OLYMPIAD ARENA
INMO HANDOUT
(Aarav Arora)
ALGEBRA
Q1. (MEXICO/2023) Find all functions
integers
,
Q2. (RUSSIA/TST/2015) Let
such that for all positive
and
be positive real numbers. Prove that
Q3. (IRAN/2024) For positive real numbers
such that
. Prove that
Q4. (CHINA/2020) Let
any
such that
and for
Determine the greatest possible value of
Q5. (IMOSL/2023) Professor Oak is feeding his
Pokémon. Each Pokémon has a
bowl whose capacity is a positive real number of kilograms. These capacities are
known to Professor Oak. The total capacity of all the bowls is
kilograms.
Professor Oak distributes
kilograms of food in such a way that each Pokémon
receives a non-negative integer number of kilograms of food (which may be larger
than the capacity of the bowl). The dissatisfaction level of a Pokémon who
received
kilograms of food and whose bowl has a capacity of kilograms is equal
to
.
Find the smallest real number such that, regardless of the capacities of the bowls,
Professor Oak can distribute food in a way that the sum of the dissatisfaction levels
over all the
Pokémon is at most .
Q6. (APMO/2008) Let
and
be given by
.
Prove that
number satisfying
Q7. (USAMO/2019) Let
for any non-integer real
.
be the set of positive integers. A
function
satisfies the equation
for all
positive integers . Given this information, determine all possible values of
Q8. (CGMO/2020) Let
be an integer and
,
.
are arbitrary real
number, find the maximum value of
Q9. (BELARUS/TST/2019) Given a quadratic trinomial
with integer
coefficients such that
is not divisible by for all integers .
Prove that there exist polynomials
and
with integer coefficients such that
Q10. (IMOSL/2022) Let
property that
be a sequence of positive real numbers with the
for all positive integers . Show that
Q11. (IMSOL/2017) Let
, and
be positive integers such that
If
, prove that the
polynomial
has no positive roots.
.
Q12. (APMO/2015) Let
denote the set of integers that are greater
than or equal to . Does there exist a function
such that
Q13. (ELMO/2015) Let
, , and
Q14. (EGMO/2023) There are
be positive integers. Prove that
positive real numbers
each
we let
(here we define to be
be ). Assume that for all and in the range to , we have
if
.
Prove that
. For
and
to
if and only
.
Q15. (CGMO/2009) Let
that
be real numbers greater than or equal to
Prove
Q16. (IMO/2019) Let be the set of integers. Determine all
functions
such that, for all integers and ,
Q17. (USAMO/2014) Let , , ,
zeros
and
be real numbers such that
and all
of the polynomial
are
real. Find the smallest value the product
take.
can
Q18. (IMOSL/2010) Let the real numbers
relations
and
satisfy the
Prove that
Q19. (INMO/2023) Suppose
polynomial for each
in
are positive reals. Consider the following
:
where
indices are taken modulo
, i.e.,
that it is impossible that each of these
for any in
. Show
polynomials has all its roots real.
Q20. (IRAN/2019) Find all function
number
, if
such that for any three real
:
NUMBER THEORY
Q1. (ELMOSL/2024) Given a positive integer whose baseis
for some integer
integers
, where
, a move consists of selecting some
, such that the digits
are not all , erasing them
from , and replacing them with a divisor of
same number of digits as
representation
(this divisor need not have the
).
Prove that for all sufficiently large even integers , we may apply some sequence of
moves to to transform it into
.
Q2. (CROATIA/TST/2016) Find all pairs
of prime numbers such that
Q3. (APMO/2023) Find all integers satisfying
and
which
denotes the sum of all positive divisors of , and
prime divisor of .
Q4. (IMO/2017) For each integer
sequence
for
as
exists a number
such that
, in
denotes the largest
, define the
Determine all values of such that there
for infinitely many values of .
Q5. (IMOSL/2021) Find all positive integers
pair
of positive integers, such that
such that there exists a
is not divisible by the cube of
any prime, and
Q6. (IMOSL/2023) Let
the
products
be
positive integers such that
form a strictly
increasing arithmetic progression in that order. Determine the smallest possible
integer that could be the common difference of such an arithmetic progression.
