A chair set
by truongphatt2668, Jul 31, 2025, 8:20 AM
During an extracurricular activity, there are 100 chairs arranged into two rows facing each other, with each row containing 50 chairs. The teacher assigns 100 students to sit on the 100 chairs, with one student per chair.
(a) Suppose each student only knows the person sitting directly opposite or the person sitting next to them. What is the minimum number of students that need to be selected such that every student knows at least one selected student?
(b) Suppose one student leaves their seat. The teacher then repeatedly performs the following operation: pick any student who is sitting in a row with at least one empty seat and move them to an empty seat. This operation can be performed multiple times. Prove that after a finite number of such operations, the teacher cannot move any student to a seat where they are sitting directly opposite their original seat.
(a) Suppose each student only knows the person sitting directly opposite or the person sitting next to them. What is the minimum number of students that need to be selected such that every student knows at least one selected student?
(b) Suppose one student leaves their seat. The teacher then repeatedly performs the following operation: pick any student who is sitting in a row with at least one empty seat and move them to an empty seat. This operation can be performed multiple times. Prove that after a finite number of such operations, the teacher cannot move any student to a seat where they are sitting directly opposite their original seat.
3-var inquality
by sqing, Jul 31, 2025, 7:44 AM
The result before GMA Problem 575
by mihaig, Jul 31, 2025, 4:54 AM
Given
Prove that
is the least constant
such that
holds true for all
with 

![$\sqrt{\left[\frac{n^2}4\right]}$](http://latex.artofproblemsolving.com/e/3/2/e3254d77401e67774e80634d3d27bab56b92c68f.png)




Prove Equidistance
by Eeightqx, Jul 30, 2025, 5:28 AM
Two circles
and
intersects at one point
. The common tangent further from
is
where
and
. A line parallel to
cuts
at four points
.
cuts
at
. If
, show that the distance from
to
is equal to the distance from
to
.


















This post has been edited 2 times. Last edited by Eeightqx, Yesterday at 1:33 PM
Peru IMO TST 2024
by diegoca1, Jul 25, 2025, 11:33 PM
Determine the minimum possible value of
, where
are positive real numbers satisfying:
![\[
\begin{cases}
x + y + z = 5, \\
\frac{1} {x} + \frac{1} {y} + \frac{1} {z} = 2.
\end{cases}
\]](//latex.artofproblemsolving.com/a/9/c/a9cdf75c556377570410d2eee58ce72267804fa7.png)


![\[
\begin{cases}
x + y + z = 5, \\
\frac{1} {x} + \frac{1} {y} + \frac{1} {z} = 2.
\end{cases}
\]](http://latex.artofproblemsolving.com/a/9/c/a9cdf75c556377570410d2eee58ce72267804fa7.png)
2025 Greece IMO TST P3
by brainfertilzer, Jul 16, 2025, 3:09 AM
Let
be a convex pentagon and let
be the midpoint of
. Suppose that segment
is tangent to the circumcircle of triangle
at
and that
lies on the circumircles of
and
. Lines
and
interesect at
, and lines
and
intersect at
. Points
and
lie on line
so that
.
Prove that lines
and
are concurrent.



















Prove that lines


I'm moving my marbles
by YaoAOPS, Jul 16, 2025, 3:01 AM
Let
and
be positive integers. James has
marbles with weights
,
,
,
. He places them on a balance scale, so that both sides have equal weight. Andrew may move a marble from one side of the scale to the other, so that the absolute difference in weights of the two sides remains at most
.
Find, in terms of
, the minimum positive integer
such that Andrew may make a sequence of moves such that each marble ends up on the opposite side of the scale, regardless of how James initially placed the marbles.








Find, in terms of


This post has been edited 1 time. Last edited by YaoAOPS, Yesterday at 5:48 PM
Iran TST P5
by TheBarioBario, Apr 2, 2022, 9:59 AM
Find all
such that every sequence of integers
which is bounded from below and for all
satisfy
is periodic.
Proposed by Navid Safaei




Proposed by Navid Safaei
This post has been edited 1 time. Last edited by TheBarioBario, Apr 12, 2022, 4:26 PM
Recursive sequence satisfies inequality
by orl, Aug 9, 2008, 2:03 PM
Let
be given, and starting
define recursively:
![\[ a_{n+1} = \left(\frac{a^2_n}{a^2_{n-1}} - 2 \right) \cdot a_n.\]](//latex.artofproblemsolving.com/d/9/b/d9bc5e0d669b8e7deaa6c426edaca778bf11dff9.png)
Show that for all integers
we have: 


![\[ a_{n+1} = \left(\frac{a^2_n}{a^2_{n-1}} - 2 \right) \cdot a_n.\]](http://latex.artofproblemsolving.com/d/9/b/d9bc5e0d669b8e7deaa6c426edaca778bf11dff9.png)
Show that for all integers


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