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satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，and any rectangular strip of width 1 covers at most two points of S.
+1 w
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
, 
, 
, 
, 
in some order.
index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs. made mc nats
and 
are erected outside an acute triangle 
Suppose that ![\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]](http://latex.artofproblemsolving.com/3/3/7/3378f0c4409d14bc0afddf2616992c13f624be3b.png)
Prove that lines 
and 
are concurrent.
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
such that 
is an integer for every positive integer 
and 
in the plane and a subset 
of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair 
of cities, they are connected with a road along the line segment if and only if the following condition holds: distinct from 
and 
, there exists 
such[/center]
is directly similar to either 
or 
.[/center]
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to , 
to , and 
to 
.
must be in the interior of 
. In other case as is convex, we can take a line passing through and not intersecting . So let be our point in the interior of quadrilateral , which we will denote with 
. Let 
be a line passing through and intersecting and 
in points and respectively. Let 
and 
be such points on that 
and 
. Let lines 
and 
intersect side in 
,
respectively. By 
we will denote area of triangle 
and the same thing for quadrilaterals... As lines 
,
and 
are lines dividing into two parts of equal area, we obtain equations ![\[ S(KQX)+S(XPL)=S(QMX)+S(NPX) \]](http://latex.artofproblemsolving.com/1/4/e/14e11850bd3579356df0fff5d47457b2060135b8.png)
but as 
we have 
which means that 
.Now let be a line passing through and parallel to this line intersects with lines 
and in points 
and respectively. As we have we have also 
,but there is also equality , and so on, if 
it would mean that 
which is impossible. We obtained that 
, so 
.
which means that is a parallelogram! 
Now , if is a parallelogram for point we can take intersection point of it's diagonals 
satisfying: 
