[/quote]
,
be a prime. For a positive integer 
,
as the unique nonnegative integer 
such that 
but 
.
,
such that for all positive integers 
,
positive integers 
satisfy![\[
\frac{m!}{p^{\nu_p(m!)}} \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/3/8/d/38defc16b4e09607b9c6c0ad3f40ddff44692e3d.png)
Proposed by Dong Zhenyu
such that: 
Prove that :
,
be the minimal positive integer, such that for any 
,
,
,
,
takes at most 
ranges over all non-empty subsets of 
.
.
vertices labelled 
.
,
total turns.
,
be the total weight of all edges at the end of the game. Alice plays to maximize 
be real numbers greater than or equal to 1. Prove that![\[ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. \]](http://latex.artofproblemsolving.com/0/2/2/02201b8c4511b22c664e7dbf6e6f1e5fcbaa5b34.png)
be an acute triangle with circumcircle 
.
be the midpoint of 
and let 
be the midpoint of 
.
be the foot of the altitude from 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
subsets of 
such that any two subsets have an intersection size of exactly one. Show that two of these subsets must be the same.
of real numbers that satisfy the system of equations![\[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\]](http://latex.artofproblemsolving.com/d/4/d/d4d6c485b4653577e732ff235448bb779914a576.png)
and circumcenter 
,
.
be the midpoint 
,
meets 
at 
.
through H meets 
.
is tangent to 
.
such that the following equality holds for any real numbers 
and 
.
Y by GA34-261
3. [Russia 1961]
Real numbers are written in an
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
Real numbers are written in an
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.This post has been edited 2 times. Last edited by CatalanThinker, May 28, 2025, 5:43 AM
