Balkan MO Shortlist 2021

A1 Find all functions f : R+ → R and g : R+ → R such that

f (x2 + y 2 ) = g(xy)

holds for all x, y ∈ R+ .

A2 Find all functions f : R → R such that

1
f (x2 + y) ≥ ( + 1)f (y)
x
holds for all x ∈ R \ {0} and all y ∈ R.

A3 Find all functions f : R+ → R+ , such that f (x + f (x) + f (y)) = 2f (x) + y for all positive reals
x, y.

A4 Let f, g be functions from the positive integers to the integers. Vlad the impala is jumping around
the integer grid. His initial position is x0 = (0, 0), and for every n ≥ 1, his jump is
xn − xn−1 = (±f (n), ±g(n)) or (±g(n), ±f (n)),
with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to
his initial location (0, 0) infinitely many times when
(a) f, g are polynomials with integer coefficients?
(b) f, g are any pair of functions from the positive integers to the integers?

A5 Find all functions f : R+ → R+ such that

f (xf (x + y)) = yf (x) + 1

holds for all x, y ∈ R+ .

Proposed by Nikola Velov, North Macedonia

A6 Find all functions f : R → R such that

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

holds for all x, y ∈ R.

C1 Let An be the set of n-tuples x = (x1 , ..., xn ) with xi ∈ {0, 1, 2}. A triple x, y, z of distinct elements
of An is called good if there is some i such that {xi , yi , zi } = {0, 1, 2}. A subset A of An is called
good if every three distinct elements of A form a good triple.
Prove that every good subset of An has at most 2( 32 )n elements.

C2 Let K and N > K be fixed positive integers. Let n be a positive integer and let a1 , a2 , ..., an be
distinct integers. Suppose that whenever m1 , m2 , ..., mn are integers, not all equal to 0, such that
| mi |≤ K for each i, then the sum
X n
mi ai
i=1

is not divisible by N . What is the largest possible value of n?

Proposed by Ilija Jovcevski, North Macedonia

C3 In an exotic country, the National Bank issues coins that can take any value in the interval [0, 1].
Find the smallest constant c > 0 such that the following holds, no matter the situation in that
country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more
than 1000, can split those coins into 100 boxes, such that the total value inside each box is at
most c.[/i]

C4 A sequence of 2n + 1 non-negative integers a1 , a2 , ..., a2n+1 is given. There’s also a sequence of

2n + 1 consecutive cells enumerated from 1 to 2n + 1 from left to right, such that initially the
number ai is written on the i-th cell, for i = 1, 2, ..., 2n + 1. Starting from this initial position, we
repeat the following sequence of steps, as long as it’s possible:
Step 1: Add up the numbers written on all the cells, denote the sum as s.
Step 2: If s is equal to 0 or if it is larger than the current number of cells, the process terminates.
Otherwise, remove the s-th cell, and shift shift all cells that are to the right of it one position to
the
left. Then go to Step 1.
Example: (1, 0, 1, 2, 0) → (1, 0, 1, 0) → (1, 1, 0) → (1, 0) → (0).
A sequence a1 , a2 , ..., a2n+1 of non-negative integers is called balanced, if at the end of this
process there’s exactly one cell left, and it’s the cell that was initially enumerated by (n + 1),
i.e. the cell that was initially in the middle.
Find the total number of balanced sequences as a function of n.
Proposed by Viktor Simjanoski, North Macedonia

C5 Problem 4. Angel has a warehouse, which initially contains 100 piles of 100 pieces of rubbish
each. Each morning, Angel performs exactly one of the following moves:
 (a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.
What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse?

C6 There is a population P of 10000 bacteria, some of which are friends (friendship is mutual),
so that each bacterion has at least one friend and if we wish to assign to each bacterion a
coloured
membrane so that no two friends have the same colour, then there is a way to do it with 2021
colours, but not with 2020 or less.
Two friends A and B can decide to merge in which case they become a single bacterion whose
friends are precisely the union of friends of A and B. (Merging is not allowed if A and B are
not friends.) It turns out that no matter how we perform one merge or two consecutive merges,
in the resulting population it would be possible to assign 2020 colours or less so that no two
friends have the same colour. Is it true that in any such population P every bacterium has at
least 2021 friends?

G1 Let ABC be a triangle with AB < AC < BC. On the side BC we consider points D
and E such that BA = BD and CE = CA. Let K be the circumcenter of triangle ADE and
let F , G be the points of intersection of the lines AD, KC and AE, KB respectively. Let ω1 be
the circumcircle of triangle KDE, ω2 the circle with center F and radius F E, and ω3 the circle
with center G and radius GD.
Prove that ω1 , ω2 , and ω3 pass through the same point and that this point of intersection lies on
the line AK.

