rows and 
columns, with 
.
,
rows of this grid together occupy at least 
s
r
t
and 
has the property for every positive integer 
,
term is greater than the mean of the first 
terms, and the sum of its first 
be the number of such sequences satisfying 
.
.
has side length 
lies on segment 
such that line 
is perpendicular to line 
.
,
,
.
and 
satisfy 
,
,
.
is equal to 
,
be a triangle. Point 
lies on side 
,
lies on side 
,
lies on side 
such that 
,
,
.
be the foot of the altitude from 
to 
,
,![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
,
can be expressed as 
and tells Bob that the three resulting remainders are 
,
,
satisfying the system of equations 
The value of 
can be expressed as 
,
and 
are positive integers such that 
.
and 
inclusive. We say a sequence 
in 
,
that are at least 
is 
.
,
denote the value of the binary number obtained by reading the binary representation of 
has at least ten positive integer solutions 
such that 
and 
for 
.
.
such that 
and 
is a multiple of 
.
,
,
,
For example, 
.
functions 
whose domain and range are the set 
,
for all 
.
be an acute triangle with 
.
and 
intersect at point 
.
be the feet of the perpendicular from 
.
and 
lie on segments 
and 
,
.![\[
\angle KAL = \frac{1}{2} \angle BAC.
\]](http://latex.artofproblemsolving.com/f/a/d/fad437704b2dcdc39ce04b23f860d26b764fd3a2.png)
ad 
be two circles intersecting at 
The tangent like closer to 
and 
and 
find 
then we could just use the ratio us between the side lengths of these triangles to find 
and 
where 
is the digit sum of 
,
.
Y by
3 Pokemon each have they own number to use on a part of a lock. Star the Pikachu has the number 87, Luna the Eevee has the number 92, and Aero the Mew has the number 79. A part of their code is found in the expression 
.
To double check if it is right then it needs to follow
and 
. If it does follow it then you need to find the sum of the coefficients with the code the code itself, but if not replace the sum of the coefficients with the lowest possible sum of coefficients where 
in 
where X, Y, and Z are positive integers more than 0.
.To double check if it is right then it needs to follow
and 
. If it does follow it then you need to find the sum of the coefficients with the code the code itself, but if not replace the sum of the coefficients with the lowest possible sum of coefficients where 
in 
where X, Y, and Z are positive integers more than 0.![\[
S=87, \quad L=92, \quad A=79
\]](http://latex.artofproblemsolving.com/7/0/6/706ecc898a59947cf1ca523ef741e51b150c7660.png)
and![\[
C \equiv (3S + 5L - 2A) \bmod 101.
\]](http://latex.artofproblemsolving.com/b/4/2/b42238e566af9c4d19f415b39b896da125131c4a.png)
![\[
3S + 5L - 2A = 3 \times 87 + 5 \times 92 - 2 \times 79 = 261 + 460 - 158 = 563.
\]](http://latex.artofproblemsolving.com/8/2/a/82ad403299fa3e70d34fabf01e0674b29b710638.png)
Modulo 101:![\[
C \equiv 563 \bmod 101 \equiv 58.
\]](http://latex.artofproblemsolving.com/f/1/1/f11d2ae75bad57c3ffd00ac90361b1de083978a3.png)
![\[
C \equiv 5 \pmod{8} \quad \Rightarrow \quad 58 \mod 8 = 2 \neq 5,
\]](http://latex.artofproblemsolving.com/4/1/3/413dcbdccfcd9a047b97186b268767a8403c5cca.png)
![\[
C \equiv 3 \pmod{11} \quad \Rightarrow \quad 58 \mod 11 = 3,
\]](http://latex.artofproblemsolving.com/f/e/4/fe44f945fdc28f3cb5f06fb10a1118b4d3ff4046.png)
so 
fails.
:![\[
C \equiv 5 \pmod{8}, \quad C \equiv 3 \pmod{11}.
\]](http://latex.artofproblemsolving.com/e/c/2/ec2f2715f16df5e4a98af5a802711f4c4da9b550.png)
.![\[
8k + 5 \equiv 3 \pmod{11} \Rightarrow 8k \equiv -2 \equiv 9 \pmod{11}.
\]](http://latex.artofproblemsolving.com/8/e/b/8eb0aa838eaf924a04d62c2af0219578bf9775b8.png)
![\[
k \equiv 9 \times 7 = 63 \equiv 8 \pmod{11}.
\]](http://latex.artofproblemsolving.com/2/b/2/2b2ec5684ed030fbb1cd071f1b6e3481f205ef96.png)
![\[
C = 8 \times 8 + 5 = 69.
\]](http://latex.artofproblemsolving.com/4/6/7/467e8efb865092b581f8e35cbce291cd1d8a348e.png)
such that:![\[
(XZ + YL - ZA) \equiv C \pmod{101}.
\]](http://latex.artofproblemsolving.com/0/f/6/0f6fad6b8deb448a3a3c37e43ea6fb575fc687ee.png)
![\[
XZ - ZA + 92Y \equiv 69 \pmod{101} \implies Z(X - 79) + 92Y \equiv 69 \pmod{101}.
\]](http://latex.artofproblemsolving.com/9/b/6/9b6a14813ad212803381e42d934c7a33c462893a.png)
(so 
)![\[
92Y \equiv 69 \pmod{101}.
\]](http://latex.artofproblemsolving.com/6/8/a/68a9a601fd5039205765f9f6f250a4ee14b325cc.png)
![\[
Y \equiv 69 \times 56 = 3864 \equiv 26 \pmod{101}.
\]](http://latex.artofproblemsolving.com/3/f/b/3fb9c0fc1c2f2173001a6436d53bf98d9a2f006b.png)
,
.![\[
X + Y + Z + C = 79 + 26 + 1 + 69 = \boxed{175}.
\]](http://latex.artofproblemsolving.com/6/6/5/6651123b5cbfd6455bd7deae1d1f12c4579bdfff.png)
