MathDB

p1. The least prime factor of

a

3

, the least prime factor of

b

7

. Find the least prime factor of

a + b

.
p2. In a Cartesian coordinate system, the two tangent lines from

P = (39, 52)

meet the circle defined by

x^2 + y^2 = 625

at points

Q

and

R

. Find the length

QR

.
p3. For a positive integer

n

, there is a sequence

(a_0, a_1, a_2,..., a_n)

of real values such that

a_0 = 11

and

(a_k + a_{k+1}) (a_k - a_{k+1}) = 5

for every

k

with

0 \le k \le n-1

. Find the maximum possible value of

n

. (Be careful that your answer isn’t off by one!)
p4. Persons

A

and

B

stand at point

P

on line

\ell

. Point

Q

lies at a distance of

10

from point

P

in the direction perpendicular to

\ell

. Both persons intially face towards

Q

. Person

A

walks forward and to the left at an angle of

25^o

with

\ell

, when he is again at a distance of

10

from point

Q

, he stops, turns

90^o

to the right, and continues walking. Person

B

walks forward and to the right at an angle of

55^o

with line

\ell

, when he is again at a distance of

10

from point

Q

, he stops, turns

90^o

to the left, and continues walking. Their paths cross at point

R

. Find the distance

PR

.
p5. Compute

\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.

p6. There is a unique real value

A

such that for all

x

with

1 < x < 3

and

x \ne 2

\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.

Compute

A

.
p7. Nine poles of height

1, 2,..., 9

are placed in a line in random order. A pole is called dominant if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly

2

dominant poles.
p8.

\tan (11x) = \tan (34^o)

and

\tan (19x) = \tan (21^o)

. Compute

\tan (5x)

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement