AoPS Academy
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.
is the center of the inscribed circle, point 
lies on the side of 
, with 
. Prove that the distance from point to line 
is equal to the diameter of the circle inscribed in triangle 
from the set of non-negative integers to itself such that 
, whenever 
, are non-negative integers satisfying 
.
and radius 
.
. and radius ..a^{2}+b^{2}+c^{2} \geq 4sqrt{3}S
complex numbers with 
such that
Find the value of 
.
and 
be complex numbers. For a complex number 
where 
, define the norm 
to be the distance of 
from the origin, or 
. Let 
be the minimum value and be the maximum value of 
for all complex numbers where 
. Find 
.
such that,
be 
, and 
. What is the sum 
? through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?
such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
be the number of positive integers such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including and . Find the remainder when is divided by 
.
ordered pairs 
, knowing that there are 2013 ordered pairs in each permutation we can conclude that the
which 
and every Number in the permutation is considered modulo 2014 ( 2014 is an excuse and we consider it 2014 ) there is exactly one index 
such that 
and 
appear consecutively ( with the correct order ) in 
which is 
( mod 2014 ) and 
uniquely determine 
and 
. Prove that
.
Let 
.
From here we must have,
Observe that thus far the RHS of the divisibility statement has been strictly positive, following from the fact that 
for 
.
to be,
,
.
.
}
and 
in this set at last in one of the permutation
permutations.
