MathDB

p10. Three rectangles of dimension

X \times 2

and four rectangles of dimension

Y \times 1

are the pieces that form a rectangle of area

3XY

where

X

and

Y

are positive, integer values. What is the sum of all possible values of

X

?
p11. Suppose we have a polynomial

p(x) = x^2 + ax + b

with real coefficients

a + b = 1000

and

b > 0

. Find the smallest possible value of

b

such that

p(x)

has two integer roots.
p12. Ten square slips of paper of the same size, numbered

0, 1, 2, ..., 9

, are placed into a bag. Four of these squares are then randomly chosen and placed into a two-by-two grid of squares. What is the probability that the numbers in every pair of blocks sharing a side have an absolute difference no greater than two?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement