Let be a acute triangle with orthocenter and circumcircle .Let intersect at respectively. is the midpoint of . intersect the at .The line parallel to through intersect second time at .The line perpendicular to at intersect at . and intersecting at .Suppose that is cyclic quadrilateral. intersect at second time.
Prove that and are tangent to each other.

[center] :trampoline: :first: Stanford Math Tournament :first: :trampoline: [/center]

----------------------------------------------------------

[center]IMAGE[/center]

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!

Find, with proof, the least integer such that if any elements are removed from the set , one can still find distinct numbers among the remaining elements with sum .

hi guys, i'm about to be an incoming freshman, does anyone have recommendations for classes to take next year and camps this summer? i am sure that i can aime qual but not jmo qual yet. ty

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 .

rant

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 .

What is the value of

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.

So, I have decided to attempt taking the AMC 12 this fall. I don't have any experience with math competitions, and I thought that here might be a good place to see if anyone who has taken the AMC 12 (or done any other math competitions) has any suggestions on what to expect, how to prepare, etc. Thank you!

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?