One old Long List No.1
by prof., May 29, 2025, 7:57 AM
A circle is inscribed in a rhombus. In each corner of the rhombus, a circle is inscribed such that it touches two sides of the rhombus and inscribed circle. These corner circles have radii
and
and the radius of the inscribed circle is
. If
and
are natural numbers and
, find the value of
and
.








4 var inequality
by SunnyEvan, May 29, 2025, 7:07 AM
Inspired by a9opsow_
by sqing, May 29, 2025, 1:49 AM
Let
Prove that
Where 







This post has been edited 2 times. Last edited by sqing, Today at 2:28 AM
Inspired by Adhyayan Jana
by sqing, May 28, 2025, 2:38 AM
Let
aand
Prove that
Let
aand
Prove that
Let
aand
Prove that 









This post has been edited 4 times. Last edited by sqing, Yesterday at 2:59 AM
Cup of Combinatorics
by M11100111001Y1R, May 27, 2025, 7:24 AM
There are
cups labeled
, where the
-th cup has capacity
liters. In total, there are
liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most
steps.
Prove that in at most
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.









This post has been edited 1 time. Last edited by M11100111001Y1R, May 27, 2025, 7:26 AM
an equation from the a contest
by alpha31415, May 21, 2025, 11:23 AM
Find all (complex) roots of the equation:
(z^2-z)(1-z+z^2)^2=-1/7
(z^2-z)(1-z+z^2)^2=-1/7
This post has been edited 1 time. Last edited by alpha31415, May 21, 2025, 11:24 AM
Reason: correct the problem
Reason: correct the problem
Shortest number theory you might've seen in your life
by AlperenINAN, May 11, 2025, 7:51 PM
Let
and
be prime numbers. Prove that if
is a perfect square, then
is also a perfect square.




This post has been edited 3 times. Last edited by AlperenINAN, May 12, 2025, 10:09 AM
Reason: Typo
Reason: Typo
Equation with primes
by oVlad, Jan 8, 2024, 6:50 PM
Let
and
be prime numbers such that
where
Find all possible values of 





IMO ShortList 1999, geometry problem 2
by orl, Nov 13, 2004, 11:47 PM
A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.
This post has been edited 1 time. Last edited by orl, Nov 14, 2004, 10:19 PM
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