Monday, June 23, 2025
Monday, June 9, 2025
2024/2025 Mathcounts, AMCs, AIMEs Competition Preparation Strategies
Hi, Thanks for visiting my blog.
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 !!
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.
Newest Mathcounts' competition problems and answer key
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.
E-mail me at thelinscorner@gmail.com if you want to learn with me. :) :) :)
Currently I'm running different levels of problem solving lessons, and it's lots of fun learning along with students from different states/countries.
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.
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 (ONLY the hardest problems), 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.
In Chinese : 《專題報導》亞裔尖子生 職場難出頭？
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.
Newest Mathcounts' competition problems and answer key
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.
Do Not Rush !!
Awesome site!!
For concepts reviewing, try the following three links.
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.
Hope this is helpful!!
Do Not Rush !!
Awesome site!!
For concepts reviewing, try the following three links.
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.
Hope this is helpful!!
Friday, June 6, 2025
2015 Mathcounts National sprint #22 level 1 +
2015 Mathcounts National
sprint #22
If six
people randomly sit down at a table with six chairs, and they do not notice
that there are name tags marking assigned seats, what is the probability that
exactly three of them sit in the seat he or she was assigned? Express your
answer as a common fraction.
Try this yourself first (extremely), then scroll down for solutions.
Method 1-Elementary Counting
Choose the three people who sit correctly:
\( \binom{6}{3}=20 \).
The remaining three must all sit incorrectly.
A quick check (or listing) shows only two ways: \((A\,B\,C)\) or \((A\,C\,B)\).
Total favourable seatings \(=20\times2=40\); total seatings \(=6!\).
Therefore \( P=\dfrac{40}{720}=\boxed{\tfrac{1}{18}} \).
Method 2 – Derangement formula
Pick the three fixed seats: \( \binom{6}{3}=20 \).
Derange the other three: \( !3 = 2 \).
Again \( 20\times2 = 40 \) good seatings, so
\[
P \;=\; \frac{20\cdot!3}{6!} \;=\; \boxed{\tfrac{1}{18}} .
\]
Derange the remaining 3 people. The number of derangements of 3 items is \[ D_3 \;=\; 3!\left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}\right) \;=\; 6\left(1 - 1 + \frac{1}{2} - \frac{1}{6}\right) \;=\; 2. \]
Wednesday, June 4, 2025
student learning log, a high school student I volunteer
each week a space
SAT 14 #2 wrong , 15 #2
SAT 16, #5, 6, wrong , 17, 3 and 6
2 weeks later
18, #5 and 5
19 vocab. words 11 - 20
SAT practice test 8, later math don't know what to do
2 weeks later
20, #5, 21, #4
more than a month later
22, #4 and 5 , 23, #4, 6, 8, 12
30, #5, 31, #9
32, #4, 5, 33, 2, 9
local math competition : 21-22 meet 1, # 1, 2, 3 right
34, #4, 35, #6 and 8
21-22 math competition meet 2, #1 and 3 , #2 almost
36 #1，2，3 need to work on harder vocabulary words
37 # 6，9
Math 2021-22 meet 2 # 1 and 3
5/6/25 This week for reading I did Crack Sat tests 38 and 39 and I only gave myself 15 minutes for each test to time myself.
For test 38 the questions I got wrong were 1 and 11 and for test 39 the questions I got wrong were 2 and 5.
Although I felt rushed I was able to complete the test while still comprehending everything so that’s good.
For math this week I realized that I struggled a lot on the algebra 2 part of the (local math competition) tests so I decided that I should learn some of the curriculum.
To learn the course I went on Khan academy and did 2 units of the course and I plan on continuing learning the course to help with (local math competition) problems in the future. Thank you.
Monday, June 2, 2025
6/2, 2025 one interesting question that has nice solution : Beginning level
2010 Mathcounts
Nationals sprint :
22. Side AB of regular
hexagon ABCDEF is extended past B to point X such that AX = 3AB. Given that
each side of the hexagon is 2 units long, what is the length of segment FX?
