MathDB

p1. Suppose

x \ne 1

10

and logarithms are computed to the base

10

. Define

y= 10^{\frac{1}{1-\log x}}

and

z = ^{\frac{1}{1-\log y}}

. Prove that

x= 10^{\frac{1}{1-\log z}}

p2. If

n

is an odd number and

x_1, x_2, x_3,..., x_n

is an arbitrary arrangement of the integers

1, 2,3,..., n

, prove that the product

(x_1 -1)(x_2-2)(x_3- 3)... (x_n-n)

is an even number (possibly negative or zero).
p3. Prove that

\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2n} < \sqrt{\frac{1}{2n + 1}}

for all integers

n = 1,2,3,...

p4. Prove that if three angles of a convex polygon are each

60^o

, then the polygon must be an equilateral triangle.
p5. Find all solutions, real and complex, of

4 \left(x^2+\frac{1}{x^2} \right)-4 \left( x+\frac{1}{x} \right)-7=0

p6. A man is

\frac38

of the way across a narrow railroad bridge when he hears a train approaching at

60

miles per hour. No matter which way he runs he can just escape being hit by the train. How fast can he run? Prove your assertion.

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

Problem Statement