[/quote]
,
board, there is initially a tree of height 
.
.
such that the gardener can ensure there are eventually 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that the equation
be a positive integer and 
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that for every positive real 
,
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
be a positive integer and let 
be a strictly increasing sequence of 
be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
be a scalene triangle with circumcircle 
,
at 
.
meets 
and 
.
intersects the 
-
,
,
.
passes through either 
be positive real numbers such that : 
. Prove that : 
![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
be real numbers less than 2.Then prove that 
and 
intersect each other at 
and 
.
and 
respectively. Let 
and 
be the reflection of 
intersect circles 
and 
(both distinct from 
is the second tangent to 
.
with center 
,
are all different and lie on line 
be the points of intersection of 
and the segment 
.
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
.
Y by
Let 
and 
be coprime integers. Call a set 
balanced if the numbers 
form an arithmetic progression with difference 
, and the numbers 
form an arithmetic progression with difference 
.
In terms of
and 
, determine the maximum size of a collection of balanced sets such that every two of them have a non-empty intersection.
and 
be coprime integers. Call a set 
balanced if the numbers 
form an arithmetic progression with difference 
, and the numbers 
form an arithmetic progression with difference 
.In terms of
and 
, determine the maximum size of a collection of balanced sets such that every two of them have a non-empty intersection.This post has been edited 1 time. Last edited by Assassino9931, Apr 9, 2025, 10:33 PM
