,
be the sum of the digits of 
such that there are infinite positive integers 
.
,
be fixed positive integers. There are 
ducks sitting in a
be the circumcenter of triangle 
.
intersects the altitude from 
at point 
.
be the midpoints of 
,
respectively. If 
intersects 
at 
,
meets 
at 
is cyclic.
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
such that ![$\lfloor \sqrt[3]{n}\rfloor\mid n$](http://latex.artofproblemsolving.com/5/6/2/562cedff73965e1dce784a360514b931e36158b1.png)
.
red balls and 
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.
where 
.
but none of my attempts worked and the fractions looked increasingly complex. I don't think finding an explicit form is possible (I would love to be disproven), so I decided to look at the relation asymptotically.
.
tends to 
,
is a fixed point. This is also obvious when looking at the original problem, since the probability of drawing a match gets closer to 
.
.
to any function.
?
,
is the greatest integer less than or equal to 
is prime?
Prove that
Let 
Prove that
be complex numbers. For a complex number 
where 
,
to be the distance of 
from the origin, or 
.
be the minimum value and 
be the maximum value of![\[
\frac{ |a + b| + |b + c| + |c + a| }{ |a| + |b| + |c| }
\]](http://latex.artofproblemsolving.com/1/7/5/1752b55a5bca8490d3dbb398fb2c4e4f9f71f47c.png)
for all complex numbers 
.
.
.
and 
are added, their sum 
is also a vector. Geometrically, this is represented by placing the tail of 
.
be the same distance from the origin as 
Substituting this in the given equation, we get
.
.
![\[
x^2 + 2|x| - 6x + 15 = 0.
\]](http://latex.artofproblemsolving.com/1/1/d/11dbf7a605e8c49bb730eb0bd3737019b277fa41.png)
(Note: the modulus of a complex number 
is 
.
and 
.
and 
.
.
![\[
\frac{ab + bc + ca}{a^2 + b^2 + c^2} + \frac{1}{6} \left( \frac{(a - b)^2}{a^2 + b^2} + \frac{(b - c)^2}{b^2 + c^2} + \frac{(c - a)^2}{c^2 + a^2} \right) \leq 1.
\]](http://latex.artofproblemsolving.com/d/8/b/d8b3869addff6715d86b73083c77395d64ab1995.png)
are the 256 roots (not necessarily distinct) of the equation 
,
.
Y by CyclicISLscelesTrapezoid, OronSH, pineapply, kiyoras_2001, golue3120, KevinYang2.71, EpicBird08, Ciobi_
Let 
be a logarithmic spiral centered at the origin (ie curve satisfying for any point 
on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
Prove that for a point
on the spiral, the polar of wrt. is tangent to the spiral.
be a logarithmic spiral centered at the origin (ie curve satisfying for any point 
on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.Prove that for a point
on the spiral, the polar of wrt. is tangent to the spiral.This post has been edited 3 times. Last edited by qwerty123456asdfgzxcvb, Apr 21, 2025, 7:33 PM
be inverflection centered at 
be polarity with respect to 
.
is preserved by 
be the tangency point of 
fixing 
are always conjugate in 
is the tangent at 
is the tangent, then 
is constant, 
is constant, and 
is constant, so the angle between 
is constant. This angle is 
when 
,
