AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
lie in a square of area 
. Prove that among the triangles there are 
that intersect in a region of area not less than 
.
be real numbers such that 
and 
Find the value of 
be real numbers such that 
and 
Find the value of 
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 
. Prove that 
for every 
.
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 
.
. Let 
be a point on side 
, and 
be a point on the extension of 
such that 
. Let 
be the circumcenter of 
, and 
be a point on the side extension of 
satisfying 
. Line BP intersects AC at point Q. Prove that 
. The length of the first jump is 
, the second 
, the third 
, and so on. The length of 
jump is equal to 
. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
over all pairs of positive integers 
.
lamps 
arranged in a circle in that order. At any given time, each lamp is either on or off. Every second, each lamp undergoes a change according to the following rule:
, if 
have the same state in the previous second, then is off right now. (Indices taken mod .) is on right now.
which is on. Prove that for infinitely many integers all the lamps will be off eventually, after a finite amount of time.
and 
, let us consider the equation
[list=a]
and 
, find the least positive integer 
satisfying this equation. and , there exist infinitely many positive integers satisfying this equation.
denotes the greatest common divisor of positive integers 
and .)
for which there exist real numbers 
satisfying 
, 
and
for 
.
such that 
holds for all positive rational numbers 
.
such that 
, find the maximum possible number of integers 
such that 
is a regular polygon with 
diagonals. Randomly select a triangle whose three vertices are the three vertices of polygon 
,![$P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$](http://latex.artofproblemsolving.com/0/5/1/0514f85850830af9a9e1a4dd5e7a4e8e68b2dff9.png)
.
polynomial with only one root.
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)
.
.
,![\[1 + 2 + 3 + 4 \equiv 0\]](http://latex.artofproblemsolving.com/3/f/c/3fcedf4231a796e7687b298c97950bc9cfa7056e.png)
![\[1 + 4 + 9 + 16 \equiv 0\]](http://latex.artofproblemsolving.com/e/5/c/e5c896dd226b0e5b529f5bb61ca854b5060613f8.png)
![\[1 + 8 + 27 + 64 \equiv 0\]](http://latex.artofproblemsolving.com/b/f/8/bf899d319e6ce0e9bbc1de191ce26f2d9bbb75dc.png)
![\[1 + 16 + 81 + 256 \equiv 4\]](http://latex.artofproblemsolving.com/d/6/b/d6bca7789fafd0eaa5e4b6014d2dc4a5be0e5e36.png)
Then, 
gives us 
again, so it's periodic. Therefore, only 
fails, so 
denote the elementary symmetric polynomials in 
.
.
,
.
.
so 
.
.
so WLOG 
.
so 
contradiction as 
![\[P_1 = S_1\]](http://latex.artofproblemsolving.com/f/1/6/f1668acc070546f7da5d344d79e2acddf64da5a8.png)
![\[P_2 = S_1P_1 - 2S_2\]](http://latex.artofproblemsolving.com/3/a/e/3ae498510c7d234169489af297b3cdb78261d58d.png)
![\[P_3 = S_1P_2 - S_2P_1 + 3S_3\]](http://latex.artofproblemsolving.com/d/3/a/d3ad7da0529aa10744fd31db33b31000ef136e5f.png)
![\[P_4 = S_1P_3 - S_2P_2 + S_3P_1 - 4S_4\]](http://latex.artofproblemsolving.com/b/3/4/b3431d1823184d6c40d56ed247d37e20487d1a5c.png)
So 
.
.
,
.
for all ![\[P_{k+4}=S_1P_{k+3}-S_2P_{k+2}+S_3P_{k+1}-S_4P_k\]](http://latex.artofproblemsolving.com/a/f/f/aff7128cf260aada87444570e8b2fd9f47430f79.png)
we also have 
.
.
,
such that 
.
.![\[P_{k+r-1}=S_1P_{k+r-2}-S_2P_{k+r-3}+S_3P_{k+r-4}-S_4P_{k+r-5}\]](http://latex.artofproblemsolving.com/2/b/1/2b14e3e5f7ad1087c123ecc2e890dbc358b0d41e.png)
In particular note that the 
term is paired with 
.
,
,
term is 
.
,
and 
.![\[S_rP_{k-1}\equiv 0\pmod {p^m}\]](http://latex.artofproblemsolving.com/b/c/d/bcdbc7f2349aa68f4ef31ed8a16250029ca0639c.png)
implying that 
.
,
.
so 
.
are 
and 
.
for either of them, so 
.
is a contradiction.
consecutive 
such that 
.
values of 
,