[/quote]
,
?
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，
lie in a square of area 
.
that intersect in a region of area not less than 
.
of positive real numbers satisfies ![\[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]](http://latex.artofproblemsolving.com/0/3/9/03909a5f60d944063182ec97cf73f283178e8d4d.png)
for every positive integer 
.
for every 
.
be the orthocenter of acute triangle 
,
be the foot of the altitude from 
to 
,
be the reflection of 
.
intersects line 
and 
.
.
is an integer for every positive integer 
,![$P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$](http://latex.artofproblemsolving.com/0/5/1/0514f85850830af9a9e1a4dd5e7a4e8e68b2dff9.png)
.
polynomial with only one root.
be a finite set of real numbers, 
be a real number, and 
be 2025 non-zero real numbers. Define![\[A =
\left\{
(x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d
\right\},\]](http://latex.artofproblemsolving.com/2/e/1/2e18b6d25bfaa3e3d52f7efe5e4cba7a9cb87db9.png)
![\[B =
\left\{
(x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0
\right\},\]](http://latex.artofproblemsolving.com/4/1/f/41f6b126cc3cf24757b37147906c767d49a0fdd9.png)
![\[C =
\left\{
(x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0
\right\}.\]](http://latex.artofproblemsolving.com/d/e/6/de6586b88004e3f5aa7774525b00ee975c072ff9.png)
Show that 
.
and 
.
![$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$](http://latex.artofproblemsolving.com/2/2/e/22e6ec2f98777db80e39333542664e903663bf81.png)
![$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$](http://latex.artofproblemsolving.com/f/c/a/fca343ee701eeff4c309b31beb3426ec96b49b55.png)
Let 
.
![$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$](http://latex.artofproblemsolving.com/6/9/5/695cc35986a391fdc15e4863d361ce373fbbf64c.png)
.
be a point on side 
such that 
.
be the circumcenter of 
,
satisfying 
.
.
,
,
,
jump is equal to 
.
over all pairs of positive integers 
.
.
and 
,
.
,
is composite, then all positive divisors of 
Y by
Let 
be a set of integers with the following properties:
contains all positive integers.
be a set of integers with the following properties:
.- If
and 
, then 
. - If for some
, 
is composite, then all positive divisors of are in .
contains all positive integers.This post has been edited 1 time. Last edited by pikapika007, Mar 21, 2025, 12:12 PM
Reason: wrong year
Reason: wrong year
,
composite spam rule #2![$\sqrt[3]{k} < p < \sqrt{k}$](http://latex.artofproblemsolving.com/c/e/6/ce6e509d7a7df8ccce799cf56b09518c4ae9939f.png)
such that 
be inverse of 
'
so we get 
also in 
works for fermat primes but i didnt realize that 
be the odd primes in 
.
is composite, so all of its prime divisors must be in 
is composite and in 
.
be the smallest number not in 
in 
for some residue 
.
is in 
,
?
.
be the smallest power of four larger than 
,
are less than 
is in 
and 
,
and 
is 
(because the latter is 
)
.![\[\nu_2(p^2-1) = \nu_2(p-1) + \nu_2(p+1) \le 1 + (2k-1) = 2k \implies 2^{\nu_2(p^2-1)} \in S\]](http://latex.artofproblemsolving.com/f/5/b/f5bac8bb37eab51045d98d70e53721e19e018eef.png)
by the claim. Now the relatively-prime generation rule gives![\[ \left( \frac{p-1}{\nu_2(p-1)}, \frac{p+1}{\nu_2(p+1)}, 2^{\nu_2(p^2-1)} \right) \implies p^2-1 \in S \]](http://latex.artofproblemsolving.com/6/6/0/660bfeb44cb582c6d8dae1dc9f387c62c1a8d2f2.png)
which consequently generates 
be prime. If 
,
or 
.
,
.
be prime between 
and 
.
similar to above. Similarly, 
so we're done, or 
.
,
,
and 
.
,
is 
,
,
,
.
,
.
and 
,
)
then 
,
is composite so assume that 
are all on 
(because 
is smol :c), base cases are given and for the inductive step just note that 
and therefore 
and also then 
but also note that 
so it can't be prime therefore it is also in 
for all 
composite and that posses one odd factor but then considering the rest of divisors of this amount by making 
for all positive integers 
for all positive integers 
are on 
for 
and using euler theorem we get that all odd prime powers are on 
.
that 
with the base cases being 
already given.
:
is composite we're already done from the third bullet.
is an (odd) prime number. We say a number is good if the prime powers in its prime factorization are all less than 
nor 
is a power of ![\[ s \coloneq p^2-1 = (p-1)(p+1) \]](http://latex.artofproblemsolving.com/6/b/7/6b76522a095ce72e0f2002ca8394a31d0bcab5b1.png)
is good. Indeed, one of the numbers has only a single factor of 
is certainly less than 
is good. As 
,
.
is odd. First we contend that the number ![\[ s' \coloneq 2^{q+1} - 1 = \left( 2^{(q+1)/2}-1 \right) \left( 2^{(q+1)/2}+1 \right) \]](http://latex.artofproblemsolving.com/1/6/a/16a703361bf5076dc25201a4ace6db41df5811b4.png)
is good. Indeed, this follows from the two factors being coprime and both less than 
is in ![\[ s \coloneq p^2-1 = (p-1)(p+1) \in S \]](http://latex.artofproblemsolving.com/8/c/d/8cdaf54033ce3efb72c46ec28d5c2af8a9f4f527.png)
as we did in the previous case, because the largest power of 
will be exactly 
which is known to be in 
,
is a Fermat prime. Then in particular, 
.![\[ s \coloneq 2p-1 \equiv 0 \pmod 3 \]](http://latex.artofproblemsolving.com/e/6/4/e64d203eb4f3d7fcc962507123ea7c8e1444eb49.png)
and moreover 
,
by Zsigmondy/Mihailescu/etc.). So 
is good, and since 
,
is sufficient as a base case; however this requires a bit more work to check. Here is how:
,
.
we get 
.
we get 
.
we get 
.
,
are sufficient as base cases for the earlier cases to finish the rest of the problem. (Case 2 works once 
,
.
is not sufficient; for example 
satisfies all the problem conditions.
.
)
)
composite? WHAT! Help Me!![$[x] \cap S = [x]$](http://latex.artofproblemsolving.com/9/4/f/94f7de57826641ffcf953b59af1f41ca0bf0a26c.png)
.
!
is NOT PRIME NUGGET! By theorem: 
WHERE: 
,
.
IS PRIME NUGGET! Consider case:
!
.
and 
is 1! Notice: 
!
!
is NOT PRIME! Because divide by prime nugget 
.
!
.
!
.
!
!
.
NUGGETS DIVIDE 
!
!
so 
.
!
!
!![\[ 2^{\mathcal G + 1} \nu \mu \in S!!!!!!!!! \]](http://latex.artofproblemsolving.com/1/f/8/1f87397cdea89bde8d071fa1136ecf7931cd5ad8.png)
Gegagedi finish: Nugget point 3 give 
divisors in 