Q7. (USAMO/TST/2021) Determine all integers
positive integers , , , such
that
and divides
for which there exist
.
Q8. (EGMO/2023) We are given a positive integer
. For each positive
integer , we define its twist as follows: write as
, where
are nonnegative integers and
, then
. For the positive integer , consider
the infinite sequence
where
and
is the twist of for each
positive integer .
Prove that this sequence contains if and only if the remainder when is divided
by
is either or .
Q9. (INDIA-IRAN/2024) Prove that there are no integers
satisfying the
equation
Q10. (IRAN/2024) Given a sequence
of positive integers, Ali proceed
the following algorithm: In the i-th step he marks all rational numbers in the
interval
which have denominator equal to . Then he write down the
number equal to the length of the smallest interval in
which both two ends
of that is a marked number. Find all sequences
with
and such
that for all
we have
Q11. (CGMO/2020) Find all the real number sequences
and
satisfy the following conditions:
(i) For any positive integer ,
;
(ii) For any positive integer ,
and
is the two roots of the
equation
.
Q12. (MEXICO/2024) Determine all pairs
1.
that
of integers that satisfy both:
2. There exists a natural number such that the numbers and
are
consecutive divisors of , in that order.
Note: Two positive integers
are consecutive divisors of , in that order, if there
is no divisor of
such that
.
Q13. (ELMOSL/2018) Consider infinite sequences
of positive integers
satisfying
and
for all positive
integers and For a given positive integer
find the maximum possible value
of
Q14. (USAMO/2024) Find all integers
such that the following property
holds: if we list the divisors of in increasing order
as
, then we have
Q15. (BALKAN/2024) Let
divisible by
and
. Prove that
be distinct positive integers such that
is
.
Q16. (KOREA/2024) On a blackboard, there are
numbers written:
. Nahyun repeatedly performs the following operations.
(Operation) Nahyun chooses two numbers from the 10 numbers on the blackboard
that are not in a divisor-multiple relationship, erases them, and writes their GCD
and LCM on the blackboard.
If every two numbers on the blackboard form a divisor-multiple relationship,
Nahyun stops the process. What is the maximum number of operations Nahyun can
perform?
(Note:
are in a divisor-multiple relationship iff
Q17. (IMO/2013) Assume that
exist positive integers
or
.)
and are two positive integers. Prove that there
such that
Q18. (VIETNAM/2024) or each positive integer , let
positive divisors of .
be the number of
a) Find all positive integers such that
.
b) Prove that there exist infinitely many positive integers such that there are
exactly two positive integers
satisfying
Q19. (SPAIN/2022) Find all triples
, satisfying simultaneously that
.
of positive integers, with
Q20. (APMO/2012) Determine all the pairs
positive integer
for which
is an integer.
of a prime number
and a
GEOMETRY
Q1. (PAKISTAN/TST/2017) Let
be a cyclic quadrilateral. The
diagonals
and
meet at , and
and
meet at . Suppose
is
perpendicular to
. Let be the midpoint of
. Prove that
is perpendicular
to
.
Q2. (IMO/2012) Given triangle
the point is the centre of the excircle
opposite the vertex This excircle is tangent to the side
at , and to the
lines
and
at and , respectively. The lines
and
meet at , and
the lines
and
meet at
Let be the point of intersection of the
lines
and
, and let be the point of intersection of the
lines
and
Prove that
is the midpoint of
Q3. (IMOTC/2024) Let
be an acute-angled triangle with
, incentre , and let
be the midpoint of major arc
. Suppose the
perpendicular line from to segment
meets lines
,
, and
at points
, , and
respectively. Prove that the
median line in
passes through
the circumcentre of
.
Q4. (IMOTC/2024) Let
be an acute-angled triangle with
, and
let
be its circumcentre and orthocentre respectively. Points
lie on
segments
respectively, such that
The
perpendicular line from
to line
meets lines
and
at
respectively.
