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 an acute, scalene triangle with circumcenter 
, circumcircle 
, orthocenter 
. Line 
meets again at 
. Let 
be the midpoint of segments 
respectively. The line through and perpendicular to 
meets line 
at 
.
meets again at 
. Prove that the intersection of line 
and the perpendicular bisector of 
lies on the circumcircle of triangle 
. and perpendicular to 
meets line at 
. Line 
meets again at 
. Let 
be the intersection of lines 
; 
be the intersection of lines 
; 
be the intersection of lines 
. Let 
be the intersection of lines 
and 
be the line through and perpendicular to 
. Prove that bisects 
.
such that 
. Show that the equation 
has no integer solution.
, 
, 
, be positive numbers. Prove that
, where 
always occurs at 
and 
are primes, then can 
be a perfect square?
a prime number with 
and 
when 
. then 
else 
.
such that 
. Find the minimum value of:
be a triangle such that 
and 
is obtuse. Point is the circumcenter of triangle , and is the reflection of 
in . Let 
be an arbitrary point on line , such that 
is in between and 
. Line 
cuts the circumcircle of in 
. Circumcircles of triangles 
and 
cut in 
respectively. Prove that 
is tangent to the circumcircle of triangle 
.
from the non-zero reals to the non-zero reals, such that ![\[f(x+y) \left(f(x) + f(y)\right) = f(xy)\]](http://latex.artofproblemsolving.com/2/f/e/2fe8a09e628f60faedaa07ceb154891375763a83.png)
for all non-zero reals 
such that 
.
that satisfy the equation
for any 
. Note that 
.
be an acute-angled scalene triangle with circumcircle 
and circumcenter . Suppose 
. Let be the orthocenter and 
be the incenter of triangle . Let 
be the midpoint of the arc of the circumcircle of triangle 
, containing .
be a point on the arc 
of not containing , such that 
. Let be the circumcenter of triangle 
. Prove that the lines 
and 
meet on .
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
vertices of the face opposite to it. Find the integer closest to 
are distinct numbers in AP such that 
are also in AP. Find the smallest possible value of 
that satisfy 
![\[
r_1 \ge r_a,\quad r_2 \ge r_b,\quad r_3 \ge r_c
\]](http://latex.artofproblemsolving.com/8/2/1/8214c052ab9ad2c979414f92b6707fa6de33664f.png)
because if any of the radii were smaller than these, the circles would not be able to reach and touch the incircle.![\[
\frac{r_1 + r_2 + r_3}{r} \ge \frac{r_a + r_b + r_c}{r}
\]](http://latex.artofproblemsolving.com/b/b/a/bbadb40e6aa197207abfa907981e0a4a88926969.png)
![\[
\frac{r_a + r_b + r_c}{r} = \frac{h_a - 2r}{h_a} + \frac{h_b - 2r}{h_b} + \frac{h_c - 2r}{h_c}
\]](http://latex.artofproblemsolving.com/5/3/2/5329fec1f02a728b5f34817e0ed43ded663516ae.png)
![\[
= \left(1 - \frac{2r}{h_a}\right) + \left(1 - \frac{2r}{h_b}\right) + \left(1 - \frac{2r}{h_c}\right)
= 3 - 2r \left( \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c} \right)
\]](http://latex.artofproblemsolving.com/1/e/0/1e087ea87566fb4cf8cd2ae0cda3389247095eea.png)
![\[
\frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c} = \frac{a + b + c}{2S}
\]](http://latex.artofproblemsolving.com/a/4/a/a4ab49131584cf2028270e7248fcbb5d2c6a7adb.png)
![\[
3 - 2r \cdot \frac{a + b + c}{2S} = 3 - \frac{4S}{a + b + c} \cdot \frac{a + b + c}{2S} = 3 - 2 = 1
\]](http://latex.artofproblemsolving.com/0/f/3/0f3ada5ca22deeecc1644d79138942caf04d5134.png)
![\[
\frac{r_1 + r_2 + r_3}{r} \ge 1
\]](http://latex.artofproblemsolving.com/7/1/e/71ea14c8fb8e3860a1db427a0afcdb3b2fd6254b.png)
of a triangle 
are positive integers. Let
for any positive integer 
.
and 
,