Express your answer in simplest radical form.
Try this question first before you scroll down for the solution.
Then when we draw line FX, and by the Pythagorean Theorem, we have \[ FX = \sqrt{(\sqrt{3})^2 + (1 + 2 + 4)^2} = \sqrt{3 + 49} = \boxed{2\sqrt{13}}. \]
Friday, May 30, 2025
Ar. Student reflection notes to keep track of progress
from a 9th grader Ar.
Hello Mrs. Lin, 4/25
I hope you are well.
Sorry for sending this a bit late, but I wanted to share my reflection for what I have done this week. I first looked over the problems we did in class. I had some issue with the last problem-I am still not 100 percent on that one. I was hoping if you could please re-explain this in class, that would be helpful. I also looked over the formula sheet. While doing some of the AMC problems you gave, I tried to really focus on the first 7 problems. I was hoping we could go over some quicker ways to think on problems 3, 5, and 6 on the AMC 10 2023 A.
Sorry for sending this late.
Thank you,
Thank you Mrs. Lin. 5/2
I wanted to share my reflection for this week. I reviewed all of the problems we went through during class, and I really understood everything. I continued doing problems from the AMC 10 2023 A test, and I redid problems 3, 5, 7. Those were the problems I struggled with last week, so I reviewed those. I also tried to go on by doing problems 7 to 13, but it took me a while to do those and I didn’t get those correct. I went back to problems 1 through 7, except for the B test.
Thank you,
Hello Mrs. Lin, 5/11
I wanted to share my reflection from this week.
I felt good about all the problems we did in class, but I wanted to just quickly go over the last problem once more. I had a question on that one. I started a new test (2022 AMC 10 A), and did questions 1-10. I was hoping to go over questions 5, 7, 8, and 10.
5/17 no notes
5/24
This week I reviewed the SAT problems we went over in class. If you could please give me some of those harder SAT problems going forward for homework that would be great. I thought that they were good practice.
I didn’t have a ton of time this week for AMC work, because I have finals for many classes coming up. However, I did do some problems from the 2016 AMC 10 A.
I had some trouble with problems 11, 12 and 9. If we could please go over those that would be great Sorry about the late reflection again.
5/31
Hi Mrs. Lin,
I wanted to send you my reflection for this week. I re-did the AMC 10 2016 A test, including the problems from last week. I also did problems 13 to 20. I had questions on 13, 17, 18, and 20.
Thanks,
Tuesday, May 27, 2025
2025 Mathcounts state sprint #22 problem and solution : level 1 +
2025 Mathcounts state sprint
#22: Let n be a positive integer less than
or equal to 1000. If the last two digits of n are reversed, the resulting
integer is exactly 85 percent of n. What is the sum of the possible values of
n?
Try this question first. Then scroll down for solution.
Let n be a positive integer less than or equal to 1000. If the last two digits of n are reversed, the resulting integer is exactly 85 percent of n. What is the sum of the possible values of n?
Let the original number be:
$$n = 100h + 10t + u$$The number formed by swapping the tens and units digits is:
$$n' = 100h + 10u + t$$According to the problem:
$$n' = \frac{17}{20}n$$So \( n \) has to be divisible by 20 (make sure you know why). This implies:
$$u = 0, \quad t \text{ is even}$$Let:
$$t = 2k, \quad 0 \leq k \leq 4$$Then:
$$n = 100h + 10t = 100h + 20k$$ $$n' = 100h + t = 100h + 2k$$Now compute the difference:
$$n - n' = 18k$$Also, from the given:
$$n - n' = n - \frac{17}{20}n = \frac{3}{20}n$$Equating both expressions:
$$18k = \frac{3}{20}n \Rightarrow n = 120k$$Since \( k \neq 0 \), we get:
$$n = 120k$$Valid values for \( k \in \{1, 2, 3, 4\} \), so the numbers are:
$$120, \quad 240, \quad 360, \quad 480$$Their sum is:
$$120 + 240 + 360 + 480 = 120(1 + 2 + 3 + 4) = 120 \times 10 = \boxed{1200}$$Sunday, May 25, 2025
V's record
V's record
weekly homework about 30 to 45 minutes
extra videos, links optional
First lesson:
2010 chapter sprint:
Hi, I got a score of 22 and got questions 16, 22, 23, 24,25,27,29, and 30 wrong.
