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.
such that 
. Let 
and 
permutations of 
. Is possible that:![$$\int_0^1[xf_1(x)+x^2f_2(x)+...+x^nf_n(x)]dx > 2024\int_0^1[xg_1(x)+x^2g_2(x)+...+x^ng_n(x)]dx \text{ }?$$](http://latex.artofproblemsolving.com/9/5/6/9566b4c17aad5990302cf641e9030e9fb86ee957.png)
![\[A=\begin{bmatrix} 1&1&1&1&1\\1&-1&1&-1&1\\?&?&?&?&?\\?&?&?&?&?\\?&?&?&?&?\end{bmatrix}\in\mathbb R^{5\times 5}\]](http://latex.artofproblemsolving.com/f/c/b/fcbdeb5107deefc1ec6441dfc0b7e93006729b9e.png)
satisfy 
Determine the maximum value of 
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 
.
and 
, determine the maximum size of a collection of balanced sets such that every two of them have a non-empty intersection.
be an integer. The sequence 
is defined via 
, 
and![\[ a_{n+2} = ka_{n+1} + a_n \]](http://latex.artofproblemsolving.com/b/5/e/b5e3e6f9f0b0709113498266e41b372299a2986f.png)
for any positive integer 
. Prove that there are finitely many pairs 
of positive integers such that 
.
is 
, maybe even with rather elementary means? As usual, 
denotes the -th prime number. The problem of that limit came up in my partial solution of this problem: https://artofproblemsolving.com/community/c7h3495516.
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and 
. The tangents to at and intersect at 
. Prove that lies on the line .
on one variable with real coefficients such that the equation 
has exactly 
ordered pairs 
as solutions?
be positive reals such that 
. Prove that 
(Shining Sun, USA)
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
![$f \colon [0,1] \to \mathbb{R} $](http://latex.artofproblemsolving.com/2/1/8/21859d37ed035abd07d0be112f49d52598fa477e.png)
be a differentiable function such that its derivative is an integrable function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, and 
. Prove that ![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
be a positive integer. Prove that there exists a positive integer 
such that exactly 
of the integers 
are squarefree.
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
cells of an 
square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell 
, one can reach any other black cell 
through a sequence 
of black cells where each consecutive pair 
are adjacent (sharing a common side) for every 
. Let 
denote the maximum possible cost over all initial colorings with exactly black cells. Determine a constant 
such that![\[
\frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha}
\]](http://latex.artofproblemsolving.com/0/a/0/0a06d4ff52479c8cb785c6450cea8748bf7769b6.png)
for any 
.
be a function having finite one-sided derivatives at all points in 
. Prove that 
is differentiable in except for a set that is at most countable.
.
