## Friday, December 8, 2017

### Harder Mathcounts State/AMC Questions

2012 Mathcounts State Sprint #30: In rectangle ABCD, shown here, point M is the midpoint of side BC, and point N lies on CD such that DN:NC = 1:4. Segment BN intersects AM and AC at points R and S, respectively. If NS:SR:RB = x:y:z, where x, y and z are positive integers, what is the minimum possible value of x + y + z?

Solution I :

$$\overline {AB}:\overline {NC}=5:4$$ [given]

Triangle ASB is similar to triangle CSN (AAA)

$$\overline {NS}:\overline {SB}= 4 : 5$$

Let $$\overline {NS}= 4a, \overline {SB}= 5a.$$

Draw a parallel line to $$\overline {NC}$$ from M and mark the interception to $$\overline {BN}$$as T.

$$\overline {MT}: \overline {NC}$$ = 1 to 2. [$$\Delta BMT$$ and $$\Delta BCN$$ are similar triangles ]

$$\overline {NT} = \overline {TB}= \dfrac {4a+5a} {2}=4.5a$$

$$\overline {ST} = 0.5a$$

$$\overline {MT} : \overline {AB}$$ = 2 to 5
[Previously we know  $$\overline {MT}: \overline {NC}$$ = 1 to 2 or 2 to 4 and  $$\overline {NC}:\overline {AB}= 4 : 5$$ so the ratio of the two lines  $$\overline {MT} : \overline {AB}$$ is 2 to 5.]

$$\overline {TB} = 4.5 a$$  [from previous conclusion]

Using 5 to 2 line ratio [similar triangles $$\Delta ARB$$ and $$\Delta MRT$$ , you get $$\overline {BR} =\dfrac {5} {7}\times 4.5a =\dfrac {22.5a} {7}$$ and $$\overline {RT} =\dfrac {2} {7}\times 4.5a =\dfrac {9a} {7}$$

Thus, x : y : z = 4a : $$\dfrac {1} {2}a + \dfrac {9a} {7}$$ : $$\dfrac {22.5a} {7}$$ = 56 : 25 : 45

x + y + z = 126

Solution II :
From Mathcounts Mini: Similar Triangles and Proportional Reasoning

Solution III:
Using similar triangles ARB and CRN , you have $$\dfrac {x} {y+z}=\dfrac {5} {9}$$.
9x = 5y + 5z ---- equation I

Using similar triangles ASB and CSN and you have $$\dfrac {x+y} {z}=\dfrac {5} {4}$$.
4x + 4y = 5z  ---- equation II

Plug in (4x + 4y) for 5z on equation I and you have 9x = 5y + (4x + 4y) ; 5x = 9y ; x = $$\dfrac {9} {5}y$$
Plug in x = $$\dfrac {9} {5}y$$ to equation II and you have z  =  $$\dfrac {56} {25}y$$

x : y : z = $$\dfrac {9} {5}y$$  : y  :  $$\dfrac {56} {25}y$$ =  45 y :  25y  :  56y

45 + 25 + 56 = 126

Solution IV : Yes, there is another way that I've found even faster, saved for my private students. :D

Solution V : from Abhinav, one of my students solving another similar question :

#### Two other similar questions from 2016 AMC A, B tests :

Solution from Abhinav

2016 AMC 10 B #19 : Solution from Abhinav