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 
distinct from 
and 
,
such[/center]
is directly similar to either 
or 
.
is directly similar to 
if there exists a sequence of rotations, translations, and dilations sending 
to 
to 
to 
.
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
and 
Prove that
be a positive integer, and let 
be nonnegative integers such that 
Prove that![\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]](http://latex.artofproblemsolving.com/c/4/9/c49d0ccfef69402d05dc44d721172498c605cf7e.png)
Note: 
for all nonnegative integers 
.
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
,
are all greater than 
is an integer for every positive integer 
be nonnegative real numbers such that 
.
Prove that
be a prime, and let 
be integers. Show that there exists an integer ![\[a_1 + k, a_2 + 2k, \dots, a_p + pk\]](http://latex.artofproblemsolving.com/7/6/2/7624ce20fc9bfb798126ffacdb34110159ef0094.png)
produce at least 
distinct remainders upon division by 
and 
be a 
grid of unit squares.
or 
rectangle. A subset 
of grid squares in 
edges of the grid squares.
Y by Rounak_iitr
Let 
be a triangle such that 
. Let 
be a point on the segment 
such that 
. The angle bisectors of 
and 
meet the segments 
and 
at 
and 
respectively. Let 
be the circumcircle of 
and 
be the midpoint of 
. The ray 
meets at 
and the line through parallel to meets again at 
. If 
meets at 
, show that 
, and are concurrent.
Authored by Nikola Velov
be a triangle such that 
. Let 
be a point on the segment 
such that 
. The angle bisectors of 
and 
meet the segments 
and 
at 
and 
respectively. Let 
be the circumcircle of 
and 
be the midpoint of 
. The ray 
meets at 
and the line through parallel to meets again at 
. If 
meets at 
, show that 
, and are concurrent.Authored by Nikola Velov
This post has been edited 3 times. Last edited by steppewolf, Feb 9, 2025, 10:41 PM
and 
are orthogonal and 
,
.
so 
.
concurrent.So if 
then 
then 
concurrent at 
and 
.
and 
we get 
.
,
is cyclic. Define 
,
Ths implies that 
.
,
,
are concurrent as needed.
at 
at 
.
take pencil from 
is harmonic.So, 
is symmedian in 
.
it means 
.
gets from 
isosceles trepazoid.
and 
concur, so 
.
is harmonic, but this follows from
