[/quote]
,
![$f(x)\in\mathbb Z[x]$](http://latex.artofproblemsolving.com/2/9/5/2953d049559ec5d4020e6777d965bdfff9ebde0f.png)
have positive integer leading coefficient. Show that there exists infinte positive integer 
such that the number of digit that doesn'r equal to 
is no more than 
,
,
positive real numbers. Prove that:![\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \geq 4+\frac{8(a-d)^2}{(a+b+c+d)^2}\]](http://latex.artofproblemsolving.com/f/b/5/fb5a179f427615c8bc2da46164ecd9a3484d6625.png)
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be the Gergonne point of the triangle 
is parallel with line 
.
has at least one pair of positive integer solution 
for each positive integer 
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
we have 
.
.
such that![\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]](http://latex.artofproblemsolving.com/3/1/9/319c9a2a80c4da5961cd5389af0606fa429ca8e2.png)
for any positive integer 
be an integer. Prove that there exists a positive integer 
(possibly depending on 
)
is divisible by 
.
and 
on 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
with 
and 
,
(i.e: sum of powers of point 
![\[
\max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1),
\]](http://latex.artofproblemsolving.com/4/c/b/4cba741142419b031efdeba713748762fa6b2aac.png)
where 
runs over the set of permutations of 
.
denote the integers modulo ![$\mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/e/5/a/e5a21492095c9116b100e2f7626a70cc26834e82.png)
be the set of polynomials with coefficients in ![$P(x) \in \mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/7/a/a/7aa574df0669ce78536b1ecc5c949120aa2b3009.png)
such that for all integers 
such that 
.
quartic polynomials in 
,
Determine 
.
Y by
A sequence of integers 
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
Find the smallest integer
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)
is call 
if it satisfies the following properties:
and 
for all indices 
.
.Find the smallest integer
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)This post has been edited 1 time. Last edited by TUAN2k8, Apr 19, 2025, 8:24 AM
Reason: some mistake
Reason: some mistake
