MathDB

p1. Let

2^k

be the largest power of

2

dividing

30! = 30 \cdot 29 \cdot 28 ... 2 \cdot 1

. Find

k

.
p2. Let

d(n)

be the total number of digits needed to write all the numbers from

1

n

in base

10

, for example,

d(5) = 5

and

d(20) = 31

. Find

d(2012)

.
p3. Jim and TongTong play a game. Jim flips

10

coins and TongTong flips

11

coins, whoever gets the most heads wins. If they get the same number of heads, there is a tie. What is the probability that TongTong wins?
p4. There are a certain number of potatoes in a pile. When separated into mounds of three, two remain. When divided into mounds of four, three remain. When divided into mounds of five, one remain. It is clear there are at least

150

potatoes in the pile. What is the least number of potatoes there can be in the pile?
p5. Call an ordered triple of sets

(A, B, C)

nice if

|A \cap B| = |B \cap C| = |C \cap A| = 2

and

|A \cap B \cap C| = 0

. How many ordered triples of subsets of

\{1, 2, · · · , 9\}

are nice?
p6. Brett has an

n \times n \times n

cube (where

n

is an integer) which he dips into blue paint. He then cuts the cube into a bunch of

1 \times 1 \times 1

cubes, and notices that the number of un-painted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of Brett’s original cube?
Note that

\lfloor x\rfloor

denotes the largest integer less than or equal to

x

.
p7. Choose two real numbers

x

and

y

uniformly at random from the interval

[0, 1]

. What is the probability that

x

is closer to

1/4

than

y

is to

1/2

?
p8. In triangle

ABC

, we have

\angle BAC = 20^o

and

AB = AC

D

is a point on segment

AB

such that

AD = BC

. What is

\angle ADC

, in degree.
p9. Let

a, b, c, d

be real numbers such that

ab + c + d = 2012

bc + d + a = 2010

cd + a + b = 2013

da + b + c = 2009

. Find

d

.
p10. Let

\theta \in [0, 2\pi)

such that

\cos \theta = 2/3

. Find

\sum_{n=0}^{\infty}\frac{1}{2^n}\cos(n \theta)

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

Problem Statement