2^a + 3^b + 1 = 6^c
by togrulhamidli2011, Mar 16, 2025, 12:34 PM
Unsolved Diophantine(I think)
by Nuran2010, Mar 14, 2025, 4:41 PM
Find all solutions for the equation
where
is a positive integer and
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)



Variable point on the median
by MarkBcc168, Jun 11, 2019, 12:23 AM
Let
be a scalene triangle with circumcircle
. Let
be the midpoint of
. A variable point
is selected in the line segment
. The circumcircles of triangles
and
intersect
again at points
and
, respectively. The lines
and
intersect (a second time) the circumcircles to triangles
and
at
and
, respectively. Prove that as
varies, the circumcircle of
passes through a fixed point
distinct from
.





















Bosnia and Herzegovina JBMO TST 2013 Problem 1
by gobathegreat, Sep 16, 2018, 8:21 PM
It is given
positive integers. Product of any one of them with sum of remaining numbers increased by
is divisible with sum of all
numbers. Prove that sum of squares of all
numbers is divisible with sum of all
numbers





IMO 2014 Problem 1
by Amir Hossein, Jul 8, 2014, 12:17 PM
Let
be an infinite sequence of positive integers. Prove that there exists a unique integer
such that
![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](//latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.


![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
Proposed by Gerhard Wöginger, Austria.
This post has been edited 1 time. Last edited by v_Enhance, Nov 5, 2023, 5:17 PM
Reason: missing < sign
Reason: missing < sign
Sequences and limit
by lehungvietbao, Jan 3, 2014, 10:32 AM
Let
be two positive sequences defined by
and
for all
.
Prove that they are converges and find their limits.


![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)

Prove that they are converges and find their limits.
IMO 2012 P5
by mathmdmb, Jul 11, 2012, 7:03 PM
Let
be a triangle with
, and let
be the foot of the altitude from
. Let
be a point in the interior of the segment
. Let
be the point on the segment
such that
. Similarly, let
be the point on the segment
such that
. Let
be the point of intersection of
and
.
Show that
.
Proposed by Josef Tkadlec, Czech Republic















Show that

Proposed by Josef Tkadlec, Czech Republic
This post has been edited 3 times. Last edited by Eternica, Jun 19, 2024, 10:03 AM
One secuence satisfying condition
by hatchguy, Sep 4, 2011, 12:18 AM
Prove that there exists only one infinite secuence of positive integers
with
,
and
for all positive integers
.





Can this sequence be bounded?
by darij grinberg, Jan 19, 2005, 11:00 AM
Let
,
,
, ... be an infinite sequence of real numbers satisfying the equation
for all
, where
and
are two different positive reals.
Can this sequence
,
,
, ... be bounded?
Proposed by Mihai Bălună, Romania







Can this sequence



Proposed by Mihai Bălună, Romania
This post has been edited 1 time. Last edited by djmathman, Sep 27, 2015, 2:12 PM
To share with readers my favorite problem I came across today :) (Shout for contrib.)
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