2^a + 3^b + 1 = 6^c
by togrulhamidli2011, Mar 16, 2025, 12:34 PM
Find all positive integers (a, b, c) such that:
![\[
2^a + 3^b + 1 = 6^c
\]](//latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
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.)
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 
.
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
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 numbersIMO 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.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](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
![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](//latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
Prove that they are converges and find their limits.
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.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
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
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 .
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
, 
, 
, ... 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
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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