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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
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
n + k are composites for all nice numbers n, when n+1, 8n+1 both squares
parmenides51 3
N
by Nuran2010
Source: 2022 Saudi Arabia JBMO TST 1.1
The positive 
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers 
?
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers 
?
3 replies
Divisibility NT
reni_wee 2
N
by reni_wee
Source: Iran 1998
Suppose that 
and 
are natural numbers such that
is a prime number. Find all possible values of ,,.
and 
are natural numbers such that
is a prime number. Find all possible values of ,,.
2 replies
Aslı tries to make the amount of stones at every unit square is equal
AlperenINAN 0
Source: Turkey JBMO TST 2025 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
