MathDB

p1. What is the largest distance between any two points on a regular hexagon with a side length of one?
p2. For how many integers

n \ge 1

\frac{10^n - 1}{9}

the square of an integer?
p3. A vector in

3D

space that in standard position in the first octant makes an angle of

\frac{\pi}{3}

with the

x

axis and

\frac{\pi}{4}

with the

y

axis. What angle does it make with the

z

axis?
p4. Compute

\sqrt{2012^2 + 2012^2 \cdot 2013^2 + 2013^2} - 2012^2

.
p5. Round

\log_2 \left(\sum^{32}_{k=0} {{32} \choose k} \cdot 3^k \cdot 5^k\right)

to the nearest integer.
p6. Let

P

be a point inside a ball. Consider three mutually perpendicular planes through

P

. These planes intersect the ball along three disks. If the radius of the ball is

2

and

1/2

is the distance between the center of the ball and

P

, compute the sum of the areas of the three disks of intersection.
p7. Find the sum of the absolute values of the real roots of the equation

x^4 - 4x - 1 = 0

.
p8. The numbers

1, 2, 3, ..., 2013

are written on a board. A student erases three numbers

a, b, c

and instead writes the number

\frac12 (a + b + c)\left((a - b)^2 + (b - c)^2 + (c - a)^2\right).

She repeats this process until there is only one number left on the board. List all possible values of the remainder when the last number is divided by 3.
p9. How many ordered triples of integers

(a, b, c)

, where

1 \le a, b, c \le 10

, are such that for every natural number

n

, the equation

(a + n)x^2 + (b + 2n)x + c + n = 0

has at least one real root?
Problems' source (as mentioned on official site) is Gator Mathematics Competition.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement