Own made functional equation
by JARP091, May 31, 2025, 4:10 PM
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
This post has been edited 1 time. Last edited by JARP091, 2 hours ago
Inequality conjecture
by RainbowNeos, May 29, 2025, 1:03 PM
Show (or deny) that there exists an absolute constant 
that, for all 
and positive real numbers 
, there is
![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](//latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
that, for all 
and positive real numbers 
, there is![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
Serbian selection contest for the IMO 2025 - P6
by OgnjenTesic, May 22, 2025, 4:07 PM
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
such that 
coprime, 
and 
.Reflected point lies on radical axis
by Mahdi_Mashayekhi, Apr 19, 2025, 11:01 AM
Given is an acute and scalene triangle 
with circumcenter 
. 
and 
intersect the altitude from 
to 
at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
with circumcenter 
. 
and 
intersect the altitude from 
to 
at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.This post has been edited 1 time. Last edited by Mahdi_Mashayekhi, Apr 19, 2025, 11:25 AM
Inequality
by SunnyEvan, Apr 1, 2025, 9:54 AM
Let 
, 
, 
be non-negative real numbers, no two of which are zero. Prove that :
, 
, 
be non-negative real numbers, no two of which are zero. Prove that :
2- player game on a strip of n squares with two game pieces
by parmenides51, Mar 26, 2024, 3:48 PM
Alice and Bob play a game on a strip of 
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares.
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.
For which can Bob ensure a win no matter how Alice plays?
For which can Alice ensure a win no matter how Bob plays?
(Karl Czakler)
squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of 
squares.The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.
For which
can Bob ensure a win no matter how Alice plays?For which
can Alice ensure a win no matter how Bob plays?(Karl Czakler)
This post has been edited 2 times. Last edited by parmenides51, Mar 26, 2024, 3:50 PM
Polynomial Application Sequences and GCDs
by pieater314159, Jun 19, 2019, 8:08 PM
Let 
be a polynomial with integer coefficients such that 
, and let 
be an integer. Define 
and 
for all integers 
. Show that there are infinitely many positive integers such that 
.
Proposed by Milan Haiman and Carl Schildkraut
be a polynomial with integer coefficients such that 
, and let 
be an integer. Define 
and 
for all integers 
. Show that there are infinitely many positive integers such that 
.Proposed by Milan Haiman and Carl Schildkraut
This post has been edited 2 times. Last edited by pieater314159, Jun 27, 2019, 9:45 PM
equal segments on radiuses
by danepale, Apr 25, 2016, 6:39 PM
Let be an acute triangle with circumcenter . Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
be an acute triangle with circumcenter . Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.Incenters and Circles
by rkm0959, Nov 2, 2014, 9:24 AM
In a triangle 
with incenter 
,
Let
= 
= incenter of 
= incenter of 
= intersection of 
and 
= intersection of 
and 
= midpoint of 
Prove that
concycle
with incenter 
,Let
= 
= incenter of 
= incenter of 
= intersection of 
and 
= intersection of 
and 
= midpoint of 
Prove that
concycleThis post has been edited 1 time. Last edited by rkm0959, Jun 7, 2015, 2:40 AM
March 2009
February 2009
January 2009