I just guessed these questions because I didn't really find a way to do them.
Second meet:
2011-12 Mathcounts handbook (40 questions total)
Warm Up 1: 8
Warm Up 2: 19,20
Warm Up 2: 19,20
Warm Up 3: 32, 39
Workout 1: 23, 24, 30
Third meet:
2010 Mathcounts school test :
Hi, I finished trying the Sprint and Target questions:
Fifth meet : review
these are the problems that I got wrong
Sprint: I got 7/15 correct.
15,16: Attempted but answer was wrong
19,20,26,27,29: Didn't attempt
30: Couldn't find a good method to do, but was able to solve it by listing out all the possibilities.
Target: I got 6/8 correct.
The first 6 were relatively easy and I could find a clear way to do them
The last 2, I couldn't find a way to approach the problem.
Fourth meet :
2011 Mathcounts school test : last 15 sprint and last 4 target
Sprint: Out of the last 15,
I got 7 correct.
The questions that I got wrong were 20,21,22,24,27,28,29,30.
I tried 21,22, and 27 but the answers were incorrect. I wasn't able to attempt the rest.
Target: I got every question other than #5.
However, questions 1 and 2,
I got wrong at first, but when I retried them,
I was able to get them. I read the problems wrong and didn't fully understand them the first time.
Fifth meet : review
Sixth meet :
2011 chapter
Hi Mrs Lin! I was able to try both the target and the sprint round questions and here are my results:
Target: I missed 3 & 8, and I didn't attempt them.
Sprint: I got 17,21,23,24,25, 27,29,30.
I didn't attempt 27,29, and 30,
2012 Mathcounts school
Hi Mrs Lin! I was able to finish both the chapter and target, and here are my results:
Target: I got 5,6,and 8 wrong. I didn't try any of them because I didn't know how to do them.
Sprint: I got 18,23,25,26, and 30
Apr. 30, 2023
2012 Mathcounts chapter
Hi Mrs. Lin! I hope your having a very good day and week! I tried both the target and the sprint.
Target: I got only number 8 wrong, but I didn't know how to do it.
Sprint: I got number 22, 24, 26,28,29, and 30 wrong. I tried to do 22, but I got it wrong. The rest I didn't know how to do them..
May 7th, review
May 13th, 2023
2013 Mathcounts School
Hi Mrs.Lin, I was able to try both the Target and the Sprint and here are my results!
Target (5/8) I got 6,7, and 8 wrong, however, I was able to figure out the answer to problem 6 when I reviewed it.
Sprint: I got 30,29,28, and 24 wrong.
May 20th, 2023
Hi Mrs Lin!
2013 Mathcounts Chapter
I was able to try both the target and sprint round tests and here are my results:
Target: 6/8. I attempted the first 6 problems and got all of them right. I understood the problems fairly well and was able to do all of them on the first try.
sprint: I got 10/15 right. I got all problems I attempted right, and didn't attempt 30,28,25,24,21, and 19.
2014 Mathcounts school
Hi Mrs Lin, I was able to finish both the target and the sprint, and here were my results!
Sprint: 22/30, I didn't get 18,19,24,25,26,28,29, and 30. I attempted number 18, but didn't get it.
Target: I got 5 out of 8 on the target, but after reviewing my answers, I was able to figure out number 5.
2015 Mathcounts school
Hi Mrs Lin, I was able to try both the sprint and target. For the sprint, I got problems 20 and 27 wrong, and I didn't attempt any problems past 24 other than 27.
For my Target I got numbers 6,7, and 8 wrong, and I didn't know how to approach any of them.
