high tech FE as J1?!
by imagien_bad, Mar 20, 2025, 12:00 PM
Let 
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.
Note: A function is bijective if for every integer 
, there exists exactly one integer 
such that 
.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.Note: A function
is bijective if for every integer 
, there exists exactly one integer 
such that 
.This post has been edited 1 time. Last edited by imagien_bad, Mar 20, 2025, 12:09 PM
Stanford Math Tournament (SMT) 2025
by stanford-math-tournament, Feb 1, 2025, 3:16 AM
Stanford Math Tournament
We are excited to announce that registration is now open for Stanford Math Tournament (SMT) 2025!
This year, we will welcome 800 competitors from across the nation to participate in person on Stanford’s campus. The tournament will be held April 11-12, 2025, and registration is open to all high-school students from the United States. This year, we are extending registration to high school teams (strongly preferred), established local mathematical organizations, and individuals; please refer to our website for specific policies. Whether you’re an experienced math wizard, a puzzle hunt enthusiast, or someone looking to meet new friends, SMT has something to offer everyone!
Register here today! We’ll be accepting applications until March 2, 2025.
For those unable to travel, in middle school, or not from the United States, we encourage you to instead register for SMT 2025 Online, which will be held on April 13, 2025. Registration for SMT 2025 Online will open mid-February.
For more information visit our website! Please email us at stanford.math.tournament@gmail.com with any questions or reply to this thread below. We can’t wait to meet you all in April!
![\[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]](http://latex.artofproblemsolving.com/2/d/2/2d248ce9756f0f09265dd538de3b16b22a944c46.png)
2v2 (bob lost the game)
by GoodMorning, Mar 23, 2023, 10:07 PM
A positive integer is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer 
on the board with 
, and on Bob's turn he must replace some even integer on the board with 
. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer 
on the board with 
, and on Bob's turn he must replace some even integer on the board with 
. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of
and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
such that if any 
,Harmonic Mean
by Happytycho, Feb 21, 2016, 11:00 AM
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of 
and 
is closest to which integer?
and 
is closest to which integer?
Metamorphosis of Medial and Contact Triangles
by djmathman, Apr 30, 2014, 9:30 PM
Let 
be a triangle with incenter 
, incircle 
and circumcircle 
. Let 
be the midpoints of sides 
, 
, 
and let 
be the tangency points of with and , respectively. Let 
be the intersections of line 
with line 
and line 
, respectively, and let 
be the midpoint of arc 
of .
(a) Prove that lies on ray 
.
(b) Prove that line
bisects 
.
be a triangle with incenter 
, incircle 
and circumcircle 
. Let 
be the midpoints of sides 
, 
, 
and let 
be the tangency points of with and , respectively. Let 
be the intersections of line 
with line 
and line 
, respectively, and let 
be the midpoint of arc 
of .(a) Prove that
lies on ray 
.(b) Prove that line
bisects 
.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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