**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**