MathDB

p1. A teenager coining home after midnight heard the hall clock striking the hour. At some moment between

15

and

20

minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then?
p2. The ratio of two positive integers

a

and

b

2/7

, and their sum is a four digit number which is a perfect cube. Find all such integer pairs.
p3. Given the integers

1, 2 , ..., n

, how many distinct numbers are of the form

\sum_{k=1}^n( \pm k)

, where the sign (

\pm

) may be chosen as desired? Express answer as a function of

n

. For example, if

n = 5

, then we may form numbers:

1 + 2 + 3- 4 + 5 = 7

-1 + 2 - 3- 4 + 5 = -1

1 + 2 + 3 + 4 + 5 = 15

, etc.
p4.

\overline{DE}

is a common external tangent to two intersecting circles with centers at

O

and

O'

. Prove that the lines

AD

and

BE

are perpendicular. https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png
p5. Find all polynomials

f(x)

such that

(x-2) f(x+1) - (x+1) f(x) = 0

for all

x

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

Problem Statement