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.
be real numbers such that 
for all 
.
.
and 
be touch points of the incenter of 
at 
and 
, respectively. Let 
and 
be the circumcenter of triangles 
and 
, respectively. Show that 
and 
concurrent.
of one variable that fullfill the following for all real numbers 
and 
:
.
be a nonempty set of positive integers such that:
if 
then 
. for any prime 
, there exists 
such that 
. is finite.
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 the period of 
. Determine the total number of points in 
of period 
.
)
![$f : [0,1] \longrightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/9/a/4/9a4b7997c35990f839742d29866122791ade8b7c.png)
is differentiable with 
. If 
for all ![$x \in [0,1]$](http://latex.artofproblemsolving.com/d/3/4/d3423a3200d0ad7660fc4265d59d5598a53d551d.png)
, then show that 
for all .
which satisfy the following equality for all 
![\[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\]](http://latex.artofproblemsolving.com/2/0/a/20ae7491750bac4c94deaed32d5b9aa66b26491f.png)
Proposed by Dániel Dobák, Budapest
such that 
for all 
and 
and coefficients in the set 
. Prove that the probability for this polynomial to be special is between 
and 
, where a polynomial 
is called special if for every 
in the sequence 
there are infinitely many numbers relatively prime with 
.
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle 
, and let 
be the midpoint of the side 
. Let 
be a point on the side and 
a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
be a function whose derivative is continuous on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
. Show that![$$\lim_{n \to \infty} \sum^n_{k = 1}\left[f\left(\frac{k}{n}\right) - f\left(\frac{2k - 1}{2n}\right)\right] = \frac{f(1) - f(0)}{2}.$$](http://latex.artofproblemsolving.com/3/1/4/314a1597acae030381f980e6e26c12432f31e069.png)
be an integer and let 
be the set of all 
invertible matrices in which their entries are 
or 
. Let 
be the number of 's in the matrix 
. Determine the minimum and maximum values of in terms of , as varies over 
.
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is a continuously differentiable function such that 
and 
for some 
. Prove that there exist two tangents to the graph of 
that form an equilateral triangle with an appropriate segment of the -axis.
such that 
(otherwise would be strictly increasing or decreasing over the whole interval, contradicting 
). Hence there exists 
for which 
, because the derivative has the intermediate value property. Now the absolute maximum of must occur at the location of such an , hence there actually exists an 
for which and 
. Then if 
denotes the slope function, note that 
and by the mean value, intermediate value properties there exists 
such that 
. The corresponding tangents to the graph of running through 
and 
(i) intersect the x-axis at 
angles, (ii) do not intersect on the x-axis, and (iii) are not parallel hence intersect (somewhere off the x-axis). This is the desired equilateral triangle then.
is continuous. And given By Rolle's theorem there exist 
such that 
. Now there is a such that 
. So 
. Now applying LMVT on 
we have a 
such that
Similarly applying LMVT on 
we have a 
such that
Now apply IVT on 
we have a 
such that 
and similarly we have a 
such that 
. Easy to see that tangents at 
makes an equilateral triangle with finite segment on x axis.
2. 
3. 
Bracket 2
2. 
3. 
Bracket 3![$$\int_0^{1}\frac{dx}{1-\sqrt[3]{1-\sqrt{x}}}$$](http://latex.artofproblemsolving.com/d/5/e/d5e8db00985944b2eca341e37a94e34661cb5c86.png)
2. 
3. 
Bracket 4
2. 
3. 
Bracket 5
2. 
3. 
Bracket 6
2. 
3. 
Bracket 7
2. 
3. 
Bracket 8
2. 
3. 
![$$\int_0^{2\pi}\sqrt[4]{\cos^4(x)-\cos(2x)}dx$$](http://latex.artofproblemsolving.com/7/e/e/7eebb4a488fec702e0130a253ef87f0f6dfedb2d.png)
2. ![$$\int_0^1 \frac{x^5}{\sqrt[3]{1-x^3}}dx$$](http://latex.artofproblemsolving.com/6/0/5/605bb9d81d0ccff54a276c9f7bf1c1726c136398.png)
3. 
Bracket 2
2. 
3. 
Bracket 3
2. ![$$\int_0^1 \frac{dx}{(\sqrt{x}+\sqrt[3]{x})^2}$$](http://latex.artofproblemsolving.com/6/9/8/69869c0ed754c0dfacf412ac6411742ffa164a6d.png)
3. 
Bracket 4
2. 
3. 
2. 
3. 
Bracket 2
2. 
3. 
2. 
3. 
2. 
3. 
4. 
5. 
. Prove that
