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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
German-Style System of Equations
Primeniyazidayi 1
N
by Primeniyazidayi
Source: German MO 2025 11/12 Day 1 P1
Solve the system of equations in 
![\begin{align*}
\frac{a}{c} &= b-\sqrt{b}+c \\
\sqrt{\frac{a}{c}} &= \sqrt{b}+1 \\
\sqrt[4]{\frac{a}{c}} &=\sqrt[3]{b}-1
\end{align*}](//latex.artofproblemsolving.com/0/c/d/0cd22d36acedbcb10d76d1b884aa35418273ed52.png)
![\begin{align*}
\frac{a}{c} &= b-\sqrt{b}+c \\
\sqrt{\frac{a}{c}} &= \sqrt{b}+1 \\
\sqrt[4]{\frac{a}{c}} &=\sqrt[3]{b}-1
\end{align*}](http://latex.artofproblemsolving.com/0/c/d/0cd22d36acedbcb10d76d1b884aa35418273ed52.png)
1 reply
gcd nt from switzerland
AshAuktober 5
N
by Siddharthmaybe
Source: Swiss 2025 Second Round
Let 
be positive integers. Prove that the expression
![\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]](//latex.artofproblemsolving.com/1/b/7/1b78a2027c3b7156fbb566901d2cfd088cc09f00.png)
is always a positive integer, and determine all possible values it can take.
be positive integers. Prove that the expression![\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]](http://latex.artofproblemsolving.com/1/b/7/1b78a2027c3b7156fbb566901d2cfd088cc09f00.png)
is always a positive integer, and determine all possible values it can take.
5 replies
Shortlist 2017/G1
fastlikearabbit 92
N
by Ilikeminecraft
Source: Shortlist 2017
Let 
be a convex pentagon such that 
, 
, and 
. Prove that the perpendicular line from 
to and the line segments and 
are concurrent.
be a convex pentagon such that 
, 
, and 
. Prove that the perpendicular line from 
to and the line segments and 
are concurrent.
92 replies
