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.![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, 
and ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
converge?
with 
is called historic if 
. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
be a set of 
distinct real numbers. Let 
be the set of numbers that occur as averages of two distinct. For a given 
, what is the smallest possible number of elements in ?
, the numbers 
are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is 
after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out., determine which player (if any) has a winning strategy
be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that is an 
matrix with entries 
, 
such that the sum of the elements in each row and each column of is equal to the corresponding sum for the matrix . Prove that 
.
. Find .
is differentiable and 
Show that for some 
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer such that 
is called period of 
Determine the total number of points 
of period 
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
.
, then 
or 
,
.
, because 
is an odd function.
.
or 
, then 
and
.
.
.![$I=8\arcsin(2\sqrt{s}-1)+C=8\arcsin(2\sqrt[4]{t}-1)+C=8\arcsin(2\sqrt[8]{x}-1)+C$](http://latex.artofproblemsolving.com/5/f/b/5fbe34c824a24e83a97a5fb8c9183f395284960b.png)
.
.
, then 
and
.![$I=\left[e^{t}\frac{\sin t+\cos t}{2} \cdot \ln(\sin t+\cos t)\right]_{0}^{\frac{\pi}{2}}-\frac{1}{2}\int_{0}^{\frac{\pi}{2}} e^{t}(\sin t+\cos t) \cdot \frac{\cos t-\sin t}{\sin t+\cos t}\ dt$](http://latex.artofproblemsolving.com/1/f/c/1fce20b6f4ffa16f433b0d9e8fe8f7b74529c6c9.png)
.![$I=\left[e^{t}\frac{\sin t+\cos t}{2} \cdot \ln(\sin t+\cos t)\right]_{0}^{\frac{\pi}{2}}-\frac{1}{2}\int_{0}^{\frac{\pi}{2}} e^{t}(\cos t-\sin t)\ dt$](http://latex.artofproblemsolving.com/1/b/2/1b2f757303e3c6d31e5a2dcbb4e72f1665dc4eec.png)
.![$I=\left[e^{t}\frac{\sin t+\cos t}{2} \cdot \ln(\sin t+\cos t)-\frac{1}{2}e^{t}\cos t \right]_{0}^{\frac{\pi}{2}}=\frac{1}{2}$](http://latex.artofproblemsolving.com/b/8/1/b81efb23bdafc70f68288db7cb1ae9c0ff181582.png)
.
.
, then 
or 
,
.
.
.
.
in the first integral and renaming gives
;
.
gives
and 
(or you can do both at once but I did it this way) to get
whichever form you prefer.![$\int_0^1 \left( \frac{\tan^2 x}{\tan x -x} - \frac{3}{x} \right) dx=\left[ \ln (\tan x -x) - 3\ln (x) \right]_0^1 = \ln (3 \tan(1) -3)$](http://latex.artofproblemsolving.com/6/a/b/6ab1232f5aff91f0a52a5195e9a6671b64458b57.png)
,
We then make the substitution 
to get
.
.
.
![$\int_0^1 \frac{dx}{(1+x) \sqrt[3]{x^2-x^3}} \overset{u=\frac{1-x}{1+x}}{=} \frac1{\sqrt[3]{2}}\int_0^1 u^{-\frac{1}{3}} (1-u)^{-\frac{2}{3}} du=\frac{\pi \csc\left(\frac{\pi}{3}\right)}{\sqrt[3]{2}}=\frac{2^{\frac{2}{3}}\pi}{\sqrt{3}}$](http://latex.artofproblemsolving.com/0/f/4/0f45e3717854dd3735f1ea5f027581b2effb198e.png)
.![$\int_0^\infty \cot^{-1}(px)\cot^{-1}(qx)dx = \frac{\pi}{2}\left[\frac{1}{p}\log\left(\frac{p+q}{q}\right)+\frac{1}{q}\log\left(\frac{p+q}{p}\right)\right]$](http://latex.artofproblemsolving.com/6/e/9/6e95151f703fc4f3002703c40e9f9bfc0eb31154.png)
.
.
:
![$\frac{\partial^2 I}{\partial p\partial q}=\int_0^\infty \frac{x^2}{(1+p^2 x^2)(1+q^2 x^2)} dx=\frac{1}{q^2-p^2}\int_0^\infty \left(\frac{1}{1+p^2 x^2}-\frac{1}{1+q^2 x^2}\right) dx = \frac{1}{q^2-p^2}\left[\frac{\tan^{-1}(px)}{p}-\frac{\tan^{-1}(qx)}{q}\right] _0^\infty=\frac{\pi}{2pq(p+q)}.$](http://latex.artofproblemsolving.com/8/7/0/870619b21ac2910cea63be89dce7f5560fa8565a.png)
,
.![$I(p,q)=-\frac{\pi}{2}\int \frac{1}{q^2}\log\left(\frac{p+q}{p}\right) dq = \frac{\pi}{2}\left[\frac{1}{p}\log\left(\frac{p+q}{q}\right)+\frac{1}{q}\log\left(\frac{p+q}{p}\right)\right]$](http://latex.artofproblemsolving.com/b/d/5/bd5893f1bea5e370d523889e011220c02a887982.png)
,
.
.
gives
gives
,
We want to find 
But I know nothing about special functions...![$$I=\int^1_0\left(\left\lfloor\frac{1}{\sqrt[69]{x}}\right\rfloor-420\left\lfloor\frac{1}{420\sqrt[69]{x}}\right\rfloor\right)dx$$](http://latex.artofproblemsolving.com/8/3/2/8328f820b2070e8bbd1d3d6aef081c4bc54a0774.png)
Closing the contour in the upper half-plane, we have inside the contour two simple poles - at 
and 
.
,
and consider
On the one hand, the function 
is analytical in all complex plane (because its Taylor's series is analytical and valid in all complex plane). On the other hand, in the upper half-plane at 
Taking 
,
_______________________
one can show via direct evaluation that
or the same result can be obtained from the formula above, just using 
in the same way,
where 
we evaluate, integrating 
along the rectangular contour 
:
Taking derivatives and putting 
Using 
and 
(taking derivatives of 
and 
)
F12:
digits of 
,
,
.
![[asy]
usepackage("amsmath, amsthm, amssymb, tikz, tikz-cd");
label("\begin{tikzcd}[ampersand replacement=\&, row sep=tiny]
{}\&\&\&\color{blue}\scriptstyle \lim\limits_{n\to\infty}\mathcal I\ =\ 1012.\color{black}\qed
\arrow[Rightarrow, color=blue, from=1-1, to=1-4]
\end{tikzcd}");
[/asy]](http://latex.artofproblemsolving.com/9/7/9/979c4772ee1e3f9f1e852bff587f761f96461e09.png)