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,
,![\[ f(x)f(yf(x))=f(x+y). \]](http://latex.artofproblemsolving.com/5/6/0/560049dd70944cafa0423e5f1229d215f78215a0.png)
Two players want to obtain a number divisible by 2023
a_507_bc 3
N
by fathalishah
Source: All-Russian MO 2023 Final stage 11.5
Initially, 
ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some 
numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by 
. If in 
moves on the board a number divisible by 
appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?
ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some 
numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by 
. If in 
moves on the board a number divisible by 
appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?
3 replies
Points on a lattice path lies on a line
navi_09220114 1
N
by pbornsztein
Source: TASIMO 2025 Day 1 Problem 3
Let 
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to . Prove that for each positive integer 
there is a line in the plane (possibly different lines for different ) which contains at least points of .
be a nonempty subset of the points in the Cartesian plane such that for each 
exactly one of 
or 
also belongs to . Prove that for each positive integer 
there is a line in the plane (possibly different lines for different ) which contains at least points of .
1 reply
Functional inequality
Jackson0423 2
N
by nitride
Show that there does not exist a function 
such that for all positive real numbers 
,
![\[
f^2(x) \geq f(x+y)\left(f(x) + y\right).
\]](//latex.artofproblemsolving.com/7/1/f/71fb0d51492ee4c3b2bf4049e8224882fec16020.png)
such that for all positive real numbers 
,![\[
f^2(x) \geq f(x+y)\left(f(x) + y\right).
\]](http://latex.artofproblemsolving.com/7/1/f/71fb0d51492ee4c3b2bf4049e8224882fec16020.png)
2 replies
Find all integers
velmurugan 3
N
by grupyorum
Source: Titu and Dorin Book Problem
Find all positive integers 
such that 
is a divisor of 
.
such that 
is a divisor of 
.
3 replies
Graph Process Problem
Maximilian113 10
N
by Ru83n05
Source: CMO 2025 P1
The 
players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left.
The players will update their votes via a series of rounds. In one round, each player
updates their vote, one at a time, according to the following procedure: At the time of the update, if is voting for 
and 
is voting for 
then updates their vote to 
(Note that 
and 
need not be distinct; if 
then 's vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds).
They repeat this updating procedure for rounds. Prove that at this time, all players will unanimously vote for the same person.
players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left.The players will update their votes via a series of rounds. In one round, each player
updates their vote, one at a time, according to the following procedure: At the time of the update, if is voting for 
and 
is voting for 
then updates their vote to 
(Note that 
and 
need not be distinct; if 
then 's vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds).They repeat this updating procedure for
rounds. Prove that at this time, all players will unanimously vote for the same person.
10 replies
Congrats to former two perfect scorer in IMO
mszew 0
Source: Where should it be posted?
Congrats to the new president of Romania...Mr. Nicuşor Dan
https://en.wikipedia.org/wiki/Nicu%C8%99or_Dan
https://www.imo-official.org/participant_r.aspx?id=1571
https://en.wikipedia.org/wiki/Nicu%C8%99or_Dan
https://www.imo-official.org/participant_r.aspx?id=1571
0 replies
Austrian Regional MO 2025 P4
BR1F1SZ 3
N
by LeYohan
Source: Austrian Regional MO
Let 
be a positive integer that is not divisible by 
. Furthermore, let 
be a positive integer. Prove that none of the numbers of the form 
is a square number.
(Walther Janous)
be a positive integer that is not divisible by 
. Furthermore, let 
be a positive integer. Prove that none of the numbers of the form 
is a square number.(Walther Janous)
3 replies
Nice concurrency
navi_09220114 3
N
by sami1618
Source: TASIMO 2025 Day 1 Problem 2
Four points 
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to 
. Points 
and 
lie on segments 
and 
respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to 
. Points 
and 
lie on segments 
and 
respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
3 replies
system in R+, four equations/variables
jasperE3 2
N
by Yiyj
Source: Bulgaria 1972 P2
Solve the system of equations:
if the following conditions are satisfied: 
, 
, 
.
H. Lesov
if the following conditions are satisfied: 
, 
, 
.H. Lesov
2 replies
