## 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

## Sunday, November 26, 2017

### 2018 Mathcounts Competition Preparation Strategies

Hi, Thanks for visiting my blog.

E-mail me at thelinscorner@gmail.com if you want to join my groups.

Currently I'm running different levels of problem solving group lessons, and it's lots of fun learning along with students from different states.

My most advanced group of  students are just AMAZING !! to say the least.
Ha ha, we are using AMC-10, 12 questions as countdown round practices and some can solve the first few AIME problems in less than a minute. Oh dear !!

So many students are not learning smart.

Problem solving is really fun (and a lot of the times very hard, yes).

Good questions are intriguing and delicious, so come join our vibrant community and have the pleasure of finding things out on your own.

There is no overnight success.

My other blogs :

thelinscorner  : Standardized test preps, books, links/videos for life-time learning

Take care and have fun learning.

Don't forget other equally interesting activities/contests, which engage your creativity  and imagination.

Some also require team work. Go for those and have fun !!

Don't just do math.

Before going full throttle mode for competition math, please spend some time reading this
well- thought-out article from BOGTRO at AoPS "Learn How to Learn".

It will save you tons of time and numerous, unnecessary hours without a clear goal, better method in mind.

Less is more. My best students make steady, very satisfactory progress in much less time than those
counterparts who spent double, triple, or even more multiple times of prep with little to show.

It's all about "deliberate practices", "tenacity", and most of all, "the pleasure of finding things out on your own".

Take care and have fun problem solving.

I have been coaching students for many years. By now, I know to achieve stellar performance you need :
Grit (from TED talk), not only that but self-awareness (so you can fairly evaluate your own progress) and a nurturing-caring environment. (Parents need to be engaged as well.)

Thanks a lot !!  Mrs. Lin

"Work Smart !!" , "Deliberate practices that target your weakness ", " Relax and get fully rested.", "Pace your time well", "Every point is the same so let go of some questions first; you can always go back to them if time permits."

"It's tremendous efforts preparing for a major event on top of mounting homework and if you are the ones who want to try that, not your parents and you work diligently towards your goal, good for you !!"

"Have fun, Mathcounts changes lives, because at middle school level at least, it's one of those rare occasions that the challenges are hard, especially at the state and national level."

Now, here are the links to get you started:

Of course use my blog.  Whenever I have time I analyze students' errors and try to find better ways (the most elegant solutions or the Harvey method I hope) to tackle a problem. Use the search button to help you target your weakness area.

Last year's Mathcounts competition problems and answer key

This year's handbook questions.
Near the end of the handbook, there is a page called  problem index (page 82 and 83 for 2013-2014 handbook).
For state/national prep, find your weakness and work on the problems backwards, from the hardest to the easiest.

Here are some other links/sites that are the best.

Mathcounts Mini : At the very least, finish watching and understanding most of the questions from 2010 till now and work on the follow-up sheets, since detailed solutions are provided along with some more challenging problems.

For those who are aiming for the state/national competition, you can skip the warm-up and go directly to "The Problems" used on the video as well as work on the harder problems afterward.

Art of Problem Solving

The best place to ask for help on challenging math problems. Some of the best students/coaches/teachers are there to help you better your problem solving skills.

Register for Alcumus and start using the great tool to practice deliberately.
Change the setting based on the levels of your proficiency of different topics.                                                                     Do Not Rush !!

Awesome site!!

For concepts reviewing, try the following three links.

Mathcounts Bible

Mathcounts Toolbox

Coach Monks's Mathcounts Playbook

You really need to understand how each concept works for the review sheets to be useful.

To my exasperation, I have kids who mix up the formulas without gaining a true understanding and appreciation of how an elegant, seemingly simple formula can answer myriads of questions.

You don't need a lot of formulas, handbook questions, or test questions to excel.

You simply need to know how the concepts work and apply that knowledge to different problems/situations.