MathDB

p1. For what integers

n

2^6 + 2^9 + 2^n

the square of an integer?
p2. Two integers are chosen at random (independently, with repetition allowed) from the set

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

. Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd.
p3. Let

X

be a point in the second quadrant of the plane and let

Y

be a point in the first quadrant. Locate the point

M

on the

x

-axis such that the angle

XM

makes with the negative end of the

x

-axis is twice the angle

YM

makes with the positive end of the

x

-axis.
p4. Let

a,b

be positive integers such that

a \ge b \sqrt3

. Let

\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3

for

n = 1,2,3,...

. i. Prove that

\lim_{n \to + \infty} \frac{a_n}{b_n}

exists. ii. Evaluate this limit.
p5. Suppose

m

and

n

are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers

a,b,c,d

, so that

m^2 = a^2 + b^2

and

n^2= c^2 + d^2

. Show than

mn

is the hypotenuse of at least two distinct Pythagorean triangles.
Hint: you may not assume that the pair

(a,b)

is different from the pair

(c,d)

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

Problem Statement