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First Poster
Last Poster
satisfying
Prove
Cup of Combinatorics
M11100111001Y1R 1
N
by Davdav1232
Source: Iran TST 2025 Test 4 Problem 2
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.
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.
1 reply
Bulgaria National Olympiad 1996
Jjesus 7
N
by reni_wee
Find all prime numbers 
for which 
divides 
.
for which 
divides 
.
7 replies
Can't be power of 2
shobber 31
N
by LeYohan
Source: APMO 1998
Show that for any positive integers 
and 
, 
cannot be a power of 
.
and 
, 
cannot be a power of 
.
31 replies
Brilliant Problem
M11100111001Y1R 4
N
by IAmTheHazard
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences 
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:
![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](//latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
4 replies
Own made functional equation
Primeniyazidayi 1
N
by Primeniyazidayi
Source: own(probably)
Find all functions 
such that 
for all 
such that 
for all 
1 reply
not fun equation
DottedCaculator 13
N
by Adywastaken
Source: USA TST 2024/6
Find all functions 
such that for all real numbers 
and 
,
![\[f(xf(y))+f(y)=f(x+y)+f(xy).\]](//latex.artofproblemsolving.com/d/2/f/d2ff6ed448cf39e7ec9bce6b944965d5e89b9878.png)
Milan Haiman
such that for all real numbers 
and 
,![\[f(xf(y))+f(y)=f(x+y)+f(xy).\]](http://latex.artofproblemsolving.com/d/2/f/d2ff6ed448cf39e7ec9bce6b944965d5e89b9878.png)
Milan Haiman
13 replies
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic 12
N
by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
For an 
table filled with natural numbers, we say it is a divisor table if:
- the numbers in the
-th row are exactly all the divisors of some natural number 
,
- the numbers in the
-th column are exactly all the divisors of some natural number 
,
-
for every 
.
A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
Proposed by Pavle Martinović
table filled with natural numbers, we say it is a divisor table if:- the numbers in the
-th row are exactly all the divisors of some natural number 
,- the numbers in the
-th column are exactly all the divisors of some natural number 
,-
for every 
.A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.Proposed by Pavle Martinović
12 replies
Geometry with fix circle
falantrng 33
N
by zuat.e
Source: RMM 2018 Problem 6
Fix a circle 
, a line 
to tangent , and another circle 
disjoint from such that and lie on opposite sides of . The tangents to from a variable point 
on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
, a line 
to tangent , and another circle 
disjoint from such that and lie on opposite sides of . The tangents to from a variable point 
on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
33 replies
USAMO 2001 Problem 2
MithsApprentice 54
N
by lpieleanu
Let 
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides 
and 
, respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by 
the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex 
is denoted by 
. Prove that 
.
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides 
and 
, respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by 
the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex 
is denoted by 
. Prove that 
.
54 replies
