IMO 2025 Comments
by Primeniyazidayi, Jul 16, 2025, 10:40 AM
What do you think about IMO 2025 exam problems and IMO 2024 Shortlist problems?
IMO
L
I miss Turbo
by sarjinius, Jul 16, 2025, 3:20 AM
Consider a
grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.

Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.
quadrilateral geo with length conditions
by OronSH, Jul 16, 2025, 3:13 AM
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
This post has been edited 1 time. Last edited by OronSH, 5 hours ago
orang NT
by KevinYang2.71, Jul 16, 2025, 3:00 AM
Find all positive integers
with the following property: for all positive divisors
of
, we have
or
is prime.





This post has been edited 1 time. Last edited by KevinYang2.71, Today at 3:17 AM
I am [not] a parallelogram
by peppapig_, Jul 16, 2025, 3:00 AM
Let
be a quadrilateral with
parallel to
and
. Lines
and
intersect at a point
. Point
distinct from
lies on the circumcircle of triangle
such that
. Point
distinct from
lies on the circumcircle of triangle
such that
. Lines
and
intersect at
.
Prove that
is parallel to
.
Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine


















Prove that


Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine
This post has been edited 6 times. Last edited by peppapig_, Today at 5:00 AM
Modula sequencia
by EeEeRUT, Jul 16, 2025, 12:57 AM
Find all ordered pair of positive integers
such that the sequence of non-negative integers
with
and
is bounded.
Note that for any positive integers
and
, define
to be a unique integer
such that
divides
.




Note that for any positive integers






This post has been edited 1 time. Last edited by EeEeRUT, Today at 12:58 AM
D1052 : How it's possible ?
by Dattier, Jul 15, 2025, 4:02 PM
Is it true for all
natural integer :
?
E=163999081217965835070356295641931525591357567735624606830696386586994976172813816839262877
50922205042989559280142547393636527346272001339597483126086699049357460700008119117240578043
46281799731794620614941989125738298381362079843446150841376016501310942563338531951229469261
017554376486801
F=666165351866558215458553224258271230186695252725433417706426521946436303489813878149680284
24137540935869820330945301911165461108917069581547697978809314332789769417680417107249591988
71252061235265894516611712110379162326930843580773177773789047845826833190774483296276708089
431095213731040922452939281280


E=163999081217965835070356295641931525591357567735624606830696386586994976172813816839262877
50922205042989559280142547393636527346272001339597483126086699049357460700008119117240578043
46281799731794620614941989125738298381362079843446150841376016501310942563338531951229469261
017554376486801
F=666165351866558215458553224258271230186695252725433417706426521946436303489813878149680284
24137540935869820330945301911165461108917069581547697978809314332789769417680417107249591988
71252061235265894516611712110379162326930843580773177773789047845826833190774483296276708089
431095213731040922452939281280
Bonza functions
by KevinYang2.71, Jul 15, 2025, 3:38 AM
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
.


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


Determine the smallest real constant




This post has been edited 6 times. Last edited by KevinYang2.71, Yesterday at 7:30 AM
Beetles on heights. If two on incircle, then third is also!
by NO_SQUARES, Jan 20, 2025, 6:02 PM
At the same time, three beetles with identical speeds began to crawl along the heights of an acute-angled non-isosceles triangle from its vertices. At some point, it turned out that the first and second beetles were on a circle inscribed in a triangle. Prove that at this moment the third beetle is also on this circle.
A. Kuznetsov
A. Kuznetsov
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