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,
,
Prove that at least one of
, 
, 
is not greater than 
.I assume that all are greater than it, but can't go more.
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Shortest number theory you might've seen in your life
AlperenINAN 2
N
by zuat.e
Source: Turkey JBMO TST 2025 P4
Let 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
2 replies
Is this even algebra or geometry
Sadigly 3
N
by Moon_settler
Source: Azerbaijan Junior NMO 2019
Alice creates the graphs 
and 
, where 
. She observes that these two graphs and 
axis divides the positive side of the plane (
) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral.
and 
, where 
. She observes that these two graphs and 
axis divides the positive side of the plane (
) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral.
3 replies
Prove or disprove the existence of such a
Sadigly 0
Source: Azerbaijan NMO
A positive number 
is given, such that could be expressed as difference of two perfect squares (
). Is it possible for 
to be expressed as difference of two perfect squares?
is given, such that could be expressed as difference of two perfect squares (
). Is it possible for 
to be expressed as difference of two perfect squares?
0 replies
Points on the sides of cyclic quadrilateral satisfy the angle conditions
AlperenINAN 1
N
by ehuseyinyigit
Source: Turkey JBMO TST 2025 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 
.
1 reply
Writing quadratic trinomials inside cells
Sadigly 0
Source: Azerbaijan Junior NMO 2019
A 
square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are 
coefficents in total, and these coefficents are chosen from the set ![$[-66;47]$](//latex.artofproblemsolving.com/d/d/9/dd92777f4ba30299d3326b285f1f73656aba544b.png)
, and each coefficient is different from each other. Prove that there exists at least one column such that the sum of the six trinomials in that column has a real root.
square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are 
coefficents in total, and these coefficents are chosen from the set ![$[-66;47]$](http://latex.artofproblemsolving.com/d/d/9/dd92777f4ba30299d3326b285f1f73656aba544b.png)
, and each coefficient is different from each other. Prove that there exists at least one column such that the sum of the six trinomials in that column has a real root.
0 replies
Product of consecutive terms divisible by a prime number
BR1F1SZ 1
N
by IndoMathXdZ
Source: 2025 Francophone MO Seniors P4
Determine all sequences of strictly positive integers 
satisfying the following two conditions:
[list]
[*]There exists an integer
such that, for all indices 
, 
.
[*]For any prime number and for any index , the number
![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](//latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of .
[/list]
satisfying the following two conditions:[list]
[*]There exists an integer
such that, for all indices 
, 
.[*]For any prime number
and for any index , the number![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of .[/list]
1 reply
1 viewing
Minimum value of a 3 variable expression
bin_sherlo 3
N
by ehuseyinyigit
Source: Türkiye 2025 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.
3 replies
Incenter is the foot of altitude
Sadigly 0
Source: Azerbaijan JBMO TST 2023
Let 
be a triangle and let 
denote the circumcircle of . The foot of altitude from 
to 
is 
. The foot of altitudes from to and 
are 
, respectively. Let 
intersect at 
, and let 
intersect at 
. Prove that is the incenter of triangle 
be a triangle and let 
denote the circumcircle of . The foot of altitude from 
to 
is 
. The foot of altitudes from to and 
are 
, respectively. Let 
intersect at 
, and let 
intersect at 
. Prove that is the incenter of triangle 
0 replies
System of equations in juniors' exam
AlperenINAN 1
N
by AlperenINAN
Source: Turkey JBMO TST 2025 P3
Find all positive real solutions 
to the following system:
to the following system:
1 reply
reals associated with 1024 points
bin_sherlo 0
Source: Türkiye 2025 JBMO TST P8
Pairwise distinct points 
, which lie on a circle, are marked by distinct reals 
. Let 
be 
good for a 
on the circle different than , if and only if 
is the greatest number on at least one of the two arcs 
. Let the score of be the number of good points on the circle. Determine the greatest 
such that regardless of the values of , there exists a point with score at least .
, which lie on a circle, are marked by distinct reals 
. Let 
be 
good for a 
on the circle different than , if and only if 
is the greatest number on at least one of the two arcs 
. Let the score of be the number of good points on the circle. Determine the greatest 
such that regardless of the values of , there exists a point with score at least .
0 replies
Three variables inequality
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#1
Apr 20, 2025, 6:58 AM
, ,Prove that at least one of
, , is not greater than .I assume that all are greater than it, but can't go more.
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#2
Apr 20, 2025, 7:20 AM
Y by
Can you give the reference? I think this is from a USAJMO contest in the 2010s.
EDIT: I was thinking of USA(J)MO 2018 #1 (#2) which can be rearranged to a similar inequality.
EDIT: I was thinking of USA(J)MO 2018 #1 (#2) which can be rearranged to a similar inequality.
This post has been edited 2 times. Last edited by programjames1, Apr 20, 2025, 7:23 AM
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#3
Apr 20, 2025, 9:01 AM
Y by
I guess that this problem is from chinese materials at 1990~2004. but I'm not sure. Thanks.
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,
from 
respectively leaves 
unchanged while reducing 
.
.![\[
(a - b)^2 - \frac{a^2 + b^2 + c^2}{2} = (a - b)^2 - \frac{a^2 + b^2 + (-a - b)^2}{2} = -3ab.
\]](http://latex.artofproblemsolving.com/1/d/7/1d7bb30efffe37945c9be6eecf3c5d852dbef4b3.png)
Moreover,![\[
(-3ab)(-3bc)(-3ca) = -27a^2b^2c^2 \leq 0,
\]](http://latex.artofproblemsolving.com/9/4/f/94f237fbafe3fd5e9c091d94e31ee539e748f40f.png)
which implies that they cannot all be greater than 0.
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get minimum at 
iff 
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