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Regional, national, and international math olympiads
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Greatest algebra ever
EpicBird08 13
N
18 minutes ago
by Assassino9931
Source: ISL 2024/A2
Let
be a positive integer. Find the minimum possible value of
where
are nonnegative integers such that
.

![\[
S = 2^0 x_0^2 + 2^1 x_1^2 + \dots + 2^n x_n^2,
\]](http://latex.artofproblemsolving.com/7/b/8/7b818fccfea8f1dcb9f80bde51dba224f91c3713.png)


13 replies


Hard sequence
straight 0
19 minutes ago
Source: Own
Consider a sequence
of real numbers.
Consider an infinite
grid
. In the first row of this grid, we place
in every square (
. In the first column of this grid, we place
in the
-th square (
.
Next, fill up the grid according to the following rule:
.
If
for all
, does this mean that
for all
?
Hint?

Consider an infinite







Next, fill up the grid according to the following rule:

If




Hint?
If you think this helps and if i'm not mistaken,
.

0 replies
The inekoalaty game
sarjinius 26
N
40 minutes ago
by straight
Source: 2025 IMO P5
Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number
which is known to both players. On the
th turn of the game (starting with
) the following happens:
[list]
[*] If
is odd, Alice chooses a nonnegative real number
such that
[*]If
is even, Bazza chooses a nonnegative real number
such that
[/list]
If a player cannot choose a suitable
, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.
Determine all values of
for which Alice has a winning strategy and all those for which Bazza has a winning strategy.
Proposed by Massimiliano Foschi and Leonardo Franchi, Italy



[list]
[*] If


![\[
x_1 + x_2 + \cdots + x_n \le \lambda n.
\]](http://latex.artofproblemsolving.com/f/5/e/f5e4f751946952d021736c6eafa0b7ec02d02055.png)


![\[
x_1^2 + x_2^2 + \cdots + x_n^2 \le n.
\]](http://latex.artofproblemsolving.com/f/f/4/ff40463336f4c5dc7381b14e71abe7d45b78a1c0.png)
If a player cannot choose a suitable

Determine all values of

Proposed by Massimiliano Foschi and Leonardo Franchi, Italy
26 replies

BMO Shortlist 2021 G6
Lukaluce 23
N
an hour ago
by Rayvhs
Source: BMO Shortlist 2021
Let
be an acute triangle such that
. Let
be the circumcircle of 
and assume that the tangent to
at
intersects the line
at
. Let
be the circle with
center
and radius
. Denote by
the second intersection point of
and
. Let
be the
midpoint of
. If the line
meets
again at
, and the line
meets
for the second
time at
, show that
, and
are collinear.
Proposed by Nikola Velov, North Macedonia




and assume that the tangent to





center






midpoint of






time at



Proposed by Nikola Velov, North Macedonia
23 replies
quadrilateral geo with length conditions
OronSH 9
N
an hour ago
by player-019
Source: IMO Shortlist 2024 G1
Let
be a cyclic quadrilateral such that
and
. Point
lies on the line through
parallel to
such that
and
lie on opposite sides of line
, and
. Point
lies on the line through
parallel to
such that
and
lie on opposite sides of line
, and
.
Prove that the perpendicular bisectors of segments
and
intersect on the circumcircle of
.
Proposed by Mykhailo Shtandenko, Ukraine

















Prove that the perpendicular bisectors of segments



Proposed by Mykhailo Shtandenko, Ukraine
9 replies
k What happened to 2025 IMO P4 post?
sarjinius 3
N
an hour ago
by AltruisticApe
Contest is over, why deleted?
3 replies
1 viewing
Periodic sequence
EeEeRUT 5
N
an hour ago
by dangerousliri
Source: Isl 2024 A5
Find all periodic sequence
of real numbers such that the following conditions hold for all
:
Proposed by Dorlir Ahmeti, Kosovo



Proposed by Dorlir Ahmeti, Kosovo
5 replies

Bonza functions
KevinYang2.71 51
N
an hour ago
by Oznerol1
Source: 2025 IMO P3
Let
denote the set of positive integers. A function
is said to be bonza if
for all positive integers
and
.
Determine the smallest real constant
such that
for all bonza functions
and all positive integers
.
Proposed by Lorenzo Sarria, Colombia


![\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]](http://latex.artofproblemsolving.com/7/b/b/7bb6b2b4fcde900ee6f7ef90b906525d61312796.png)


Determine the smallest real constant




Proposed by Lorenzo Sarria, Colombia
51 replies

Next term is sum of three largest proper divisors
vsamc 3
N
an hour ago
by KevinYang2.71
Source: 2025 IMO P4
A proper divisor of a positive integer
is a positive divisor of
other than
itself.
The infinite sequence
consists of positive integers, each of which has at least three proper divisors. For each
, the integer
is the sum of the three largest proper divisors of
.
Determine all possible values of
.



The infinite sequence




Determine all possible values of

3 replies

2024 International Math Olympiad Number Theory Shortlist, Problem 3
brainfertilzer 10
N
2 hours ago
by vsamc
Source: 2024 ISL N3
Determine all sequences
of positive integers such that for any pair of positive integers
, the arithmetic and geometric means
are both integers.


![\[ \frac{a_m + a_{m+1} + \cdots + a_n}{n-m+1}\quad\text{and}\quad (a_ma_{m+1}\cdots a_n)^{\frac{1}{n-m+1}}\]](http://latex.artofproblemsolving.com/2/0/b/20b6f3097b2a8fa8e3fbed54af1f7f892f4ca2fc.png)
10 replies
