AoPS Academy
be a triangle with circumcircle 
and let 
be the 
-excircle. Let 
and 
be the intersection points of and . Let 
and 
be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle 
intersects the tangent line at to the circumcircle of the triangle 
at a point 
. Prove that 
.
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
, where 
. Each of the 
subsets of 
is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set 
, we then write 
for the number of subsets of 
that are blue.
and 
of , ![\[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]](http://latex.artofproblemsolving.com/1/8/1/18184e9196e6c73601653d85f1cfd6d9798a8e43.png)
is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer 
on the board with 
, and on Bob's turn he must replace some even integer on the board with 
. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends. and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
+3 w
where 
is an odd prime, and let![\[
S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)}
\]](http://latex.artofproblemsolving.com/c/0/b/c0bf0826fbe8897ebaf0d4b3c861120d502baefd.png)
Prove that if 
for integers 
and , then 
is divisible by .
+1 w
be the sequence defined by 
and 
for each integer 
. Suppose that 
is a prime and 
is a positive integer. Prove that some term of the sequence is divisible by 
.
. Perform an inversion about with radius 
. Let the -median intersect the circle with diameter 
at 
and 
at 
. Note that this inversion simply swaps 
and 
, and and . Morever, 
inverts to the nine-point circle of . Finally, the line 
inverts to itself, so these three objects all pass through , a point on , so inverting back yields the desired conclusion.
be the foot of the perpendicular from to . Since the reflection of lies on the circumcircle of 
, it follows that 
and are congruent. In particular, the reflection of over and lie on . Denote the reflection of over and as and 
, respectively. By power of a point, 
. It follows that 
are concyclic. Since 
, it follows that 
as well.
be the foot of the altitude from to , and 
be the ortic triangle.
and 
are tangent to 
, 
is a harmonic quad. Project it through to line , and goes to , goes to 
, goes to 
, and goes to the harmonic conjugate of wrt. (say, ). But by ceva-menelao, 
and concurr at . Moreover, we can see that is the radical center of , and 
and thus 
is the rad ax of and which implies is in , but since it's also in the median, we have that 
and we're done.
and 
: Prove that :
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
,
.
and let 
are midpoints of 
,respectively.We know that 
is cyclic or 
, so 
lies on radical axis of the circles 
and 
,hence 
or 
.But we have by 
lies also radical axis 
,so we are done !
,
.
and 
.
is a parallelogram 
and 
.
.
.
is cyclic 
