University of British Columbia
Quarter Putnam Contest—November 10, 2015
Your solutions should be clear, concise, and complete. “Educated guesses” will earn little credit.
Write your solutions as if this were the actual Putnam examination.
Name:
QP1. Evaluate
Z
2
4
p
ln(9 − x) dx
p
p
.
ln(9 − x) + ln(x + 3)
QP2. Find all real numbers x such that
35 j
35 k
35 k
35 j
x + bx c + bx + bx cc = x +
+ x+
+ x+
+ x+
.
4
4
4
4
2
2
2
2
(Here, byc is the greatest integer that is less than or equal to y; for example, bπc = 3, b−πc = −4,
and b−1c = −1.)
QP3. Let
N = 2015! = 2015 × 2014 × 2013 × · · · × 3 × 2 × 1.
Prove that N is the sum of exactly 2015 distinct divisors of N .