MathDB

Let

ABC

be an acute scalene triangle with incenter

I

, and let

M

be the circumcenter of triangle

BIC

. Points

D

B'

, and

C'

lie on side

BC

so that

\angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ}

. Define

P = \overline{AB} \cap \overline{MC'}

Q = \overline{AC} \cap \overline{MB'}

S = \overline{MD} \cap \overline{PQ}

, and

K = \overline{SI} \cap \overline{DF}

, where segment

EF

is a diameter of the incircle selected so that

S

lies in the interior of segment

AE

. It is known that

KI=15x

SI=20x+15

BC=20x^{5/2}

, and

DI=20x^{3/2}

, where

x = \tfrac ab(n+\sqrt p)

for some positive integers

a

b

n

p

, with

p

prime and

\gcd(a,b)=1

. Compute

a+b+n+p

.
Proposed by Evan Chen

Given vectors

v_1, \dots, v_n

and the string

v_1v_2 \dots v_n

, we consider valid expressions formed by inserting

n-1

sets of balanced parentheses and

n-1

binary products, such that every product is surrounded by a parentheses and is one of the following forms:
1. A "normal product''

ab

, which takes a pair of scalars and returns a scalar, or takes a scalar and vector (in any order) and returns a vector. \\
2. A "dot product''

a \cdot b

, which takes in two vectors and returns a scalar. \\
3. A "cross product''

a \times b

, which takes in two vectors and returns a vector. \\
An example of a valid expression when

n=5

(((v_1 \cdot v_2)v_3) \cdot (v_4 \times v_5))

, whose final output is a scalar. An example of an invalid expression is

(((v_1 \times (v_2 \times v_3)) \times (v_4 \cdot v_5))

; even though every product is surrounded by parentheses, in the last step one tries to take the cross product of a vector and a scalar. \\
Denote by

T_n

the number of valid expressions (with

T_1 = 1

), and let

R_n

denote the remainder when

T_n

is divided by

4

. Compute

R_1 + R_2 + R_3 + \ldots + R_{1,000,000}

.
Proposed by Ashwin Sah

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