MathDB

1

2001 Stanford Math Tournament Team #15

ABC

be an isosceles triangle with

\angle{ABC} = \angle{ACB} = 80^\circ

. Let

D

be a point on

AB

such that

\angle{DCB} = 60^\circ

and

E

be a point on

AC

such that

\angle{ABE} = 30^\circ

. Find

\angle{CDE}

in degrees.

14

1

2001 Stanford Math Tournament Team #14

Find the prime factorization of

\textstyle\sum_{1\le i < j \le 100}ij

.

13

1

2001 Stanford Math Tournament Team #13

You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 ﬁves. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die?

12

1

2001 Stanford Math Tournament Team #12

A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.

11

1

2001 Stanford Math Tournament Team #11

Christopher and Robin are playing a game in which they take turns tossing a circular token of diameter 1 inch onto an inﬁnite checkerboard whose squares have sides of 2 inches. If the token lands entirely in a square, the player who tossed the token gets 1 point; otherwise, the other player gets 1 point. A player wins as soon as he gets two more points than the other player. If Christopher tosses ﬁrst, what is the probability that he will win? Express your answer as a fraction.

10

1

2001 Stanford Math Tournament Team #10

You know that the binary function

\diamond

takes in two non-negative integers and has the following properties:
\begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*}

\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].

Find a general formula for

x\diamond y

, assuming that y\gex>0.

9

1

2001 Stanford Math Tournament Team #9

What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction.

8

1

2001 Stanford Math Tournament Team #8

Janet and Donald agree to meet for lunch between 11:30 and 12:30. They each arrive at a random time in that interval. If Janet has to wait more than 15 minutes for Donald, she gets bored and leaves. Donald is busier so will only wait 5 minutes for Janet. What is the probability that the two will eat together? Express your answer as a fraction.

7

1

2001 Stanford Math Tournament Team #7

The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median.

6

1

2001 Stanford Math Tournament Team #6

Find the least

n

such that any subset of

{1,2,\dots,100}

with

n

elements has 2 elements with a diﬀerence of 9.

5

1

2001 Stanford Math Tournament Team #5

What quadratic polynomial whose coeﬃcient of

x^2

is

1

has roots which are the complex conjugates of the solutions of

x^2 -6x+ 11 = 2xi-10i

? (Note that the complex conjugate of

a+bi

is

a-bi

, where a and b are real numbers.)

4

1

2001 Stanford Math Tournament Team #4

For what values of

a

does the system of equations

x^2 = y^2,(x-a)^2 +y^2 = 1

have exactly 2 solutions?

3

1

2001 Stanford Math Tournament Team #3

Find the 2000th positive integer that is not the diﬀerence between any two integer squares.

2

1

2001 Stanford Math Tournament Team #2

How many positive integers between 1 and 400 (inclusive) have exactly 15 positive integer factors?

1

2001 Stanford Math Tournament Team #1

ABCD

is a square with sides of unit length. Points

E

and

F

are taken on sides

AB

and

AD

respectively so that

AE = AF

and the quadrilateral

CDFE

has maximum area. What is this maximum area?

2001 Stanford Mathematics Tournament

Subcontests

2001 Stanford Math Tournament Team #15

2001 Stanford Math Tournament Team #14

2001 Stanford Math Tournament Team #13

2001 Stanford Math Tournament Team #12

2001 Stanford Math Tournament Team #11

2001 Stanford Math Tournament Team #10

2001 Stanford Math Tournament Team #9

2001 Stanford Math Tournament Team #8

2001 Stanford Math Tournament Team #7

2001 Stanford Math Tournament Team #6

2001 Stanford Math Tournament Team #5

2001 Stanford Math Tournament Team #4

2001 Stanford Math Tournament Team #3

2001 Stanford Math Tournament Team #2

2001 Stanford Math Tournament Team #1