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 
is a divisor of 
.
, diagonals 
and 
are equal, and they intersect at 
. Perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
, and perpendicular bisectors of 
and 
intersect at point 
lying inside triangle 
. Prove that 
.
denote the set of positive rational numbers. Find, with proof, all functions 
such that, for all positive rational numbers 
and 
we have ![\[f(x)=f(x+y)+f(x+x^2f(y)).\]](http://latex.artofproblemsolving.com/9/e/a/9ea71d2ee277dc3c689ba2d791b451e29353edb3.png)
be a triangle with circumcenter 
. Let be a point on the perpendicular bisector of (see figure) and , 
be the intersections of the perpendicular bisectors of and , respectively, with 
and 
. Prove that the midpoints of 
and 
and the point are collinear.
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
, where 
and 
are integers, 
. A subset 
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to . Find the number of valuable subsets of set .
rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to coins in the th row. Initially, all of the coins show heads (H). Carley plays a game in which, on each turn, she chooses three mutually adjacent coins and flips these three coins over. To win the game, all of the coins must be showing tails (T) after a sequence of turns. An example game with 4 rows of coins after a sequence of two turns is shown. for which it is possible to win the game starting with rows of coins.
satisfying 
, prove that![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
and natural number ,
on the side , a point P is chosen such that 
. Points 
and 
are symmetric to point wrt vertices and 
. It turned out that 
. What is the angle of the original triangle?
on the side , a point P is chosen such that . Points and are symmetric to point wrt vertices and . It turned out that . What is the angle of the original triangle?
![[asy]
pair[] tri = {dir(0), dir(130), dir(270)};
string[] tri_l = {"A","B","C"};
for(int i = 0; i < tri.length; ++i) { draw(tri[i] -- tri[(i+1) % tri.length]); label("$" + tri_l[i] + "$", tri[i], dir(tri[i])); }
pair p = (tri[0] + tri[2])/2; dot("$P$", p, dir(p));
pair p1 = (tri[0]+p)/2; label("$x$", p1, dir(p1));
pair p2 = (tri[1]+tri[2])/2; label("$2x$", p2, dir(p2)+5NE/2);
pair x = 2*tri[0]-p; dot("$X$", x, dir(x));
pair y = 2*tri[2]-p; dot("$Y$", y, dir(y));
draw(p -- tri[2]); label("$y$", p -- tri[2]);
draw(tri[0] -- x); label("$x$", tri[0] -- x);
draw(tri[2] -- y); label("$y$", tri[2] -- y);
draw(tri[1] -- x); label("$z$", tri[1] -- x, N);
draw(tri[1] -- y); label("$z$", tri[1] -- y);
[/asy]](http://latex.artofproblemsolving.com/8/c/0/8c00164f9b410ef4206c492ae67a614388fc63bd.png)
satisfy that: 