Let the tangents to the circumcircle of
at points and meet at point
. Prove that
are concyclic.
Q5. (APMO/2022) Let
be a right triangle with
. Point lies on
the line
such that is between and . Let be the midpoint of
and
let be the second intersection point of the circumcircle of
and the
circumcircle of
. Prove that as varies, the line
passes through a fixed
point.
Q6. (IMO/2023) Let
be an acute-angled triangle with
. Let be
the circumcircle of
. Let be the midpoint of the arc
of containing
. The perpendicular from to
meets
at and meets again at
. The line through parallel to
meets line
at . Denote the circumcircle of
triangle
by . Let meet again at
. Prove that the line tangent
to at meets line
on the internal angle bisector of
.
Q7. (IMO/2021) Let be a circle with centre , and
a convex quadrilateral
such that each of the segments
and
is tangent to . Let be the
circumcircle of the triangle
. The extension of
beyond meets at , and
the extension of
beyond meets at . The extensions
of
and
beyond meet at and , respectively. Prove that
Q8. (IGO/INTER/2024) Let
be an acute triangle with a point on
side
. Let be a point on side
such that
, and be the
circumcircle of triangle
. The line
intersects again at a point
, and is the feet of the altitude from to
.
Prove that if
, then the line perpendicular to
through is tangent
to .
Q9. (ELMO/2023) Let
be an acute scalene triangle with orthocenter
. Line
intersects
at and line
intersects
at . Let be the foot
of the perpendicular from to the line through parallel to
. Point
lies on
line
such that
that
is parallel to
is parallel to
, and point
. Prove that points
Q10. (USAMO/2023) In an acute triangle
. Let be the foot of the perpendicular from
circumcircle of triangle
intersects line
. Let
be the midpoint of
,
lies on line
,
,
such
are concyclic.
, let
be the midpoint of
to
. Suppose that the
at two distinct points and
. Prove that
.
Q11. (EGMO/2021) Let
be a triangle with incenter and let be an
arbitrary point on the side
. Let the line through perpendicular
to
intersect
at . Let the line through perpendicular
to
intersect
at . Prove that the reflection of across the line
lies on
the line
.
Q12. (IMO/2020) Consider the convex quadrilateral
interior of
. The following ratio equalities hold:
. The point
is in the
Prove that
the following three lines meet in a point: the internal bisectors of
angles
and
and the perpendicular bisector of segment
.
Q13. (IRAN/2024) et
be a parallelogram and let
and
altitudes from to
, respectively. A line
bisects
meets
at
. Similarly, a line
bisects
and
meets
at
. Show that the circumcircles of
and
tangent to each other
be the
and
are
Q14. (IMOSL/2023) Let
be a triangle with
let be the
circumcircle of
and let be its radius. Point is chosen on
such
that
and point is the foot of the perpendicular from to
. Ray
meets again at . Point is chosen on line
such
that
and
lie on a line in that order. Finally, let be a point
satisfying
and
. Prove that
lies on .
Q15. (IMOSL/2022) In the acute-angled triangle
, the point is the foot of
the altitude from , and is a point on the segment
. The lines
through parallel to
and
meet
at and , respectively.
Points
and
lie on the circles
and
, respectively, such
that
and
.
Prove that
and are concyclic
Q16. (USAMO/2019) Let
satisfying
point on side
satisfying
be a cyclic quadrilateral
. The diagonals of
intersect at . Let be a
. Show that line
bisects
.
Q17. (ELMOSL/2024) Let
be a triangle, and let
be centered at
,
and tangent to line
at , respectively. Let line
intersect again
at
and let line
intersect again at . If is the other intersection of the
circumcircles of triangles
and
, then prove that lines
,
, and
either concur or are all parallel.
Q18. (SHARYGIN/2024) Let
be a non-isosceles triangle, be its incircle.
Let
and be the points at which the incircle of
touches the
sides
and
respectively. Let
be the point on ray
such
that
. Let
be the point on ray
such that
. Let the
circumcircles of
and
intersect again at and respectively.
Prove that
and
concur.
