and 
. Prove that 
where 
are lengths of parallel sides and 
are other two sides.
. Prove that
be real numbers satisfying 
and ![\[a_{i+1} - 2a_i + a_{i-1} = a^2_i\]](http://latex.artofproblemsolving.com/c/d/8/cd878e4ac230853182da75c54dcbf7be05b305bc.png)
for 
Prove that 
for 
people, 
are wearing a jacket and 
are wearing a hat. If 
people are wearing neither, how many people are wearing both 
red marbles and 
blue marbles. The rules of the game are as follows:
t
Doing this, we get that 
Now, we can solve this by factoring. By using RRT, we get that 
is a factor. Upon division, we get 
is the factored form. Now, we know that our only positive solutions are 
so our values of 
are 
Summing these gives our answer of 
to 
in or something like that idk my friend explained it pretty wonkily idk if he even solved it right but anyhow we press on. Transformations suck.
be positive integers such that 
and 
Determine the value of 
Furthermore, note that 
Thus, the answer is just 
is on point 
and 
and 
is a point on 
and 
Given that 
calculate 
so 
Cross multiplying and solving this quadratic, we get that our solution is 
Thus, 
given that 
We know from 
that 
so substituting this gives us 
Now, plugging this in to 
and 
gives us 
Solving these equations gives us 
Thus, by Vieta's, the sum of the roots is 
thus by median properties (the 2:1 ratio split thingy), we know that 
(For reference, I missed this part in-contest) Now, we can calculate 
by the Pythagorean Theorem to be 
and thus 
Now, by Pythagorean theorem, we know that the altitude from 
is 
so our area is just 
Thus, simplifying and substituting 
we get that 
Simple guess and check gives us 
as our solutions. My dumb self forgot the former solution. Thus, 
giving us 
as our sum.![\[f(x) =
\begin{cases}
x - 9, & \text{if } x > 100 \\
f(f(x + 10)), & \text{if } x \leq 100
\end{cases}\]](http://latex.artofproblemsolving.com/4/e/5/4e5b4486030a59e74df648819a1a59f82622b3b7.png)
.
we get that 
Now if we try finding 
we will realize that 
this 
pattern continues until we once again reach 
thus 
Now return to the problem statement; 
but each inner function will just become 
hence the answer is 
be real numbers such that 
Find 
Thus, simplifying, we get that 
or that 
Plugging this in to the first equation, we get that 
so this means that 
Now, 
thus 
are on circle 
and 
Determine the path length from 
formed by segment 
and arc 
is the diameter, thus we can draw 
and so by Special Right Triangles we do NOT in fact get 
but rather 
is also an integer.
.
to be an integer. Plugging in factors of 
we see that there are 
integers that work.
such that 
is a multiple of 
Plugging in numbers gives us 
as our smallest value.
are real numbers such that 
and 
Calculate 
Furthermore, 
so 
meaning that 
.
and thus 
giving us 
will follow the rules such that![\[
p_1 = (0,0), \quad p_{i+1} = (x_i + 1, y_i) \text{ or } (x_i, y_i + 1), \quad p_{10} = (4,5).
\]](http://latex.artofproblemsolving.com/6/0/3/6036ef2619ffb1e91db0ab310388217e472490f2.png)
How many sequences 
are possible such that 
is the only point with equal coordinates?
Furthermore, we know that 
so our answer is just 
in the summation. Essentially, we have 
.
In other words, we want to find 
.
.
in such a polynomial is 
.
,
.
.
,
.
and 
,
and divide by 
.
and 
in our sum, subtract by 
.
term look kind of weird? Well, we actually notice 
,
.
,
,
is our answer!
,
and 
.
,
,
.
,
for when 
terms! Anyways, the answer is 
.
so our coefficient is just 
Find 
Thus, our answer is just 
on the positive integers using the rule that for 
For all prime 
and for all other 
Find the smallest possible value of 
can be written as the sum of two distinct, non-negative integer powers of 
.
etc.
we get that it is 
Thus, we note that for an integer 
Since 
we know that all values of 
satisfy our conditions. Thus, our answer is just 
be the set of positive integers of 
for some other positive integer 
Find the only three-digit value of 
is our solution. For these Pell Equations, all solutions are of the form 
for 
Thus, we plug in 
giving us 
as our answer.
Find the value of 
and let 
be our three-digit number that we remove. Thus, we can write 
By our condition, we know that 
or that 
Furthermore, we know that 
does not work, so we try 
We know that 
by the fact that it must be divisible by a 
Noting that 
doesn't work, we try 
or that 
This, miraculously, works, and thus our value of 
Y by
Let 
be the orthocenter of 
and 
its circumcenter. Suppose that is the midpoint of the altitude 
. Line perpendicular to 
trough intersects sides 
and 
at 
and 
, respectively. Prove that the midpoints of segments 
, 
and point are collinear.
be the orthocenter of 
and 
its circumcenter. Suppose that is the midpoint of the altitude 
. Line perpendicular to 
trough intersects sides 
and 
at 
and 
, respectively. Prove that the midpoints of segments 
, 
and point are collinear.
![[asy]
size(6cm);
defaultpen(fontsize(10pt));
pair B = (0,0);
pair D = (2,0);
pair C = (5,0);
pair H = (2,sqrt(3));
pair A = 2*H-D;
pair O = circumcenter(A,B,C);
pair M = (B+C)/2;
pair D1 = 2*O-H;
pair P = extension(H,rotate(90,H)*O,A,B);
pair Q = extension(H,rotate(90,H)*O,A,C);
pair X = extension(P,P+A-C,A,H);
fill(A--P--X--cycle,lightgray);
draw(A--B--C--cycle,linewidth(1.5));
draw(H--D1^^P--Q);
draw(P--X--A,dashed);
draw(B--H--C,dashed);
draw(O--M);
dot("$A$",A,N);
dot("$B$",B,SW);
dot("$C$",C,SE);
dot("$D$",D,dir(310));
dot("$M$",M,dir(280));
dot("$D'$",D1,S);
dot("$H$",H,NW);
dot("$O$",O,NE);
dot("$P$",P,NW);
dot("$Q$",Q,NE);
dot("$X$",X,S);
[/asy]](http://latex.artofproblemsolving.com/6/0/e/60e301202932805b91615f7728a142b4185181ca.png)
,
be the reflection of 
across 
,
.
and 
)
.
on 
such that 
.
as the corresponding sides are perpendicular. Hence,
so we are done.
,
and 
.
.
and 
.
.
.
,
and we have 
.
then midpoints of 
and 
are collinear, this is a classical result that can be deduce using complex numbers.
