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Brilliant Problem
M11100111001Y1R 7
N
by flower417477
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences 
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:
![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](//latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
7 replies
In Cyclic Quadrilateral ABCD, find AB^2+BC^2-CD^2-AD^2
Darealzolt 1
N
by Beelzebub
Source: KTOM April 2025 P8
Given Cyclic Quadrilateral 
with an area of 
, with 
. If 
, Hence find the value of 
.
with an area of 
, with 
. If 
, Hence find the value of 
.
1 reply
Another FE
M11100111001Y1R 3
N
by AndreiVila
Source: Iran TST 2025 Test 2 Problem 3
Find all functions 
such that for all 
we have:
such that for all 
we have:
3 replies
Iran TST Starter
M11100111001Y1R 4
N
by flower417477
Source: Iran TST 2025 Test 1 Problem 1
Let 
be a sequence of positive real numbers such that for every 
, we have:
![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](//latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:
![\[
a_n < \frac{1}{1404}
\]](//latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
4 replies
Inequality with abc=1
tenplusten 10
N
by Adywastaken
Source: JBMO 2011 Shortlist A7
Let 
be positive reals such that 
.Prove the inequality 
10 replies
A game of cutting
k.vasilev 11
N
by NicoN9
Source: All-Russian Olympiad 2019 grade 10 problem 2
Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be 
pieces of equal weights. Can Vova prevent Pasha's win?
pieces of equal weights. Can Vova prevent Pasha's win?
11 replies
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
Cup of Combinatorics
M11100111001Y1R 8
N
by sansgankrsngupta
Source: Iran TST 2025 Test 4 Problem 2
There are 
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
8 replies
Prove that
Let 
Prove that
Prove that
