text: pow:start
  
  # Problem of the Week
  
  The Problem of the Week will return in the Fall. We thank everyone who participated and we congratulate Yuqiao Huang, who is an undisputed leader this semester. For the Summer, we suggest reviewing all the problems from the Spring and working on the additional problems posted at the bottom of the provided solutions, and the following problem.
  
  Summer Challenge (due August 31)
  
  Fix a circle $\Gamma$. Let $T=\triangle ABC$ be a triangle inscribed in the circle $\Gamma$ which is not a right triangle and let $H$ be the orthocenter of $T$. The line $HA$ intersects the circle $\Gamma$ at $A$ and at a second point $A_1$ (which can be $A$ if the line is tangent to the circle). Likewise, the line $HB$ intersects $\Gamma$ at a second point $B_1$, and the line $HC$ intersects $\Gamma$ at a second point $C_1$. The triangle $\triangle A_1B_1C_1$ is again inscribed in $\Gamma$. We denote this triangle by $\Phi(T)$. Warning: $\Phi(T)$ can be a right triangle.
  
  a) Show that triangles $T$ and $\Phi(T)$ are congruent if and only if either $T$ is equilateral or the angles of $T$ are $\pi/7$, $2\pi/7$, $4\pi/7$.
  
  b) For every integer $k>0$ find the number $t_k$ of non-congruent triangles $T$ inscribed in $\Gamma$ such that $\Phi^k(T)$ and $T$ are congruent. Here $\Phi^k$ denotes the composition $\Phi\circ \Phi \circ \ldots \circ\Phi$ of $\Phi$ with itself $k$ times. Thus, according to a), we have $t_1=2$.
  
  c) Is it true that if $\Phi^k(T)$ and $T$ are congruent then $\Phi^m(T)=T$ for some $m$?
  
  d) Find and prove your own results about $\Phi$.
  
  ## Overview
  
  Every other Monday (starting 02/03/20), we will post a problem to encourage students (both undergraduate and graduate) to enjoy mathematics outside of the classroom and engage our mathematical community in the problem solving activity. If you have a solution and want to be a part of it, e-mail your solution to Marcin Mazur (mazur@math.binghamton.edu) by the due date. We will post solutions (from us) as well as novel solutions from participants and record the names of those who've got the most number of solutions throughout each semester.
explanation: The document is a problem of the week announcement and challenge for a mathematics community. It contains a detailed geometric problem with multiple parts (a, b, c, d) and an overview of the problem-solving activity. The document presents a math question but does not provide an answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: pow:start
  
  # Problem of the Week
  
  The Problem of the Week will return in the Fall. We thank everyone who participated and we congratulate Yuqiao Huang, who is an undisputed leader this semester. For the Summer, we suggest reviewing all the problems from the Spring and working on the additional problems posted at the bottom of the provided solutions, and the following problem.
  
  Summer Challenge (due August 31)
  
  Fix a circle $\Gamma$. Let $T=\triangle ABC$ be a triangle inscribed in the circle $\Gamma$ which is not a right triangle and let $H$ be the orthocenter of $T$. The line $HA$ intersects the circle $\Gamma$ at $A$ and at a second point $A_1$ (which can be $A$ if the line is tangent to the circle). Likewise, the line $HB$ intersects $\Gamma$ at a second point $B_1$, and the line $HC$ intersects $\Gamma$ at a second point $C_1$. The triangle $\triangle A_1B_1C_1$ is again inscribed in $\Gamma$. We denote this triangle by $\Phi(T)$. Warning: $\Phi(T)$ can be a right triangle.
  
  a) Show that triangles $T$ and $\Phi(T)$ are congruent if and only if either $T$ is equilateral or the angles of $T$ are $\pi/7$, $2\pi/7$, $4\pi/7$.
  
  b) For every integer $k>0$ find the number $t_k$ of non-congruent triangles $T$ inscribed in $\Gamma$ such that $\Phi^k(T)$ and $T$ are congruent. Here $\Phi^k$ denotes the composition $\Phi\circ \Phi \circ \ldots \circ\Phi$ of $\Phi$ with itself $k$ times. Thus, according to a), we have $t_1=2$.
  
  c) Is it true that if $\Phi^k(T)$ and $T$ are congruent then $\Phi^m(T)=T$ for some $m$?
  
  d) Find and prove your own results about $\Phi$.
  
  ## Overview
  
  Every other Monday (starting 02/03/20), we will post a problem to encourage students (both undergraduate and graduate) to enjoy mathematics outside of the classroom and engage our mathematical community in the problem solving activity. If you have a solution and want to be a part of it, e-mail your solution to Marcin Mazur (mazur@math.binghamton.edu) by the due date. We will post solutions (from us) as well as novel solutions from participants and record the names of those who've got the most number of solutions throughout each semester.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a problem of the week announcement and challenge for a mathematics community. It contains a detailed geometric problem with multiple parts (a, b, c, d) and an overview of the problem-solving activity. The document presents a math question but does not provide an answer.
  
  Label: question