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thank you !
Nakumi 0
Given two non-constant polynomials 
such that for every real number 
, 
is a perfect square if and only if 
is a perfect square. Prove that 
is the square of a polynomial with real coefficients.
such that for every real number 
, 
is a perfect square if and only if 
is a perfect square. Prove that 
is the square of a polynomial with real coefficients.
0 replies
Same divisor
sam-n 16
N
by AbdulWaheed
Source: IMO Shortlist 1997, Q14, China TST 2005
Let 
be positive integers such that 
and 
Prove that if 
and 
have the same prime divisors, then 
is a power of 2.
be positive integers such that 
and 
Prove that if 
and 
have the same prime divisors, then 
is a power of 2.
16 replies
sum of gcd over sets is more then sum of gcd over union
Miquel-point 3
N
by Jupiterballs
Source: KoMaL A. 882
Let 
be non-empty subsets of the positive integers, and let 
denote their union. Prove that
![\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]](//latex.artofproblemsolving.com/1/c/d/1cdf561d3af92f524628e0724e0c5ee24d8d6a09.png)
Proposed by Dávid Matolcsi, Berkeley
be non-empty subsets of the positive integers, and let 
denote their union. Prove that![\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]](http://latex.artofproblemsolving.com/1/c/d/1cdf561d3af92f524628e0724e0c5ee24d8d6a09.png)
Proposed by Dávid Matolcsi, Berkeley
3 replies
Erasing the difference of two numbers
BR1F1SZ 5
N
by Jupiterballs
Source: Austria National MO Part 1 Problem 3
Consider the following game for a positive integer 
. Initially, the numbers 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
, or 
as 
, or 
as 
.)
For which values of is it possible to end with only one number remaining on the board?
(Michael Reitmeir)
. Initially, the numbers 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
, or 
as 
, or 
as 
.)For which values of
is it possible to end with only one number remaining on the board?(Michael Reitmeir)
5 replies
Find the value
sqing 10
N
by Sadigly
Source: 2024 China Fujian High School Mathematics Competition
Let 
and 
Find the value of 
and 
Find the value of 
10 replies
inequality
mathematical-forest 5
N
by mathematical-forest
For positive real intengers 
, such that 
proof:
, such that 
proof:
5 replies
Graph Theory
ABCD1728 0
Can anyone provide the PDF version of "Graphs: an introduction" by Radio Bumbacea (XYZ press), thanks!
0 replies
Prove that
IMO ShortList 2002, algebra problem 2
orl 28
N
by ezpotd
Source: IMO ShortList 2002, algebra problem 2
Let 
be an infinite sequence of real numbers, for which there exists a real number with 
for all 
, such that ![\[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \]](//latex.artofproblemsolving.com/a/4/d/a4d87553590a450365b6525ca62a5d0416fe41cc.png)
Prove that 
.
be an infinite sequence of real numbers, for which there exists a real number with 
for all 
, such that ![\[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \]](http://latex.artofproblemsolving.com/a/4/d/a4d87553590a450365b6525ca62a5d0416fe41cc.png)
Prove that 
.
28 replies
Prove that
