AoPS Academy
. Prove that
Let 
. Prove that
Let . Prove that
Let . Prove that
= 2∠AMP.
Let ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
denote the set of natural numbers less than or equal to 
![$f(m,n) = \sum_{(x_1,x_2,x_3, \cdots, x_m) \in [n]^{m}} \frac{x_1}{x_1 + x_2 + x_3 + \cdots + x_m} \binom{n}{x_1} \binom{n}{x_2} \binom{n}{x_3} \cdots \binom{n}{x_m} 2^{\left(\sum_{i=1}^{m} x_i\right)}$](http://latex.artofproblemsolving.com/f/d/f/fdf7b2fcfb960e93d59f3f84cef4372262c69499.png)
?
so that both and 
have odd numbers of divisors.
which is defined as all the natural numbers listed in order, side by side, behind a decimal point, without commas? If so, what is it?
and 
are primes, then can 
be a perfect square?
be a trapezoid such that 
lie on a circle with center 
, and side 
is parallel to side 
. Diagonals 
and 
intersect at point 
, and 
. It is given that 
. It is also known that the difference in length between and can be expressed in the form 
, where 
and 
are positive integers and is square-free. Compute the value of 
.
be a triangle and let 
be a circle in its plane passing through 
and 
Suppose there exist circles 
such that for 
is externally tangent to 
and passes through 
and 
where 
for all 
. Prove that 
is determined by its center 
, since we know that it passes through 
. We also know that for all 
are collinear. We must prove that in these conditions, 
.
, the angles 
and 
are equal (where 
is the circumcenter of 
). The result would then follow, since 
would be an involution on the perpendicular bisector 
of 
, mapping a point on this line to its harmonic conjugate wrt and the intersection of the tangents in 
to the circumcircle of . Since 
, we must have , as desired.
be any point on 
, and put 
. Then, 
make equal angles with 
.
.
be a triangle and let 
be a circle in its plane passing through 
and 
Suppose there exist circles 
such that for 
is externally tangent to 
and passes through 
and 
where 
for all 
. Prove that 
and 
are externally tangent at 
, we see that is the midpoint of 
and 
.
and 
and those through and 
, though . . . maybe there is a way to multiply the coordinates by these factors such that it works out?
. Then, measures of the angles 
are easy to get, we find that 
, and we also need 
on the same side of 
(which follows from the fact that we need 
between 
and 
, blah.
by 
.As usual denote the side opposite to vertex 
by 
for 
.
,where is the centre of .
.
.
.Also because 
we note that 
passes through and 
.We thus obtain 
.
of for each . To this end, let 
, 
and 
.![[asy] size(8cm); defaultpen(fontsize(9pt)); pair A2 = (0,0); pair O1 = (-8,0); pair O2 = (5,0); pair A1 = abs(A2-O1)*dir(285)+O1; pair A3 = abs(A2-O2)*dir(245)+O2; pair O3 = extension(O2, A3, midpoint(A1--A3), midpoint(A1--A3)+dir(A1-A3)*dir(90));
filldraw(circle(O1,abs(A1-O1)), invisible, deepgreen); filldraw(circle(O2,abs(A2-O2)), invisible, deepgreen); filldraw(circle(O3,abs(A3-O3)), invisible, deepgreen); filldraw(A1--A2--A3--cycle, invisible, red);
pair J = (0,-3.2); draw(O1--O2--O3, deepcyan+dotted);
dot("$O_1$", O1, dir(90), deepcyan); dot("$O_2$", O2, dir(90), deepcyan); dot("$O_3$", O3, dir(-90), deepcyan); dot("$A_1$", A1, dir(A1-J), red); dot("$A_2$", A2, dir(A2-J), red); dot("$A_3$", A3, dir(A3-J), red);
label("$\alpha$", A1, 2.4*dir(J-A1), red); label("$\beta$", A2, 2*dir(J-A2), red); label("$\gamma$", A3, 2*dir(J-A3), red); [/asy]](http://latex.artofproblemsolving.com/f/c/4/fc494765ec438d8696fe8408d517d9a258a93ac6.png)
. Then we compute 
and repeating the same calculation another round gives ![\[ \measuredangle O_7 A_2 A_1 = k-(k-\theta) = \theta \]](http://latex.artofproblemsolving.com/0/f/a/0fa7599bb0b3a50b16fe7243944801c9e4a62bf5.png)
with 
. This implies 
, so 
.
