and 
of non-isosceles triangle 
are found such points 
and 
,
= 
= 
.
at point 
intersects the line 
.
inscribed to triangle 
.
and 
be two different primes. Prove that![\[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]](http://latex.artofproblemsolving.com/9/0/3/903486996aa105ac1a28795e55596680d0884e3c.png)
and 
are the feet of the altitudes through 
and 
respectively. The incenters of the triangles 
and 
are 
and 
respectively; the circumcenters of the triangles 
and 
are 
and 
respectively. Prove that 
and 
are parallel.
where 
.
is the circumcenter of triangle 
is the projection of point 
onto line 
.
,
are 
,
,
,
intersects 
and 
at points 
and 
,
.
be an integer. Let 
and 
be polynomials with real coefficients such that the points 
in 
are the vertices of a regular 
-
,
statements are equivalent:
is differentiable, with continuous first derivative.
and for any two sequences 
,
,
for any positive integer 
,
is convergent.
Answer the following questions. You may assume that 
is known.
Determine the number of real solutions of equation 
for each 
For 
, let 
be the segond largest real solution of 
Find 
such that 
Find 
,
is defined as
has property 
if :![\[
\begin{cases}
\text{the set } A \text{ has at least } 4 \text{ elements} \\
\text{the element } 1+1 \text{ is invertible}\\
x+x^4=x^2+x^3 \text{ holds for all } x \in A
\end{cases}
\]](http://latex.artofproblemsolving.com/b/6/9/b6993ee76eb3792e08d441f5d5c463247c392c4a.png)
a) Prove that if a ring 
are distinct elements, such that 
are units, then 
is also a unit, but 
is not a unit.
be two complex numbers such that 
and 
,
.
satisfy the relation 
.
and 
is solvable over 
for every 
.
,
.
be collinear points (in order) and 
a point in plane. Consider the disc 
of center 
,
.![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
.![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
![$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)
,
.![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
Y by Exponent11
There are 12 gold stars arranged in a circle on a blue background. Giorgi wants to label each star with one of the letters 
, 
, or 
, such that no two consecutive stars have the same letter.
Determine the number of distinct ways Giorgi can label the stars.
Proposed by Giorgi Arabidze, Georgia
, 
, or 
, such that no two consecutive stars have the same letter.Determine the number of distinct ways Giorgi can label the stars.
Proposed by Giorgi Arabidze, Georgia
or 
is the same as the solution to the above problem.
be amount of ways we can write 
and 
.
be the answer to our new problem. Since we want last letter to be the same as the first one, we cant have 
-
.
.
