# Admissions Quiz

## Instructions

The admissions quiz is an opportunity for you to show us how you think and approach problems. In particular, you should work on the problems below, by yourself. Do not consult or get help from others for any of the problems. Additionally, please do not post the problems online or discuss them with anyone until after the application deadline.

You should justify your claims and prove to us that your solution is correct. We are more interested in your reasoning than in whether you obtain the correct answer. Any claims unaccompanied by an explanation will be worth very little.

We've purposely designed the quiz to be fairly challenging, so we do not necessarily expect you to fully solve all problems. In the past, we've admitted students who solved fewer than two problems, though of course the cutoff changes every year. The problems may take longer than you're used to, especially if you are not so used to Olympiad math. There is much room for partial progress! So, even if you don't solve a problem, we care very much to hear about any ideas, thoughts, or conjectures that you may have. You are encouraged to try all the problems and write up either a proof if you solve it or your ideas if you don't. Partial progress will be worth points.

None of our problems will require a computer or the internet. In fact, we recommend working on the problems offline, with your usual math tools (pen/pencil, paper, compass, straightedge, etc.). You may not use GeoGebra. If you do consult any external resources, please cite them (a link or book title is sufficient).

When you are done with the quiz, scan your work and submit it here. Make sure you also submit the informational form. The application is due March 26, 2023 at 11:59pm PDT.

Questions about the quiz or application should be sent to g2mathprogram@gmail.com.

A PDF version of the quiz can be found here: G2_2023_Admissions_Quiz.pdf.

## Problems

Let a1, . . . , an be nonnegative integers with max(a1, . . . ,an) = q. Show that one can pick the signs in s = ± a1 ± a2 ± … ± an such that 0 ≤ s ≤ q.

Let m, n be positive integers with m < n. Consider an n x n grid, where the top right m x m squares have been removed, leaving an L-shaped board. For which pairs (m, n) can this board be tiled with 1 x 3 and 3 x 1 dominoes? Dominoes cannot overlap, and have to be entirely contained within the board.

Let a,b,c be nonnegative integers with a+b+c > 1. Let x > 0 be a real number satisfying ax2 + bx - c = 0. Show that x ≥ 1/(a+b+c-1).

In triangle ABC, let M be the midpoint of BC, and let the line through M perpendicular to the internal angle bisector of <BAC intersect AC at R. Let Q lie on AC such that RQ = AB, such that Q lies on the same side of R as A. Let S be the intersection of MQ with AB. Prove that SA = RC.

Given a prime p ≥ 3, find the minimum degree of an integer polynomial Q such that Q(n) takes exactly 3 distinct values modulo p, where n ranges over the integers.

Is it possible to label each lattice point in the plane with a positive integer such that the following two properties hold:

there are infinitely many positive integers used, and

for any line not passing through the origin that passes through at least two lattice points, the assigned numbers on that line form a periodic sequence?