Show Your Work, Or, How My Math Abilities Started to Decline
Show your work, or, how my math abilities started to decline
I think it's problematic the way schools teach Algebra. Those meaningless show-your-work approaches, without knowing what Algebra is truly about. The overuse of calculators and the piecemeal way of teaching without the unification of the math concepts are detrimental to our children's ability to think critically and logically.
Of course eventually, it would be beneficial to students if they show their work with the much more challenging word problems (harder Mathcounts state team round, counting and probability questions, etc...), but it's totally different from what some schools ask of our capable students.
How do you improve problem solving skills with tons of worksheets by going through 50 to 100 problems all look very much the same? It's called busy work.
Quote: "Insanity: doing the same thing over and over again and expecting different results."
Quotes from Richard Feynman, the famous late Nobel-laureate physicist. Feynman relates his cousin's unhappy experience with algebra:
My cousin at that time—who was three years older—was in high school and was having considerable difficulty with his algebra. I was allowed to sit in the corner while the tutor tried to teach my cousin algebra. I said to my cousin then, "What are you trying to do?" I hear him talking about x, you know."Well, you know, 2x + 7 is equal to 15," he said, "and I'm trying to figure out what x is," and I say, "You mean 4." He says, "Yeah, but you did it by arithmetic. You have to do it by algebra."And that's why my cousin was never able to do algebra, because he didn't understand how he was supposed to do it. I learned algebra, fortunately, by—not going to school—by knowing the whole idea was to find out what x was and it didn't make any difference how you did it. There's no such a thing as, you know, do it by arithmetic, or you do it by algebra. It was a false thing that they had invented in school, so that the children who have to study algebra can all pass it. They had invented a set of rules, which if you followed them without thinking, could produce the answer. Subtract 7 from both sides. If you have a multiplier, divide both sides by the multiplier. And so on. A series of steps by which you could get the answer if you didn't understand what you were trying to do.
So I was lucky. I always learnt things by myself.
I think it's problematic the way schools teach Algebra. Those meaningless show-your-work approaches, without knowing what Algebra is truly about. The overuse of calculators and the piecemeal way of teaching without the unification of the math concepts are detrimental to our children's ability to think critically and logically.
Of course eventually, it would be beneficial to students if they show their work with the much more challenging word problems (harder Mathcounts state team round, counting and probability questions, etc...), but it's totally different from what some schools ask of our capable students.
How do you improve problem solving skills with tons of worksheets by going through 50 to 100 problems all look very much the same? It's called busy work.
Quote: "Insanity: doing the same thing over and over again and expecting different results."
Quotes from Richard Feynman, the famous late Nobel-laureate physicist. Feynman relates his cousin's unhappy experience with algebra:
My cousin at that time—who was three years older—was in high school and was having considerable difficulty with his algebra. I was allowed to sit in the corner while the tutor tried to teach my cousin algebra. I said to my cousin then, "What are you trying to do?" I hear him talking about x, you know."Well, you know, 2x + 7 is equal to 15," he said, "and I'm trying to figure out what x is," and I say, "You mean 4." He says, "Yeah, but you did it by arithmetic. You have to do it by algebra."And that's why my cousin was never able to do algebra, because he didn't understand how he was supposed to do it. I learned algebra, fortunately, by—not going to school—by knowing the whole idea was to find out what x was and it didn't make any difference how you did it. There's no such a thing as, you know, do it by arithmetic, or you do it by algebra. It was a false thing that they had invented in school, so that the children who have to study algebra can all pass it. They had invented a set of rules, which if you followed them without thinking, could produce the answer. Subtract 7 from both sides. If you have a multiplier, divide both sides by the multiplier. And so on. A series of steps by which you could get the answer if you didn't understand what you were trying to do.
So I was lucky. I always learnt things by myself.
Sunday, May 18, 2025
The Grid Technique in Solving Harder Mathcounts Counting Problems : from Vinjai
The following notes are from Vinjai, a student I met online. He graciously shares and offers the tips here on how to tackle those harder Mathcounts counting problems.
The
point of the grid is to create a bijection in a problem that makes it easier to
solve. Since the grid just represents a combination, it can be adapted to work
with any problem whose answer is a combination.
For
example, take an instance of the classic 'stars and bars' problem (also known
as 'balls and urns', 'sticks and stones', etc.):
Q: How
many ways are there to pick an ordered triple (a, b, c) of nonnegative integers
such that a+b+c = 8? (The answer is 10C2 or 45 ways.)
Solution I:
This
problem is traditionally solved by thinking of ordering 8 stars and 2 bars. An
example is:
*
* * | | * * * * *
^
^ ^
a
b c
This
corresponds to a = 3, b = 0, c = 5.
Solution II:
But
this can also be done using the grid technique:
The red path corresponds to the same arrangement: a = 3, b = 0, c = 5. The increase corresponds to the value: a goes from 0 to 3 (that is an increase of 3), b goes from 3 to 3 (that is an increase of 0), and c goes from 3 to 8 (that is an increase of 5). So a = 3, b = 0, c = 5.
Likewise,
using a clever 1-1 correspondence, you can map practically any problem with an
answer of nCk to fit the grid method. The major advantage of this is that it is
an easier way to think about the problem (just like the example I gave may be
easier to follow than the original stars and bars approach, and the example I
gave in class with the dice can also be thought of in a more numerical sense).
Wednesday, May 14, 2025
Similar Triangles: Team question : Beginning level
9. In the figure below, quadrilateral CDEG is a square with CD = 3, and quadrilateral BEFH is a rectangle. If EB = 5, how many units is BH? Express your answer as a mixed number
Triangle BED is a 3-4-5 right triangle and is similar to triangle GEF.
BE : ED = GE : EF = 5 : 3 = 3 : FE
EF = 9/5 = BH The answer
Wednesday, May 7, 2025
Ay reflection notes
Ay
5/7
1) Reviewed the attached problems and solutions and I understand them well
2) Attempted the 2024 AMC 12A in 60 minutes. I attempted #1 to #19 and got them all correct.
I had a headache after that so I couldn't attempt the rest of the questions. Skipped -#20 to #25
Not sure how many I could have actually done.
Sunday, May 4, 2025
An, a 7th grader sample student reflection note, or report
from a 7th grader A.
2025 chapter test
total 34 correct
21-30 wrong on sprint
8th was wrong on target
Sent from my iPhone
2024 chapter test
Sprint Round
2 Q19 Algebra Silly mistake Didn't set up the equation correctly Underline important parts of the question
3 Q23 Geometry Didn't know how to do it Learn how to do it
4 Q25 Geometry Didn't know how to do it Learn how to do it
5 Q26 Algebra Didn't know how to do it Learn how to do it
6 Q27 Time, Rate, Distance Didn't know how to do it Learn how to do it
7 Q28 Number Theory Didn't know how to do it Learn how to do it
8 Q29 Probability Silly mistake Didn't count all possible scenarios Learn how to do a faster way to solve these probability problems
9 Q30 Geometry Didn't know how to do it Learn how to do it
10 Target
11 Q5 Alegbra Wasn't sure on how to do it. Learn how to do it
12 Q6 Number Theory Didn't know how to do it Learn how to do it
13 Q8 Number Theory Wasn't sure on how to do it. Learn how to do it
14
15 Total score: 32
Sent from my iPhone
Monday, February 17, 2025
Harder Mathcounts State/AMC Questions: Intermediate level if you can solve in less than 2 mins.
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 :
Solution I :
\(\overline {AB}:\overline {NC}=5:4\) [given]
Triangle ASB is similar to triangle CSN (AAA)
\(\overline {NS}:\overline {SB}= 4 : 5\)
Let \(\overline {NS}= 4a, \overline {SB}= 5a.\)
\(\overline {MT}: \overline {NC}\) = 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 :
2016 AMC 10 A, #19 : Solution from Abhinav
2016 AMC 10 B #19 : Solution from Abhinav
Subscribe to:
Posts (Atom)