Q19. (SHARYGIN/2024) Let
and
be the bisectors of a
triangle
. The segments
and
meet at point . Let be the
projection of to
. Points and on sides
and
respectively are such
that
. Prove that
.
Q20. (IMOSL/2017)
Let be the circumcenter of an acute triangle
. Line
intersects the
altitudes of
through and at and , respectively. The altitudes meet
at . Prove that the circumcenter of triangle
lies on a median of
triangle
.
COMBINATORICS
Q1. (BRAZIL/2024) A partition of a set is a family of non-empty subsets of
, such that any two distinct subsets in the family are disjoint, and the union of all
subsets equals . We say that a partition of a set of integers is separated if each
subset in the partition does not contain consecutive integers. Prove that, for every
positive integer , the number of partitions of the set
number of separated partitions of the set
For example,
hand,
since
is equal to the
.
is a separated partition of the set
. On the other
is a partition of the same set, but it is not separated
contains consecutive integers.
Q2. (BULGARIA/2022)
A white equilateral triangle with side length
is divided into equilateral
triangles with side (cells) by lines parallel to the sides of . We'll call two
cells
if they have a common vertex. Ivan colours some of the cells in black.
Without knowing which cells are black, Peter chooses a set of cells and Ivan tells
him the parity of the number of black cells in . After knowing this, Peter is able to
determine the parity of the number of
cells of different colours. Find all
possible cardinalities of such that this is always possible independent of how Ivan
chooses to colour the cells.
Q3. (CANADA/2021) At a dinner party there are hosts and guests, seated
around a circular table, where
. A pair of two guests will chat with one
another if either there is at most one person seated between them or if there are
exactly two people between them, at least one of whom is a host. Prove that no
matter how the
people are seated at the dinner party, at least pairs of guests
will chat with one another.
Q4. (FRANCE/TST/2012) Let and be two positive integers. Consider a group
of people such that, for each group of people, there is a
-th person that
knows them all (if knows then knows ).
1) If
, prove that there exists a person who knows all others.
2) If
, give an example of such a group in which no-one knows all others.
Q5. (HONGKONG/TST/2024) Given is an
board, with an integer written in
each grid. For each move, I can choose any grid, and add to all
numbers in
its row and column. Find the largest
, such that for any initial choice of
integers, I can make a finite number of moves so that there are at least
even
numbers on the board.
Q6. (IRAN/2022) We have many
subsets of a
set. We
know that the union of every of them has at least
elements. Find the most
possible value for the number of these subsets.
Q7. (KAZAKHASTAN/2024) Given an integer
. The board
is colored
white and black in a chess-like manner. We call any non-empty set of different cells
of the board as a figure. We call figures and
similar, if can be obtained
from
by a rotation with respect to the center of the board by an angle multiple
of
and a parallel transfer. (Any figure is similar to itself.) We call a
figure connected if for any cells
there is a sequence of
cells
such that
,
, and also and
have a common
side for each
. Find the largest possible value of such that for any
connected figure consisting of cells, there are figures
similar to such
that has more white cells than black cells and
has more black cells than white
cells in it.
Q8. (MEXICO/2023) Let
be a positive integer. For every number
from to , there is a card with this number and which is either black or white.
A magician can repeatedly perform the following move: For any two tiles with
different number and different colour, he can replace the card with the smaller
number by one identical to the other card.
For instance, when
and the initial configuration is
, the magician can choose
on the first move to
obtain
and then
on the second move to
obtain
.
Determine in terms of all possible lengths of sequences of moves from any possible
initial configuration to any configuration in which no more move is possible.
Q9. (USAMO/2010) There are students standing in a circle, one behind the other.
The students have heights
. If a student with height is
standing directly behind a student with height
or less, the two students are
permitted to switch places. Prove that it is not possible to make more than
switches before reaching a position in which no further switches are possible.
such
Q10. (RUSSIA/2024) Let and be positive integers. On a straight line, white
segments and black segments are given, with
and
. Suppose that
no two segments of the same colour intersect (and do not have common ends).
Moreover, suppose that for any choice of white segments and black segments,
some pair of selected segments will intersect. Prove
that
.
