AoPS Academy
has points 
,
,
on segment 
in that order, where is between 
and , and 
and 
trisect angle 
. If 
, 
, and 
, find 
.
has the property that the sum of its first 
terms is divisible by for every positive integer 
. Let 
be the number of such sequences satisfying 
. Compute the remainder when is divided by 
.
has side length 
. Point 
lies on segment 
such that line 
is perpendicular to line 
. Given that 
, the area of can be expressed as 
, where 
and are relatively prime positive integers. Compute 
.
and 
satisfy 
, 
, and 
. If the number of possible ordered pairs 
is equal to 
, compute the remainder when is divided by . be a triangle. Point 
lies on side , point 
lies on side 
, and point 
lies on side 
such that 
, 
, and 
. Let 
be the foot of the altitude from 
to . Given that 
, 
, and ![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
, the value of 
can be expressed as , where and are relatively prime positive integers. Compute .
and tells Bob that the three resulting remainders are 
, 
, and , in some order. For how many values of is it possible for Bob to uniquely determine ?
satisfying the system of equations 
The value of 
can be expressed as 
, where 
and 
are positive integers such that is square-free. Compute 
. be the set of all monotonically increasing six-term sequences whose terms are all integers between 
and 
inclusive. We say a sequence 
in is symmetric if for every integer 
, the number of terms of 
that are at least 
is 
. The probability that a randomly chosen element of is symmetric is 
, where and are relatively prime positive integers. Compute ., let 
denote the value of the binary number obtained by reading the binary representation of from right to left. Find the smallest positive integer 
such that the equation 
has at least ten positive integer solutions . be a quadratic polynomial with a positive leading coefficient. There exists a positive real number 
such that 
and 
for 
. Compute 
.
such that 
and 
is a multiple of 
.
is cut from equilateral triangle 
![[asy]
draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1));
draw((2,0)--(3,0)--(2.5,.87));
label("3", (0.75,1.3), NW);
label("1", (2.5, 0), S);
label("1", (2.75,.44), NE);
label("A", (1.5,2.6), N);
label("B", (3,0), S);
label("C", (0,0), W);
label("D", (2.5,.87), NE);
label("E", (2,0), S);[/asy]](http://latex.artofproblemsolving.com/c/8/3/c83a21ce759a694a577eb8fc114c18cc14a92710.png)
because we are adding 1 and subtracting 2 from the triangle's perimeter.
which is just![\[ AC+CE+ED+DA=3+2+1+2=\boxed{\textbf{(E)} \ 8}. \ \blacksquare\]](http://latex.artofproblemsolving.com/6/4/b/64bdfd779f0b9b2fd6d98a5ade5b961f6865395c.png)
