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 .

[![Triangle with angles 40°, 30°, and 10°, and AB = EC][1][1](https://imgur.com/a/6oIVAbI )]!

I came across this geometry problem while practicing triangle properties. It involves given angles and an equality between two segments: .

I have tried using the angle sum property in triangles and the isosceles triangle theorem (since ), but I am unsure how to proceed to find . I also considered exterior angles but could not establish a clear relationship.

I find this problem interesting because it seems to involve multiple geometric properties, and I would like to understand the best approach to solve it. Any hints or explanations would be greatly appreciated!

Why are AMC, AIME, USAMO problems posted in C&P rather than where they should ideally be i. e. HSM and HSO?
Is it not simply more efficient to have everything in the same forum? Or is there some secret lore behind this?

Let be a positive integer, and let be nonnegative integers such that Prove that
Note: for all nonnegative integers .

Let and be positive integers. Prove that there exists a positive integer such that for every odd integer , the digits in the base- representation of are all greater than .

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

Let be nonnegative real numbers such that . Find min

Let Prove that

Hi Everyone,

USA(J)MO is tomorrow. I am a Junior, so this is my last chance. I made USAMO by ZERO points but I've actually been studying oly seriously since JMO last year. I am more stressed than I was before AMC/AIME because I feel Olympiad is more unpredictable and harder to prepare for. I am fairly confident in my ability to solve 1/4 but whether I can solve the rest really leans on the topic distribution.

Anyway, I'm just super stressed and not sure what to do. All tips are welcome!

Thanks everyone! Good luck tomorrow!

What is your opinion for MOHS for day 1?

JMO 1:
JMO 2/AMO 1:
JMO 3:
AMO 2:
AMO 3:

Let be a prime, and let be integers. Show that there exists an integer such that the numbers
produce at least distinct remainders upon division by .

Proposed by Ankan Bhattacharya

Let and be positive integers, and let be a grid of unit squares.

A domino is a or rectangle. A subset of grid squares in is domino-tileable if dominoes can be placed to cover every square of exactly once with no domino extending outside of . Note: The empty set is domino tileable.

An up-right path is a path from the lower-left corner of to the upper-right corner of formed by exactly edges of the grid squares.

Determine, with proof, in terms of and , the number of up-right paths that divide into two domino-tileable subsets.