be the center of for 
.![\[\measuredangle O_7A_1A_2=-\measuredangle A_2A_1A_3-\measuredangle A_3A_1O_6=-\measuredangle A_2A_1A_3-\measuredangle O_6A_3A_1=-\measuredangle A_2A_1A_3+\measuredangle A_1A_3A_2+\measuredangle A_2A_3O_5=\]](http://latex.artofproblemsolving.com/8/5/f/85f120aa56a3a988f80dddf9a0c33e58f857ef36.png)
![\[-\measuredangle A_2A_1A_3+\measuredangle A_1A_3A_2+\measuredangle O_5A_2A_3=-\measuredangle A_2A_1A_3+\measuredangle A_1A_3A_2-\measuredangle A_3A_2A_1-\measuredangle A_1A_2O_4.\]](http://latex.artofproblemsolving.com/e/6/3/e6375008fcddfadf9dd7fb940c8e2ca8c3c1cb39.png)
A similar angle chase finishes.
be the center of . Also let 
. Observe that is uniquely determined by 
, and is uniquely determined by 
, so if we can show those are equal we're done. Now, let 
. By basic angle chasing (since 
are collinear), we can find:
hence 
as desired. 
with center and ratio 
(it's not important). Define 
and 
. Note that 
is a line for 
because 
pass through . Also note that and 
are parallels for 
because and are tangents in . We want that 
, it's sufficient to prove that 
. Now we will use directed angles. Let 
:
and 
are parallels as desired.
be the center of . Define 
Define 
It suffices to show 
In fact, note that
Note that this process is the exact same for 
, and since function 
is cyclic with order 
, we are done.
, where is the center of . Furthermore, let 
be the angles of the triangle. Then, check that we have the sequence 
Upon applying another sequence of three transformations, this returns the identity 
, so the centers of and coincide, which suffices.
and for 
. Name these 
. Since 
is just 
reflected over the perpendicular bisecter of some side, and because the internal tangent is not dependent on the circle (it is in fact what makes the next circle), we just have to prove that 
.
, 
, and 
, , and 
is less than (otherwise 
does not exist). Thus we call this 
and we know that the angle formed by , , and 
is 
we can see that this is equal to the angle formed by , , and 
, so we know that the angle formed by , , is 
. Continuing this pattern for subsequent angles we find that the next angles are 
, 
, 
, and . Thus we find that 
and we are done. 
, let be the center of and 
. Then for 
, the tangency condition tells us that 
, and are collinear, so
Therefore, applying this identity in succession, we have
Thus, 
and 
. This is enough to imply that 
since we would otherwise have 
(using non-directed angles), which is impossible since 
. So, we are done.
denote the center of each circle . Let 
and 
denote respectively the perpendicular bisectors of segments 
and 
. Note that for all 
, the center 
of lies on the line 
. First, note that since is a center of a circle passing through points 
and 
, it must lie on 
. Further, since it is externally tangent to , which clearly passes through as well, also lies on 
.
and 
are collinear since then and thus as desired. To see why, simply note that![\[\measuredangle O_1A_1O_4 = \measuredangle O_4A_2O_1 = \measuredangle O_5A_2O_2 = \measuredangle O_2A_3O_5 = \measuredangle O_3A_3O_6 = \measuredangle O_6A_1O_3\]](http://latex.artofproblemsolving.com/7/5/2/7520c96bafde55ccfe30e6ef4b233f2722524045.png)
which implies that and are collinear finishing the proof.
be the angles of the triangle, and 
Then observe we obtain the sequence 
Doing this one more time yields 
the original angle so we are done. QED
and 
Prove that
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 
.
.
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 
(in degrees).
that satisfy 
map to 
.
and 
are both tangent and pass through 
and 
.
is a line through 
and similarly, 
passes through 
.
.
passes through both 
passes through 
.
and passes through 
is a line passing through 
is a line parallel to 
passes through 
is a circle tangent to 
is a line through 
.
and 
respectively and let 
and 
respectively. Since 
and 
are tangents to 
.
,
.![\[ \angle PA_2'A_3' = \angle PA_3'A_2'\implies \angle PA_3'A_2' = \angle QA_2'A_3'\implies QA_2'\parallel PA_3'\implies \omega_1'\parallel \omega_6'\implies \omega_1' = \omega_7'\]](http://latex.artofproblemsolving.com/4/b/8/4b89bde7aabab5eee9d9c2d6933e39d17a98f810.png)
and 
,
be 
.
are externally tangent at 
,
.
,
,
,
.
,
is the perpendicular bisector of the next side).
,
bisecting 
is possible)
as the centers of each respective circle. Let 
.
We can apply this transformation twice to get that,![\[\angle{A_{2}A_{1}O_{7}} = \alpha\implies \omega_{1} = \omega_{7}\]](http://latex.artofproblemsolving.com/1/3/a/13ae292ef9ffdc235200134f3e1d5e70660c53c7.png)