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.
satisfy the equation 
. Prove that there exist positive integers 
such that
be a given integer. Prove that infinitely many terms of the sequence 
, defined by ![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
, 
denotes the largest integer not exceeding .)
there exists an -digit number divisible by 
all of whose digits are odd.
, 
, 
, 
be distinct positive integers, 
. Prove that there exist distinct indices 
and 
such that 
does not divide any of the numbers 
, 
, , 
.
be real numbers that satisfy![\[
p^2+q^2=r^2+s^2
\]](http://latex.artofproblemsolving.com/5/1/b/51b19eb831481a9181fee897d22e7f71015425c3.png)
![\[
p^2+s^2-ps=q^2+r^2+qr
\]](http://latex.artofproblemsolving.com/3/7/c/37cc3145b42eb86b7019f13f1014ab0908753d28.png)
Hence find the value of 
, the value of 
is 
. Find all complex numbers .![\begin{align*}
(a+bi)\cdot(a-bi) +[(a+bi)+(a-bi)] =a^2+b^2+2a.
\end{align*}](http://latex.artofproblemsolving.com/4/0/9/409b0aed187f4f2cbc8e3f204f8b5fac9bcb44b0.png)
and 
. when the complex numbers lie on the circle centered at 
with radius 
.
red balls and blue balls. A draw consists of randomly selecting 
of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
, 
, 
be fixed positive integers. There are 
ducks sitting in a ducks ducks picking paper, and ducks picking scissors., , and , the maximum number of moves which could take
be the circumcenter of triangle 
. Line 
intersects the altitude from 
at point 
. Let 
be the midpoints of 
, 
respectively. If 
intersects 
at 
, and the circumcircle of triangle 
meets 
at 
, prove that 
is cyclic. be a triangle with incentre 
and circumcentre . Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
be the intersection of the diagonals of a cyclic quadrilateral 
. Find the length of 
, if it is known that 
mm , 
mm, 
mm, and 
.
if
,
,
?
,
in terms of 
.
to z because we already know the value of 
.
(in the equation 
)
.
.
and cross multiply.
.
.
with a radius of 
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
and 
.
.