such that 
.
and 
.
be such that the midpoint of 
is 
.
be the point on the opposite side of 
as 
on the circumcircle of 
and let 
be the circumcircle of 
.
at 
and the tangent to 
at 
,
is parallel to 
red balls and 
of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
and 
,
in degrees as 
ranges over all real numbers.
and let 
(ie. 
with 
removed). Determine the number of injective* functions 
such that 
for all 
.
for all distinct 
and 
in the domain of 
possible triangles instead of 
)
be the centroid of 
as the centroid of 
and define 
and 
similarly. Points 
and 
are selected randomly from the sets 
,
,
,![$\frac{[G_AG_BG_c]}{[ABC]}.$](http://latex.artofproblemsolving.com/d/3/8/d38c1ec527c34f1e1519bba772a0765849e6fe7d.png)
chessboard, 
white knights and 
different weighted coins of probability 
and 
of coming up heads. Alice and Bob each choose a random, not necessarily distinct coin and flip it three times. Given that the two players flipped the same sequence of heads and tails, find the probability that they chose the same coin.
is a monic polynomial of smallest degree satisfying 
.
.
denotes the fractional part function)
bigger than 
water if the last four digit is 
.
be a positive integer. Prove that![\[
\underbrace{99\dots 99}_{k-1}4\underbrace{00\dots 00}_{k-1}9
\]](http://latex.artofproblemsolving.com/2/f/c/2fc13c37764c96401d76f6288476d7372c17435b.png)
can't be a prime number.
,![\[
x^2+2x-3\underbrace{99\dots 99}_{2m}.
\]](http://latex.artofproblemsolving.com/5/4/d/54d311ebb1f6d92ffc1d2b8f4d6d5de225b349e9.png)
be the set of positive integers less than 
?
,
Y by
Let 
be a positive integer. Consider a figure of a equilateral triangle of side and splitted in 
small equilateral triangles of side 
. One will mark some of the 
vertices of the small triangles, such that for every integer 
, there is not any trapezoid(trapezium), whose the sides are 
, with all the vertices marked. Furthermore, there are no small triangle(side ) have your three vertices marked. Determine the greatest quantity of marked vertices.
be a positive integer. Consider a figure of a equilateral triangle of side and splitted in 
small equilateral triangles of side 
. One will mark some of the 
vertices of the small triangles, such that for every integer 
, there is not any trapezoid(trapezium), whose the sides are 
, with all the vertices marked. Furthermore, there are no small triangle(side ) have your three vertices marked. Determine the greatest quantity of marked vertices.This post has been edited 1 time. Last edited by mathisreal, Feb 7, 2022, 1:54 PM
.
.
if 
or 
if 
.
that if 
is the answer for ![\[f(n) \leq f(n-2)+n+1\]](http://latex.artofproblemsolving.com/d/e/4/de47c9fb8d386c1b610049419f518a2373ee8058.png)
we have maximum 
marked vertices, and we have 
more points, we consider these points in a path (the red path in the picture).
where 
is the number of packets of size 1 between the packets of size 2.![\[2a+b+ (a+b-1)+(a-1) = 2(2a+b)-2 \leq 2n+1\]](http://latex.artofproblemsolving.com/f/9/6/f96258009802b09fea4da8a7e6a4860f230936e7.png)
The marked points, the spaces between each packet and the "extra" not marked point between the packets of size 2.
but 
is integer then 
,
,
.
and then if ![\[2+4+\ldots+(2k+2) = 2 \left(1+2+\ldots+k+1\right)=(k+1)(k+2) \]](http://latex.artofproblemsolving.com/5/4/6/546410523d16f7067d0fc66067b282a3561aba40.png)
and if ![\[1+3+\ldots+ 2k+1 = (k+1)^2\]](http://latex.artofproblemsolving.com/b/1/a/b1a84d998729fae6ef35c10a254294ad426c32bf.png)
and 
depending of the parity of 