Q11. (IRAN-SINGAPORE-TAIWAN/2022) Let be the set of lattice points
whose both coordinates are positive integers no larger than
. i.e.,
. We put a card with one gold side
and one black side on each point in . We call a rectangle "good" if:
(i) All of its sides are parallel to the axes and have positive integer coordinates no
larger than
.
(ii) The cards on its top-left and bottom-right corners are showing gold, and the
cards on its top-right and bottom-left corners are showing black.
Each "move" consists of choosing a good rectangle and flipping all cards
simultaneously on its four corners. Find the maximum possible number of moves one
can perform, or show that one can perform infinitely many moves.
Q12. (THAILAND/2024) In a table with
rows and
columns, each cell is
colored either purple or yellow. Suppose that for each yellow cell ,
Where
is the number of purple cells that lie in the same row as , and
is
the number of purple cells that lie in the same column as .
Find the least possible number of cells that are colored purple.
Q13.(TURKEY/TST/2023) Let
have elements. For every
values of
when
and
be a positive integer and
and
we have
be a set of sets which
. Find all
.
Note:
Q14. (VIETNAM/2021) A student divides all marbles into boxes
numbered
(after being divided, there may be a box with no marbles).
a) How many ways are there to divide marbles into boxes (are two different ways if
there is a box with a different number of marbles)?
b) After dividing, the student paints those marbles by a number of colors (each
with the same color, one color can be painted for many marbles), so that there are
no marbles in the same box. have the same color and from any boxes it is
impossible to choose marbles painted in colors. Prove that for every division, the
student must use no less than colors to paint the marbles.
c) Show a division so that with exactly colors the student can paint the marbles
that satisfy the conditions in question b).
Q15. (IMO/2003) Let
set
be a
-element subset of the
. Prove that there exist numbers
,
in such that the sets
are pairwise disjoint.
Q16. (IMO/2022) The Bank of Oslo issues two types of coin: aluminium (denoted A)
and bronze (denoted B). Marianne has aluminium coins and bronze coins
arranged in a row in some arbitrary initial order. A chain is any subsequence of
consecutive coins of the same type. Given a fixed positive integer
, Gilberty
repeatedly performs the following operation: he identifies the longest chain
containing the
coin from the left and moves all coins in that chain to the left end
of the row. For example, if
and
, the process starting from the
ordering
would
be
Find all pairs
with
such that for every initial ordering, at some
moment during the process, the leftmost coins will all be of the same type.
Q17. (ELMOSL/2024) Let
be distinct positive integers. In an infinite grid
of unit squares, each square is filled with exactly one real number so that



In each
square, the sum of the numbers in the
cells is equal.
In each
square, the sum of the numbers in the cells is equal.
There exist two cells in the grid that do not contain the same number.
Let be the set of numbers that appear in at least one square on the grid. Find, in
terms of
and , the least possible value of .
Q18. (ELMOSL/2023) Elmo has 2023 cookie jars, all initially empty. Every day, he
chooses two distinct jars and places a cookie in each. Every night, Cookie Monster
finds a jar with the most cookies and eats all of them. If this process continues
indefinitely, what is the maximum possible number of cookies that the Cookie
Monster could eat in one night?
Q19. (APMO/2022) Let and be positive integers. Cathy is playing the following
game. There are marbles and boxes, with the marbles labelled to . Initially,
all marbles are placed inside one box. Each turn, Cathy chooses a box and then
moves the marbles with the smallest label, say , to either any empty box or the box
containing marble
. Cathy wins if at any point there is a box containing only
marble .
Determine all pairs of integers
such that Cathy can win this game.
Q20. (EGMO/2023) Turbo the snail sits on a point on a circle with circumference
. Given an infinite sequence of positive real numbers
, Turbo
successively crawls distances
around the circle, each time choosing to
crawl either clockwise or counterclockwise.
Determine the largest constant
with the following property: for every
sequence of positive real numbers
with
for all , Turbo can
(after studying the sequence) ensure that there is some point on the circle that it will
never visit or crawl across.