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 
.
be an acute triangle such that 
Let 
be the midpoint 
the orthocenter of 
the midpoint of 
and 
the circumcenter of 
Prove that 
is a parallelogram.
with orthocenter 
and be the midpoint of 
, if 
is the reflection of over , then 
is diameter of 
.”
, we get that 
. is the midpoint of 
and 
is the midpoint of 
.
.
be .
is .
so 
. Hence, 
, which proves the Lemma. 
. By Lemma 1, 
and 
. Hence, 
.
.
and 
be on 
respectively such that 
and 
. Since 
is cyclic, 
so 
(ratios of diameters of circumcircles=ratios of sides). 
. Hence, 
and 
so is a parallelogram. 
and 
bisect each other but just note that is the orthocenter of and that is the diameter of the nine-point circle of . It's well known that the circumcenter of the nine-point circle is the midpoint of the line that joins the circumcenter and the orthocenter, in this case , and we're done.
and 
are both perpendicular to and thus parallel. Further we have 
, so 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
?
to 
and 
because 
,
and 
.
and 
.
because 
,
,
and all of them have modulus equal to 
.
so