be non-negative real numbers such that 
Prove that
be an acute non-isosceles triangle with circumcircle 
,
and orthocenter 
.
through 
through 
and 
lie on a circle, which is tangent to 
and intouch triangle 
.
be the foot of the perpendicular from 
onto 
.
,
intersect the sides 
respectively. Finally, let the rays 
,
meet the circumcircle of 
in 
,
respectively. Prove that the tangent from 
meet on the circumcircle of 
be the set of sequences of length 2018 whose terms are in the set 
and sum to 3860. Prove that the cardinality of ![\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]](http://latex.artofproblemsolving.com/d/e/8/de857941803349e275e9c9aa968a201f2988cb8a.png)
where 
denotes the Fibonacci sequence.**
be a sequence such that 
and 
.
and 
)
is a derivable function with the property 
,
,
be a function that admits a primitive 
such that 
if 
such that 
be holomorphic functions on 
,
.
given by 
takes a maximum value in 
be constant on 
Y by
Let 
be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number 
in ![$[0, 1]$](//latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
. At the start of the game, the only information Minivan has about is in . He needs to now learn about based on the following protocols: at each turn of his, Minivan chooses a number 
and submits to Megavan, where Megavan replies immediately with one of 
, 
, or 
, subject to two rules:
The answers in the form of and must be truthful;
Define the score of a round, known only to Megavan, as follows: 
if the answer is in the form and , and 
if in the form . Then for every positive integer 
and every consecutive rounds, at least one round has score no more than 
.
Minivan's goal is to produce numbers
such that 
and 
. Let 
be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that 
Proposed by Anzo Teh Zhao Yang
be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number 
in ![$[0, 1]$](http://latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
. At the start of the game, the only information Minivan has about is in . He needs to now learn about based on the following protocols: at each turn of his, Minivan chooses a number 
and submits to Megavan, where Megavan replies immediately with one of 
, 
, or 
, subject to two rules:
The answers in the form of and must be truthful; Define the score of a round, known only to Megavan, as follows: 
if the answer is in the form and , and 
if in the form . Then for every positive integer 
and every consecutive rounds, at least one round has score no more than 
.Minivan's goal is to produce numbers
such that 
and 
. Let 
be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that 
Proposed by Anzo Teh Zhao Yang
This post has been edited 2 times. Last edited by navi_09220114, Saturday at 12:59 PM
,
.
-![$x\in \left[m_{i-1}-\frac12s_{i-1}, m_{i-1}+\frac12s_{i-1}\right]$](http://latex.artofproblemsolving.com/2/a/3/2a3e318fab753af517b655ad60708d410894faea.png)
,
on the 
-
accordingly.
or 
,![$x\in \left[m_{i-1}-\frac12s_{i-1}, m_{i-1}\right]$](http://latex.artofproblemsolving.com/5/6/7/567d7594b31f16043e53db6c83f844eee982d5c7.png)
or ![$x\in \left[m_{i-1}, m_{i-1}+\frac12s_{i-1}\right]$](http://latex.artofproblemsolving.com/a/d/c/adc6f068578f06fc005fc00201a43b618296a13a.png)
so we can write 
and set 
depending on whether 
,
.
as the guess, so ![$x\in\left[m_{i-1}-\frac1{k+1}, m_{i+1}+\frac1{k+1}\right]$](http://latex.artofproblemsolving.com/1/b/7/1b77a2ba523b1db3f18d2a906a5953d81c6c0c64.png)
.
.
,
,
is the 
.
this is trivial. Now induct.![\[s_{t_i-1}\le \min\left\{s_{t_{i-1}}, \frac{2}{t_i-t_{i-1}}\right\}\]](http://latex.artofproblemsolving.com/9/4/c/94cba7e30e7126e552ee7e1890021d9cf6f30c7a.png)
![\[\frac1{s_{t_i-1}}\ge \max\left\{\frac1{s_{t_{i-1}}}, \frac{t_i-t_{i-1}}{2}\right\}\ge\frac12\left(\frac1{s_{t_{i-1}}}+ \frac{t_i-t_{i-1}}{2}\right)\]](http://latex.artofproblemsolving.com/a/4/e/a4e7e9d0389a779778ce4daa01078cf25092f73a.png)
![\[\frac1{s_{t_i}}\ge \frac1{s_{t_{i-1}}}+ \frac{t_i-t_{i-1}}{2}\ge \frac{t_{i-1}+2}2+\frac{t_i-t_{i-1}}2=\frac{t_i+2}2\]](http://latex.artofproblemsolving.com/3/7/c/37c7498d6804e0707968e73a289395048e5844c8.png)
hence 
.
be the largest such that 
.
,
hence done. Otherwise, note that 
,
.
be Minivan's first guess, and WLOG 
.
.
and 
queries we will show that any value of 
works. For any 
,
which is clear as 
are in 
that Minivan produces must satisfy 
,
.
