[/quote]
,
such that 
be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
be all the positive integers smaller than 
such that
for all 
.
satisfied 
Find minimum and maximum of 
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
and 
and 
.![\[
\frac{1}{x} + \frac{1}{y} + 2 \sqrt{\frac{2}{x^2 + y^2}} + 4 \geq 4 \left( \sqrt{\frac{2}{x^2 + 1}} + \sqrt{\frac{2}{y^2 + 1}} \right).
\]](http://latex.artofproblemsolving.com/1/b/8/1b8ab2e343a3043271f5a392d32b12aca748f1b9.png)
be all the positive integers smaller than ![\[gcd(N, c_i + c_{i + 1}) \neq 1\]](http://latex.artofproblemsolving.com/1/0/7/1075290d442e8c1f54237aad8ad07126f455f1f1.png)
for all 
of positive integers is called central if for every positive integer 
of positive integers such that for every central sequence 
,
.
,
,
are concurrent and the point of concurrence is 
,
are on sides 
such that 
is a cyclic quadrilateral. Let 
and 
.
such that 
.
be the midpoints of 
.
.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
?
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of 
are concyclic.
Y by Adventure10
Let 
be positive integers, where 
. It is given that there exist a positive integer 
such that
holds for all integers
. Prove that the above equation is true for all positive integers 
. (Here 
is the sum of digits of 
taken in base 
).
be positive integers, where 
. It is given that there exist a positive integer 
such that
holds for all integers
. Prove that the above equation is true for all positive integers 
. (Here 
is the sum of digits of 
taken in base 
).
.
where 
is some large number and 
.
(
is too small to interfere with 
)
.
.
and so 
.
from 
to 
condition and the 
condition are irrelevant and seems very strange that such a problem would appear in a China TST.
for 
and 
.
