AoPS Academy
be positive real numbers . Prove that :
Each circle 
is inscribed within an equilateral triangle whose vertices are points on 
Given 
has a radius of 
what is the radius of 
?
sums to 
A new sequence starting with the same term has common ratio 
The sum of the new sequence produced is 
What was the common ratio of the original sequence?
and 
be five equally spaced points on a line in that order. Let 
and 
all be on the same side of line 
such that triangles 
and 
are equilateral with side length 
Let 
be the region consisting of the interiors of all four triangles. Compute the length of segment 
that is contained in 
determine 
such that no two consecutive numbers have the same remainder when divided by 
?
cards of lined up in a row. He puts a multiplication sign between two of the s and calculates the product of the two strings of s. For example, one possible result is 
Let be the sum of all possible distinct results (note that 
yields the same result as 
). What is the sum of digits of ?
identical brown balls into 
bins labeled 
. He must distribute the balls among the bins so that each odd-labeled bin contains an odd number of balls, and each even-labeled bin contains an even number of balls (with 
considered even). In how many ways can Bruno do this?
be the number obtained by increasing every prime factor in 
by one. For instance, 
What is the lowest 
such that 
divides 
where 
denotes the th iteration of ?
do not end in a 
? guests. Each pair of party guests are either friends or enemies. Each guest has exactly 
enemies. All guests believe the following: the friend of an enemy is an enemy. Calculate the sum of all possible values of 
(Please note: Bruno is not a guest at his own party)
, let 
be the foot of the altitude from 
to 
and 
be the circumcenter. Suppose that the area of 
is equal to the area of 
Given that 
and 
compute 
gifts 
and friends 
Gift 
can be given to friend 
if![\[i -j =-1,0, \text{ or } 1 \pmod{10}\]](http://latex.artofproblemsolving.com/8/c/2/8c2d541d315f4f7ab9ba2893867a85fb8f04c4da.png)
How many ways are there for Alice to pair the gifts with the friends such that each friend receives one gift? be an equilateral triangle with side length A real number 
is selected uniformly
Points and 
lie on sides 
and 
respectively, such that 
and 
Let be the intersection of lines 
and 
Consider line 
passing through both points of intersection of the circumcircles of triangles 
and 
is the circumcenter of 
Line intersects line 
at point 
and point 
lies on 
such that 
What is the probability that the line segment 
has length less than 
?
such that 
and 
for all positive integers Find the value of![\[\frac{1}{\cos(a_1)\cos(a_2)\cos(a_3)\cdots\cos(a_{16})}.\]](http://latex.artofproblemsolving.com/f/4/e/f4ed97b8ef6f0552fba00ad81cca08dcc89f2efa.png)
to be the fractional part of 
For example, 
and 
Let
where denotes the fractional part of Compute rounded to the nearest integer.
and natural number such that 
.
and 
intersect at exactly one point. Prove that for all real numbers 
, the inequality 
is always valid.
be a function satisfying the relation 
for any real 
. Find the value of 
to the nearest integer. (Here 
denotes the set of real numbers.)
we get 
.
Hence 
gives