AY = ∠ZAX.
Proposed by David Brodsky, Uzbekistan

G6 Let ABC be an acute triangle with AB < AC, and let Γ be the circumcircle of ABC. Points X
and Y lie on Γ so that XY and BC meet on the external angle bisector of ∠BAC. Suppose that
the tangents to Γ at x and Y intersect at a point T on the same side of BC as A, and that T X
and T Y intersect BC at U and V , respectively. Let J be the centre of the excircle of triangle
T U V opposite the vertex T .
Prove that AJ bisects ∠BAC.

G7 Let ABC be a triangle with incenter I such that AB < AC < BC. The second intersections of
AI, BI, and CI with the circumcircle of triangle ABC are MA , MB , and MC , respectively. Lines
AI and BC intersect at D and lines BMC and CMB intersect at X. Suppose the circumcircle
of triangles XMB MC and XBC intersect again at S ̸= X. Lines BX and CX intersect the
circumcircle of triangle SXMA again at P ̸= X and Q ̸= X, respectively.
Prove that the circumcenter of triangle SID lies on P Q.
Proposed by Thailand

G8 Let ABC be a triangle with AB < AC < BC, and let D be a point in the interior of segment
BC. Let E be a point on the circumcircle of triangle ABC such that A and E lie on opposite
sides of the line BC and ∠BAD = ∠EAC. Let I, IB , IC , JB and JC be the incenters of triangles
ABC, ABD, ADC, ABE, and AEC, respectively. Prove that IB , IC , JB , and JC are concyclic if
and only if AI, IB JC , and JB IC concur.

© 2025 AoPS Incorporated 5

AoPS Community 2024 ISL

Proposed by Haozhe Yang, Canada

– Number Theory

N1 Find all positive integers n with the following property: for all positive divisors d of n, we have
d + 1 | n or d + 1 is prime.

N2 Determine all finite, nonempty sets S of positive integers such that for every a, b ∈ S there exists
c ∈ S with a | b + 2c.

N3 Determine all sequences a1 , a2 , . . . of positive integers such that for any pair of positive integers
m ⩽ n, the arithmetic and geometric means
am + am+1 + · · · + an 1
and (am am+1 · · · an ) n−m+1
n−m+1
are both integers.

N4 Determine all pairs (a, b) of positive integers for which there exist positive integers g and N such
that
gcd(an + b, bn + a) = g
holds for all integers n ⩾ N. (Note that gcd(x, y) denotes the greatest common divisor of inte-
gers x and y.)
Proposed by Valentio Iverson, Indonesia

N5 Let S be a finite nonempty set of prime numbers. Let 1 = b1 < b2 < . . . be the sequence of
all positive integers whose prime divisors all belong to S. Prove that, for all but finitely many
positive integers n, there exist positive integers a1 , a2 , . . . , an such that
 
a1 a2 an 1 1 1
+ + ··· + = + + ··· +
b1 b2 bn b1 b2 bn

N6 Let n be a positive integer. We say a polynomial P with integer coefficients is n-good if there
exists a polynomial Q of degree 2 with integer coefficients such that Q(k)(P (k) + Q(k)) is never
divisible by n for any integer k.
Determine all integers n such that every polynomial with integer coefficients is an n-good poly-
nomial

N7 Let Z>0 denote the set of positive integers. Let f : Z>0 → Z>0 be a function satisfying the
following property: for m, n ∈ Z>0 , the equation

f (mn)2 = f (m2 )f (f (n))f (mf (n))

© 2025 AoPS Incorporated 6

AoPS Community 2024 ISL

holds if and only if m and n are coprime.

For each positive integer n, determine all the possible values of f (n).

© 2025 AoPS Incorporated 7

Art of Problem Solving is an ACS WASC Accredited School.