MathDB

2023 Geometry #4

Source:

4/12/2023

Let

ABCD

be a square, and let

M

be the midpoint of side

BC

. Points

P

and

Q

lie on segment

AM

such that

\angle BPD=\angle BQD=135^\circ

. Given that

AP<AQ

, compute

\tfrac{AQ}{AP}

.

HMMT Feb 2023 Team p4

Source:

2/20/2023

Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair

(x, y)

of positive integers such that

x \leq 20

and

y \leq 23

. (Philena knows that Nathan’s pair must satisfy

x \leq 20

and

y \leq 23

.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair

(a, b)

of positive integers and tells it to Nathan; Nathan says YES if

x \leq a

and

y \leq b

, and NO otherwise. Find, with proof, the smallest positive integer

N

for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most

N

rounds.

2023 Combinatorics #4

Source:

4/17/2023

The cells of a

5\times5

grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through

9

cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly

3

red cells, exactly

3

white cells, and exactly

3

blue cells no matter which route he takes.

so many P(1) (2023 HMMT A4)

Source:

2/25/2023

Suppose

P (x)

is a polynomial with real coefficients such that

P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))

for all real numbers

t

. Compute the largest possible value of

P(P(P(P(1))))

.

HMMT

4

Problems(4)