MathDB

2021 Algebra/NT #8: LCM's and GCD's

Source:

5/30/2021

For positive integers

a

and

b

, let

M(a,b) = \tfrac{\text{lcm}(a,b)}{\gcd(a,b)},

and for each positive integer

n \ge 2,

define

x_n = M(1, M(2, M(3, \dots , M(n - 2, M(n - 1, n))\cdots))).

Compute the number of positive integers

n

such that

2 \le n \le 2021

and

5x_n^2 + 5x_{n+1}^2 = 26x_nx_{n+1}.

number theory

2021 Combo #8: Fill in the grid

Source:

5/30/2021

Compute the number of ways to fill each cell in a

8 \times 8

square grid with one of the letters

H, M,

or

T

such that every

2 \times 2

square in the grid contains the letters

H, M, M, T

in some order.

Combo

2021 Geo #8: Right triangle in two circles

Source:

5/30/2021

Two circles with radii

71

and

100

are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

geometry

2021 Team #8

Source:

6/27/2021

For each positive real number

\alpha

, define

\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}.

Let

n

be a positive integer. A set

S\subseteq \{1,2,\ldots,n\}

has the property that: for each real

\beta >0

,

\text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor.

Determine, with proof, the smallest positive size of

S

.

combinatoricsfloor functionnumber theory

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