卷六-17題

可能的答案

May 8th 2014 Problem E of POTW Waterloo U

It is a long time to solve some math problem. I had found a new problem of week resource from Waterloo University. These problems focus on students from primary to secondary, maybe up to high school. Here I am trying to solve some of these problem, and hoping me keep on doing this. By the way, my English is suck, so the reader should be patient with my words.

This week problem is.

A square has coordinates A(0; 0), B(-9; 12), C(3; 21) and D(12; 9). The line l passes through A and intersects CD at point T(r; s) splitting the square so that the area of square ABCD is three times the area of triangle ATD. Determine the equation of line l. 

Because the area of triangle ATD is 1/3 of the square, and area of triangle ACD if 1/2 of the square, these mean DT : CD = 2 : 3. And CT =  CD - DT, then the ratio of CT : TD = 1 : 2.

Using the ratio we have equations: 

$ (3 - r) : (r - 12) = 1 : 2 $

$ (21 - s) : (s - 9) = 1 : 2 $.

Change them to fraction form are:

$ \frac{3-r}{r-21} = \frac{1}{2}$

$ \frac{21-s}{s-9} = \frac{1}{2}$

Solve them will get the values of r and s. 

Problem 3 Fall 2012

Back to math problem again. Hope my teaching thing will be more easy to handle.

There are M gold fish and K silver fish in a lake. They are caught and eaten one at a time at random until only one color of fish remains in the lake. One of the silver fish is named George. Find the probability George is not eaten. (Answer should be in a very simple form.)

 The probability of George is not eaten equal to it is not caught, which equals to 1 minus probability that George be caught.

$P=1-\frac{C_{K}^{1}}{C_{M+K}^{1}}$