Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "G285 2021 Fall Problem Set"

m (→‎Problem 1)
m
Line 3: Line 3:
 
==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 $15$ problems, $10$ multiple-choice, and $5$ free-response.

Problem 1

Larry is playing a logic game. In this game, Larry counts $1,2,3,6, \cdots$, and removes the number $r+p$ for every $r$th move, skipping $r+jp$ for $j \neq 0 \mod 3$, and then increments $p$ by one. If $(r,p)$ starts at $(1,3)$, what is $r+p$ when Larry counts his $100$th integer? Assume $\{r,p,j \} \in \mathbb{N}$

Problem 2

Let $\triangle ABC$ be a right triangle with right angle at $B$, and $AC=12$. Let $D$ denote the intersection of the cevian dropped from $B$ onto $AC$ such that $DA=DC$. If the reflection of point $B$ across $D$ lies on the circumcircle of $\triangle ABC$ as $E$, and $\sin(BAC)<\frac{5}[8}$ (Error compiling LaTeX. Unknown error_msg), find the smallest possible integer radius of the circumcircle of $\triangle EBA$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=G285_2021_Fall_Problem_Set&oldid=157590"