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Aslı tries to make the amount of stones at every unit square is equal
AlperenINAN 0
Source: Turkey JBMO TST P2
Let 
be a positive integer. Aslı and Zehra are playing a game on an 
grid. Initially, 
stones are placed on some of the unit squares of this grid.
On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.
For which values of, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
be a positive integer. Aslı and Zehra are playing a game on an 
grid. Initially, 
stones are placed on some of the unit squares of this grid.On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square.
For which values of
, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
0 replies
Minimum value of a 3 variable expression
bin_sherlo 2
N
by Tamam
Source: Türkiye JBMO TST P6
Find the minimum value of
![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](//latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
2 replies
ISI UGB 2025 P8
SomeonecoolLovesMaths 5
N
by MathematicalArceus
Source: ISI UGB 2025 P8
Let 
and let 
be positive integers such that 
. Prove that 
and determine when equality holds.
and let 
be positive integers such that 
. Prove that 
and determine when equality holds.
5 replies
2n^2+4n-1 and 3n+4 have common powers
bin_sherlo 1
N
by Burmf
Source: Türkiye JBMO TST P5
Find all positive integers such that a positive integer power of 
equals to a positive integer power of 
.
such that a positive integer power of 
equals to a positive integer power of 
.
1 reply
Trigo relation in a right angled. ISIBS2011P10
Sayan 9
N
by sanyalarnab
Show that the triangle whose angles satisfy the equality
![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](//latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
is right angled.
![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](http://latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
is right angled.
9 replies
Pentagon with given diameter, ratio desired
bin_sherlo 0
Source: Türkiye JBMO TST P7
is a pentagon whose vertices lie on circle 
where 
. Let 
and 
intersect at 
, 
meet 
at 
. 
is the midpoint of arc 
on , not containing 
. If 
holds, then what is the value of 
?
0 replies
Points on the sides of cyclic quadrilateral satisfy the angle conditions
AlperenINAN 0
Source: Turkey JBMO TST P1
Let 
be a cyclic quadrilateral and let the intersection point of lines 
and 
be 
. Let the points 
and 
be arbitrary points on sides and respectively, which satisfy the conditions
Prove that 
.
be a cyclic quadrilateral and let the intersection point of lines and 
be 
. Let the points 
and 
be arbitrary points on sides and respectively, which satisfy the conditions
Prove that 
.
0 replies
ISI UGB 2025 P2
SomeonecoolLovesMaths 4
N
by MathsSolver007
Source: ISI UGB 2025 P2
If the interior angles of a triangle 
satisfy the equality, 
prove that the triangle must have a right angle.
satisfy the equality, 
prove that the triangle must have a right angle.
4 replies
IMO ShortList 1998, combinatorics theory problem 5
orl 47
N
by mathwiz_1207
Source: IMO ShortList 1998, combinatorics theory problem 5
In a contest, there are 
candidates and judges, where 
is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most 
candidates. Prove that ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](//latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
candidates and judges, where 
is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most 
candidates. Prove that ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](http://latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
47 replies
Cyclic equality implies equal sum of squares
blackbluecar 34
N
by Markas
Source: 2021 Iberoamerican Mathematical Olympiad, P4
Let 
be real numbers such that
![\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]](//latex.artofproblemsolving.com/7/e/b/7ebfb2f663d390714310d3c43b9c68e1b101d632.png)
Show that
.
be real numbers such that![\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]](http://latex.artofproblemsolving.com/7/e/b/7ebfb2f663d390714310d3c43b9c68e1b101d632.png)
Show that
.
34 replies