G2 Let I and O be the incenter and the circumcenter of a triangle ABC, respectively, and let sa be
the exterior bisector of angle ∠BAC. The line through I perpendicular to IO meets the
lines BC and sa at points P and Q, respectively. Prove that IQ = 2IP .

G3 Let ABC be a triangle with AB < AC. Let ω be a circle passing through B, C and assume that
A is inside ω. Suppose X, Y lie on ω such that ∠BXA = ∠AY C. Suppose also that X and C lie
on opposite sides of the line AB and that Y and B lie on opposite sides of the line AC. Show
that, as X, Y vary on ω, the line XY passes through a fixed point.

G4 Let ABC be a right-angled triangle with ∠BAC = 90◦ . Let the height from A cut its side BC at
D. Let I, IB , IC be the incenters of triangles ABC, ABD, ACD respectively. Let also EB, EC
be the excenters of ABC with respect to vertices B and C respectively. If K is the
point of intersection of the circumcircles of EC IBI and EB ICI , show that KI passes through
 the midpoint M of side BC.

G5 Let ABC be an acute triangle with AC > AB and circumcircle Γ. The tangent from A
to Γ intersects BC at T . Let M be the midpoint of BC and let R be the reflection of A in B.
Let S be a point so that SABT is a parallelogram and finally let P be a point on line SB such
that M P is parallel to AB.
Given that P lies on Γ, prove that the circumcircle of 4ST R is tangent to line AC.
Proposed by Sam Bealing, United Kingdom

G6 Let ABC be an acute triangle such that AB < AC. Let ω be the circumcircle of ABC
and assume that the tangent to ω at A intersects the line BC at D. Let Ω be the circle with
center D and radius AD. Denote by E the second intersection point of ω and Ω. Let M be the
midpoint of BC. If the line BE meets Ω again at X, and the line CX meets Ω for the second
time at Y , show that A, Y , and M are collinear.
Proposed by Nikola Velov, North Macedonia

G7 Let ABC be an acute scalene triangle. Its C-excircle tangent to the segment AB meets AB at
point M and the extension of BC beyond B at point N . Analogously, its B-excircle
tangent to the segment AC meets AC at point P and the extension of BC beyond C at point Q.
Denote by A1 the intersection point of the lines M N and P Q, and let A2 be defined as the
point, symmetric to A with respect to A1 . Define the points B2 and C2 , analogously. Prove
that 4ABC is similar to 4A2 B2 C2 .

G8 Let ABC be a scalene triangle and let I be its incenter. The projections of I on BC, CA,
and AB are D, E and F respectively. Let K be the reflection of D over the line AI, and let L
be the second point of intersection of the circumcircles of the triangles BF K and CEK. If
3 BC = AC − AB, prove that DE = 2KL.
1

N1 Let n ≥ 2 be an integer and let

 
a1 + a2 + ... + ak
M= : 1 ≤ k ≤ n and 1 ≤ a1 < . . . < ak ≤ n
k
be the set of the arithmetic means of the elements of all non-empty subsets of {1, 2, ..., n}. Find
min{|a − b| : a, b ∈ M with a 6= b}.

N2 Denote by l(n) the largest prime divisor of n. Let an+1 = an + l(an ) be a recursively
defined sequence of integers with a1 = 2. Determine all natural numbers m such that there
exists some i ∈ N with ai = m2 .
Proposed by Nikola Velov, North Macedonia
N3 Let n be a positive integer. Determine, in terms of n, the greatest integer which divides
every number of the form p + 1, where p ≡ 2 mod 3 is a prime number which does not divide n.

N4 Can every positive rational number q be written as

a2021 + b2023
,
c2022 + d2024
where a, b, c, d are all positive integers?
Proposed by Dominic Yeo, UK

N5 A natural number n is given. Determine all (n − 1)-tuples of nonnegative integers a1 , a2 , ..., an−1
such that
m 2m + a1 22 m + a2 23 m + a3 2n−1 m + an−1
b c+b n c+b n c+b n c + ... + b c=m
2n
−1 2 −1 2 −1 2 −1 2n − 1
holds for all m ∈ Z.

N6 Let a, b and c be positive integers satisfying the equation (a, b) + [a, b] = 2021c . If |a − b| is a
prime number, prove that the number (a + b)2 + 4 is composite.

N7 A super-integer triangle is defined to be a triangle whose lengths of all sides and at least
one height are positive integers. We will deem certain positive integer numbers to be good with
the condition that if the lengths of two sides of a super-integer triangle are two (not necessarily
different) good numbers, then the length of the remaining side is also a good number. Let 5 be
a good number. Prove that all integers larger than 2 are good numbers.