Difference between revisions of "G285 2021 Fall Problem Set"
Geometry285 (talk | contribs) m (→Problem 2) |
Geometry285 (talk | contribs) m (→Problem 2) |
||
Line 5: | Line 5: | ||
==Problem 2== | ==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>, | + | 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>, <math>\sin(BAC)<\frac{5}{8}</math>, and the circumradius of <math>\triangle ABC</math> is an integer, find the smallest possible value of <math>AB^2+AE^2</math>. |
Revision as of 14:09, 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 the circumradius of is an integer, find the smallest possible value of .