MathDB

Stewart (2023 HMMT February Geo #8)

Source:

2/20/2023

Let

\triangle ABC

be a triangle with

\angle BAC>90^{\circ}

,

AB=5

and

AC=7

. Points

D

and

E

lie on segment

BC

such that

BD=DE=EC

. If

\angle BAC+\angle DAE=180^{\circ}

, compute

BC

.

HMMT

HMMT Feb 2023 team p8

Source:

2/20/2023

Find, with proof, all nonconstant polynomials

P(x)

with real coefficients such that, for all nonzero real numbers

z

with

P(z)\neq 0

and

P(\frac{1}{z}) \neq 0

we have

\frac{1}{P(z)}+\frac{1}{P(\frac{1} {z})}=z+\frac{1}{z}.

2023 Combinatorics #8

Source:

2/28/2024

A random permutation

a = (a_1, a_2,...,a_{40})

of

(1, 2,...,40)

is chosen, with all permutations being equally likely. William writes down a

20 \times 20

grid of numbers

b_{ij}

such that

b_{ij} = \max (a_i, a_{j+20})

for all

1 \le i, j \le 20

, but then forgets the original permutation

a

. Compute the probability that, given the values of

b_{ij}

alone, there are exactly

2

permutations

a

consistent with the grid.

combinatorics

2023 Algebra/NT #8: Clean bijective NT sum

Source:

11/27/2023

Let

S

be the set of ordered pairs

(a, b)

of positive integers such that

\gcd(a, b) = 1

. Compute

\sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor.

floor functionnumber theory

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