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 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.
on the spiral, the polar of wrt. is tangent to the spiral.
such that every positive integer appears exactly once and the product of the first 
terms is a perfect 
power ?
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape by playing optimally. (The perimeter of shape is defined as the total length of the boundary segments between a black and a white cell.), assuming both players play optimally?
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers ?
decide whether there exist pairwise distinct positive integers 
such that for every 
, 
divides 
.
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
be real numbers such that 
find the largest value of 
such that 
and 
are all perfect square.
be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
be a regular polygon, and let 
be its set of vertices. Each point in is colored red, white, or blue. A subset of is patriotic if it contains an equal number of points of each color, and a side of is dazzling if its endpoints are of different colors. is patriotic and the number of dazzling edges of is even. Prove that there exists a line, not passing through any point in , dividing into two nonempty patriotic subsets.
be a rectangle with 
Point 
is located on the side 
, and the perpendicular bisector of ![$[MC]$](http://latex.artofproblemsolving.com/1/d/a/1da22e2a1539bb3094111c561363eae8cbf24f8b.png)
intersects the line 
at the point 
. Let 
. Knowing that 
, show that the quadrilateral is a square.
be a rectangle with Point is located on the side , and the perpendicular bisector of intersects the line at the point . Let . Knowing that , show that the quadrilateral is a square.
be point on side 
and point 
on side 
such that 
.
is right angle triangle, so 
, triangle 
is isosceles.
be on side 
such that 
, since 
is cyclic, 
be on side such that 
and 
, since is isosceles, this means that 
meaning that this rectangle is a square. 
.
, then 
, thus 
is 
excenter of 
and 
is the angle bisector of 
, that is, is a square.
