MathDB

p1. How many ordered triples of integers

(a, b, c)

where

1 \le a, b, c \le 10

are such that for every natural number, the equation

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

has at least one real root?
p2. Find the smallest integer

n

such that we can cut a

n \times n

grid into

5

rectangles with distinct side lengths in

\{1, 2, 3..., 10\}

. Every value is used exactly once.
p3. A plane is flying at constant altitude along a circle of radius

12

miles with center at a point

A

.The speed of the aircraft is v. At some moment in time, a missile is fired at the aircraft from the point

A

, which has speed v and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before colliding with the aircraft?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement