My dream is to be on IMO team ik thats not going to happen b/c the kids that make it are like 6th mop quals :play_ball:. I somehow got a index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs. made mc nats

Rectangles and are erected outside an acute triangle Suppose that Prove that lines and are concurrent.

I recently mocked 2 AMC 10’s, and 2 AIME’s. My scores for the AMC 10 were both 123 and my AIME scores were 8 and 9 for 2010 I and II. What should I study for 2025-2026 AMCs? Goal is JMO.

So someone pointed out to me that the last five problems on my previous practice AMC 10 test were rather low quality. Here are some problems that are (hopefully) better.

21.
A partition of a positive integer n is writing n as the sum of positive integer(s), where order does not matter. Find the number of partitions of 6.
A - 10, B - 11, C - 12, D - 13, E - 14

22.
Let n be the smallest positive integer that satisfies the following conditions:
- n is even
- The last digit of n is not 2 or 8
- n^2 + 1 is composite
Find the sum of the digits of n.
A - 3, B - 5, C - 8, D - 9, E - 10

23.
Find the sum of the coordinates of the reflection of the point (6, 9) over the line x + 2y + 3 = 0.
A - (-17.7), B - (-17.6), C - (-17.5), D - (-17.4), E - (-17.3)

24.
Find the number of ordered pairs of integers (a, b), where both a and b have absolute value less than 69, such that a^2 + 42b^2 = 13ab.
A - 21, B - 40, C - 41, D - 42, E - 69

25.
Let f(n) be the sum of the positive integer factors of n, where n is an integer. Find the sum of all positive integers n less than 1000 such that f(f(n) - n) = f(n).
A - 420, B - 530, C - 690, D - 911, E - 1034

Let be the orthocenter of acute triangle , let be the foot of the altitude from to , and let be the reflection of across . Suppose that the circumcircle of triangle intersects line at two distinct points and . Prove that is the midpoint of .

Determine, with proof, all positive integers such that is an integer for every positive integer

Hey guys, do you guys have any good AIME/USAJMO Level Number Theory book suggestions?
I'm trying to get 10+ on next year's AIME and hopefully qual for USAJMO.

Hi AOPSers,

I have finished writing a mock geometry test for fun and practice for the real college competitions like HMMT/PUMaC/CMIMC... There would be 10 questions and you should finish the test in 60 minutes, the test would be close to the actual test (hopefully). You could sign up under this thread, PM me your answers!. The submission would close on March 31st at 11:59PM PST.

I would create a private discussion forum so everyone could discuss after finishing the test. This is the first mock I've written, please sign up and enjoy geometry!!

~Bluesoul

Discussion forum: Discussion forum

Leaderboard

Alice the architect and Bob the builder play a game. First, Alice chooses two points and in the plane and a subset of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair of cities, they are connected with a road along the line segment if and only if the following condition holds:
[center]For every city distinct from and , there exists such[/center]
[center]that is directly similar to either or .[/center]
Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.

Note: is directly similar to if there exists a sequence of rotations, translations, and dilations sending to , to , and to .

Howd you all do?

Also will a 766 make bronze -> silver?