MathDB

B1

Part of 2020 Princeton University Math Competition

Problems(5)

2020 PUMaC Algebra B1

Source:

1/1/2022
The function f(x)=x2+(2a+3)x+(a2+1)f(x) = x^2 + (2a + 3)x + (a^2 + 1) only has real zeroes. Suppose the smallest possible value of aa can be written in the form p/qp/q, where p,qp, q are relatively prime integers. Find ∣p∣+∣q∣|p| + |q|.
algebra
2020 PUMaC Combinatorics B1

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1/1/2022
Runey is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Runese consist of anywhere between 11 and 55 letters, inclusive. As well, Runey can choose to add emphasis on any letter(s) that he chooses in a given word, hence making it a totally distinct word! What is the maximum number of possible words in Runese?
combinatorics
2020 PUMaC Geometry B1

Source:

12/31/2021
You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let θ\theta be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is ⌊100⋅θ⌋\lfloor 100 \cdot \theta \rfloor ?
floor functiongeometry
2020 PUMaC NT B1

Source:

1/1/2022
The number 20212021 leaves a remainder of 1111 when divided by a positive integer. Find the smallest such integer.
number theory
2020 PUMaC Individual Finals B1

Source:

1/1/2022
Find all pairs of natural numbers (n,k)(n, k) with the following property: Given a k×kk\times k array of cells, such that every cell contains one integer, there always exists a path from the left to the right edges such that the sum of the numbers on the path is a multiple of nn.
Note: A path from the left to the right edge is a sequence of cells of the array a1,a2,...,ama_1, a_2, ... , a_m so that a1a_1 is a cell of the leftmost column, ama_m is the cell of the rightmost column, and aia_{i}, ai+1a_{i+1} share an edge for all i=1,2,...,m−1i = 1, 2, ... , m -1.
combinatorics