by AnhQuang_67, Apr 24, 2025, 1:44 PM 
be an even prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer 
satisfying for all positive integers 
, then 
.
is "not congruence"by giangtruong13, Apr 24, 2025, 1:44 PM ![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
by eulerleonhardfan, Apr 24, 2025, 1:29 PM 
is given, 
, 
are graphs on the same set of nodes, having 
connected components respectively. Prove that 
has at least 
connected components.by sqing, Apr 24, 2025, 12:24 PM 
, 
, 
are pairwise distinct real numbers satisfying 
Prove that
Let , , are pairwise distinct real numbers satisfying 
Prove that
by FireBreathers, Apr 23, 2025, 10:27 PM 
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).by Jackson0423, Apr 13, 2025, 8:39 AM 
and 
be positive integers. is a divisor of 
and is a divisor of 
.
.by Kyj9981, Mar 18, 2025, 11:53 AM 
and 
intersect at points and . A line through intersects and at points 
and 
, respectively. Line 
intersects at point 
, and line 
intersects at point 
. If 
is the circumcenter of 
, prove that 
.by taptya17, Dec 13, 2024, 8:24 AM be a positive integer. Initially the sequence 
( times) is written on the board. In each round, Ananya choses an integer 
and a subset of the numbers written on the board and adds to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing?by scls140511, Sep 8, 2024, 1:15 PM 
and 
be the two foci of ellipse 
. 
is a point on . Let be the center of the excircle of 
. When 
, find the minimum eccentricity of .by sqing, Apr 24, 2019, 1:01 PM 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
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