MathDB

p1. A fraction is called Toba-

n

if the fraction has a numerator of

1

and the denominator of

n

. If

A

is the sum of all the fractions of Toba-

101

, Toba-

102

, Toba-

103

, to Toba-

200

, show that

\frac{7}{12} <A <\frac56

.
p2. If

a, b

, and

c

satisfy the system of equations

\frac{ab}{a+b}=\frac12

\frac{bc}{b+c}=\frac13

\frac{ac}{a+c}=\frac17

Determine the value of

(a- c)^b

.
p3. Given triangle

ABC

. If point

M

is located at the midpoint of

AC

, point

N

is located at the midpoint of

BC

, and the point

P

is any point on

AB

. Determine the area of the quadrilateral

PMCN

. https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png
p4. Given the rule of motion of a particle on a flat plane

xy

as following:

N: (m, n)\to (m + 1, n + 1)

T: (m, n)\to (m + 1, n - 1)

, where

m

and

n

are integers. How many different tracks are there from

(0, 3)

(7, 2)

by using the above rules ?
p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing

30

marbles. For each take, the player can take the least a minimum of

1

and a maximum of

6

marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking

3

marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.

day 1

Problems(1)