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.
such that for all positive real numbers 
,![\[x^2[f(x)+f(y)]=(x+y)f(yf(x)).\]](http://latex.artofproblemsolving.com/2/7/c/27c95af9b083a2a2827d428da24a5152fd926473.png)
Determine all sets 
containing 
and having the property that 
is prime, for all distinct 
be positive real number satisfy that: 
.Find the minimum: 
be real numbers so that 
and 
Show that ![$a\in[0,\frac{4}{3}]$](http://latex.artofproblemsolving.com/0/e/c/0ecb933d7b008114469c3915dc2f8cbf43e362dc.png)
,![$b\in[0,\frac{2}{3}]$](http://latex.artofproblemsolving.com/8/4/1/841685820e03663b8ba4b65615e7ecf40fc31bcc.png)
,![$c\in[0,\frac{4}{9}]$](http://latex.artofproblemsolving.com/3/3/4/3349e77972554eb938563cef5aed9425367cd48d.png)
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.
denote the number of -tuples of positive integers 
such that for each 
, we have ![\[ 1 \leq a_i \leq 7 \quad\text{and}\quad a_{a_i} = a_i. \]](http://latex.artofproblemsolving.com/5/1/7/51741f4725c64c1ad29b14949ad890f131b89687.png)
Compute the remainder when is divided by 
. be nonnegative real numbers such that 
. Find min
such that the number 
is a perfect square for all positive integers .
and 
be two similar right-angled triangles (in the same plane) with 
such that and 
lie on opposite sides of the line 
. If 
, calculate the ratio 
.
. Prove that:
2/ Let . Prove that: 
![$f:[0,1]\rightarrow [0,1].$](http://latex.artofproblemsolving.com/d/0/7/d076b5242cc57c3c4af7971c9cb104023effca65.png)
I'm interested in what solutions there are other than constant functions.
, 
, 
, 
and the diagonals 
intersect at point . The line 
passes through parallel to 
and has its endpoints on the sides and 
respectively. Express 
in terms of 
and 
.
?
and 
.![\[ab+bc+ac\leqslant 27\]](http://latex.artofproblemsolving.com/c/b/5/cb5e2a1efd462abd2354f4136f5fa64408f3138c.png)
. Prove that
to the height of 
is 
.
of the way from 
.