is an obvious solution. Now, let 
.
or 
.
.
Y by
Find the integer coefficients after expanding Newton's binomial:
This post has been edited 1 time. Last edited by Cobedangiu, Apr 4, 2025, 6:20 AM
G
Topic
First Poster
Last Poster
Number Theory
fasttrust_12-mn 5
N
by GreekIdiot
Source: Pan African Mathematics Olympiad p6
Find all integers 
for which 
is the square of an integer
for which 
is the square of an integer
5 replies
Maximum number of nice subsets
FireBreathers 0
Given a set 
of natural numbers with elements with odd number. A nonempty subset 
of is called 
if the product of the elements of divisible by the sum of the elements of , but not by its square. It is known that the set itself is good. Determine the maximum number of subsets (including itself).
of natural numbers with elements with odd number. A nonempty subset 
of is called 
if the product of the elements of divisible by the sum of the elements of , but not by its square. It is known that the set itself is good. Determine the maximum number of subsets (including itself).0 replies
Floor double summation
CyclicISLscelesTrapezoid 52
N
by lpieleanu
Source: ISL 2021 A2
Which positive integers make the equation ![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](//latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
make the equation ![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
52 replies
Polynomial
Z_. 1
N
by rchokler
Let 
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation
![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](//latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](http://latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
1 reply
Existence of perfect squares
egxa 2
N
by pavel kozlov
Source: All Russian 2025 10.3
Find all natural numbers 
for which there exists an even natural number 
such that the number
![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](//latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
2 replies
IMO 2014 Problem 4
ipaper 169
N
by YaoAOPS
Let 
and 
be on segment 
of an acute triangle 
such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of 
and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .
Proposed by Giorgi Arabidze, Georgia.
and 
be on segment 
of an acute triangle 
such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of 
and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .Proposed by Giorgi Arabidze, Georgia.
169 replies
Inequalities
Scientist10 1
N
by Bergo1305
If 
, then prove that the following inequality holds:
![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](//latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
, then prove that the following inequality holds:![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
1 reply
Tangents forms triangle with two times less area
NO_SQUARES 1
N
by Luis González
Source: Kvant 2025 no. 2 M2831
Let 
be triangle, inscribed in parabola. Tangents in points 
forms triangle . Prove that 
. (
is area of triangle 
).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
be triangle, inscribed in parabola. Tangents in points 
forms triangle . Prove that 
. (
is area of triangle 
).From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
1 reply
FE solution too simple?
Yiyj1 9
N
by jasperE3
Source: 101 Algebra Problems from the AMSP
Find all functions 
such that the equality 
holds for all pairs of real numbers 
.
I feel like my solution is too simple. Is there something I did wrong or something I missed?
such that the equality 
holds for all pairs of real numbers 
.I feel like my solution is too simple. Is there something I did wrong or something I missed?
9 replies
interesting function equation (fe) in IR
skellyrah 2
N
by jasperE3
Source: mine
find all function F: IR->IR such that 
2 replies
.problem.
G
H
G
H
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#4
Apr 5, 2025, 9:45 AM
Y by
Lankou wrote:
The coefficient is an integer when
Coefficient
integer coefficients?
not integerThis post has been edited 1 time. Last edited by Cobedangiu, Apr 5, 2025, 9:46 AM
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
N Quick Reply
G
H
