4

Part of 2015 Harvard-MIT Mathematics Tournament

Problems(2)

2015 Algebra #4: Sums Divisible by 203

Source:

Compute the number of sequences of integers

(a_1,\ldots,a_{200})

such that the following conditions hold.
[*]

0\leq a_1<a_2<\cdots<a_{200}\leq 202.

[*] There exists a positive integer

N

with the following property: for every index

i\in\{1,\ldots,200\}

there exists an index

j\in\{1,\ldots,200\}

a_i+a_j-N

is divisible by

203

algebraSequencesDivisibility

2015 Geometry #4

Source:

ABCD

be a cyclic quadrilateral with

AB=3

BC=2

CD=2

DA=4

. Let lines perpendicular to

\overline{BC}

B

C

\overline{AD}

B'

C'

, respectively. Let lines perpendicular to

\overline{BC}

A

D

\overline{AD}

A'

D'

, respectively. Compute the ratio

\frac{[BCC'B']}{[DAA'D']}

[\overline{\omega}]

denotes the area of figure

\overline{\omega}