Difference between revisions of "G285 2021 Fall Problem Set"
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==Problem 1== | ==Problem 1== | ||
Larry is playing a logic game. In this game, Larry counts <math>1,2,3,6, \cdots </math>, and removes the number <math>r+p</math> for every <math>r</math>th move, skipping <math>r+jp</math> for <math>j \neq 0 \mod 3</math>, and then increments <math>p</math> by one. If <math>(r,p)</math> starts at <math>(1,3)</math>, what is <math>r+p</math> when Larry counts his <math>100</math>th integer? Assume <math>\{r,p,j \} \in \mathbb{N}</math> | Larry is playing a logic game. In this game, Larry counts <math>1,2,3,6, \cdots </math>, and removes the number <math>r+p</math> for every <math>r</math>th move, skipping <math>r+jp</math> for <math>j \neq 0 \mod 3</math>, and then increments <math>p</math> by one. If <math>(r,p)</math> starts at <math>(1,3)</math>, what is <math>r+p</math> when Larry counts his <math>100</math>th integer? Assume <math>\{r,p,j \} \in \mathbb{N}</math> | ||
+ | |||
+ | ==Problem 2== | ||
+ | Let <math>\triangle ABC</math> be a right triangle with right angle at <math>B</math>, and <math>AC=12</math>. Let <math>D</math> denote the intersection of the cevian dropped from <math>B</math> onto <math>AC</math> such that <math>DA=DC</math>. If the reflection of point <math>B</math> across <math>D</math> lies on the circumcircle of <math>\triangle ABC</math> as <math>E</math>, and <math>\sin(BAC)<\frac{5}[8}</math>, find the smallest possible integer radius of the circumcircle of <math>\triangle EBA</math>. |
Revision as of 01:35, 9 July 2021
Welcome to the Fall Problem Set! There are problems,
multiple-choice, and
free-response.
Problem 1
Larry is playing a logic game. In this game, Larry counts , and removes the number
for every
th move, skipping
for
, and then increments
by one. If
starts at
, what is
when Larry counts his
th integer? Assume
Problem 2
Let be a right triangle with right angle at
, and
. Let
denote the intersection of the cevian dropped from
onto
such that
. If the reflection of point
across
lies on the circumcircle of
as
, and $\sin(BAC)<\frac{5}[8}$ (Error compiling LaTeX. Unknown error_msg), find the smallest possible integer radius of the circumcircle of
.