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Diophantine equation !
ComplexPhi 6
N
by Mr.Sharkman
Source: Romania JBMO TST 2015 Day 1 Problem 4
Solve in nonnegative integers the following equation :
6 replies
Cool sequence problem
AlephG_64 1
N
by FarrukhKhayitboyev
Source: 2025 Finals Portuguese Mathematical Olympiad P3
A computer science teacher has asked his students to write a program that, given a list of 
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.
Next, the teacher asked Alexandre to run the program for a list of
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.Next, the teacher asked Alexandre to run the program for a list of
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.1 reply
such that 
Prove that 
Geometry Problem
Itoz 2
N
by Itoz
Source: Own
Given 
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.
Prove that
is tangent to 
.
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.Prove that
is tangent to 
.
2 replies
1 viewing
IMO Shortlist 2014 N6
hajimbrak 27
N
by cj13609517288
Let 
be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](//latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .
Proposed by Serbia
be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .Proposed by Serbia
27 replies
FE inequality from Iran
mojyla222 3
N
by amir_ali
Source: Iran 2025 second round P5
Find all functions 
such that for all 
such that for all 
3 replies
binomial sum ratio
thewayofthe_dragon 2
N
by P162008
Source: YT
Someone please evaluate this ratio inside the log for any given n(I feel the sum doesn't have any nice closed form).
2 replies
Maximum Area of a triangle formed by 3 Lines
Kunihiko_Chikaya 1
N
by Mathzeus1024
Let 
In the 
plane with the origin 
, the line 
intersects the lines 
, and 
at the points 
, respectively. Find the maximum value of the area of 
In the 
plane with the origin 
, the line 
intersects the lines 
, and 
at the points 
, respectively. Find the maximum value of the area of 
1 reply
Ring out the Old Year and ring in the New.
Kunihiko_Chikaya 1
N
by Mathzeus1024
Let 
be positive real numbers.
Prove that
![$$\sqrt[3]{\left(\frac{a^{2022}-a}{b}+\frac{2021}{a^{\frac{a}{b}}}+1\right)\left(\frac{b^{2022}-b}{c}+\frac{2021}{b^{\frac{b}{c}}}+1\right)\left(\frac{c^{2022}-c}{a}+\frac{2021}{c^{\frac{c}{a}}}+1\right)}$$](//latex.artofproblemsolving.com/c/4/9/c4923b16be8a8faf86422477a6ed6b4b60eb2552.png)
Proposed by Kunihiko Chikaya/December 31, 2021
be positive real numbers.Prove that
![$$\sqrt[3]{\left(\frac{a^{2022}-a}{b}+\frac{2021}{a^{\frac{a}{b}}}+1\right)\left(\frac{b^{2022}-b}{c}+\frac{2021}{b^{\frac{b}{c}}}+1\right)\left(\frac{c^{2022}-c}{a}+\frac{2021}{c^{\frac{c}{a}}}+1\right)}$$](http://latex.artofproblemsolving.com/c/4/9/c4923b16be8a8faf86422477a6ed6b4b60eb2552.png)
Proposed by Kunihiko Chikaya/December 31, 2021
1 reply
Stronger inequality than an old result
KhuongTrang 21
N
by arqady
Source: own, inspired
Problem. Find the best constant satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](//latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
21 replies
