[/quote]
,
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 
be the midpoint of side 
.
intersect 
at 
.
is tangent to 
.
so that both 
have odd numbers of divisors.
be real numbers such that 
.
of side length 
lies in the union of five equilateral triangles of side length 
,
is a polynomial of degree 
such that
.
.
is called good if there exists a permutation 
of the numbers 
,
and 
have different parities for every 
is a quadratic residue modulo 
for every 
is considered a quadratic residue modulo 
such that 
.
be a circle passing through 
and 
.
be the intersection of 
and 
and 
with the tangent to 
at 
.
be the second intersection of 
and 
and 
,
is the midpoint of 
.
is an arbitrary point on 
.
be the intersection of 
and 
.
intersects with the circumcircle of 
,
.
is the circumcenter of triangle 
.
.
where 
are integers such that 
for all 
.
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?
![\[
r_1 \ge r_a,\quad r_2 \ge r_b,\quad r_3 \ge r_c
\]](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)
Y by
Let 
be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, then
the oriented angle
is equal to the oriented angle DCB.
2.If
lies on the segment 
, then
the oriented angle
plus the oriented angle 
equals 
.
3.If the oriented angle
plus the oriented angle 
equals 0°, then
lines
and are parallel.
4.If the oriented angle plus the oriented angle equals , then
lines and are parallel.
be four distinct points in the plane.Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, thenthe oriented angle
is equal to the oriented angle DCB.2.If
lies on the segment 
, thenthe oriented angle
plus the oriented angle 
equals 
.3.If the oriented angle
plus the oriented angle 
equals 0°, thenlines
and are parallel.4.If the oriented angle
plus the oriented angle equals , thenlines
and are parallel.
