Pathfinder

From Mathcounts Mini :

Counting/Paths Along a Grid

From Art of Problem Solving

Counting Paths on a Grid 

Math Principles : Paths on a Grid : Two Approaches 

Question #1: How many ways to move the dominoes on a 6 by 6 checker board if you can only move the dominoes to the right or to the bottom starting from the upper left and you can't move the dominoes diagonally? 

Solution :
You can move the dominoes 5 times to the right at most and 5 down to
the bottom at most, so the answer is \(\dfrac {\left( 5+5\right) !} {5! \times 5!}\) = 252 ways

Question # 2: How many ways can you  move from A to B if you can only move downward and to right? 

Solution : There are \(\dfrac {\left( 4+4\right) !} {4!\times 4!}\) * 2 * \(\dfrac {\left( 4+4\right) !} {4!\times 4!}\) = 9800 ways from A to B

This Week's Work : Week 53 for Inquisitive Young Mathletes

For this week's work, review 1991 Mathcounts National sprint and target round questions you got wrong or not fast enough. Please try once more to see if you now can solve them at ease.

For this week's review, try the following Mathcounts Mini :

Similar Triangles and Proportional Reasoning

Try the follow-up problems

Detailed solutions

I'll send you notes/solutions/links to some of the hardest problems later.

Take care and happy problem solving !!

This Week's Work : Week 52 for Inquisitive Young Mathletes

Finish 2000 Mathcounts National sprint and target rounds according to the rules.
Check your e-mail.

E-mail me your scores + what questions you skip, couldn't solve fast, preferably with
the wrong answers put down so I can tell if you have some ideas.

Constructive Counting from Mathcounts Mini

Try some questions from the activity sheet till you fully understand the concepts.

Activity sheet solutions 

This Week's Work : Week 51 for Inquisitive Young Mathletes

For this week's work :

Try 2013 Mathworks Math Contest problems from Texas State University

Answer key and statistics can be viewed here. 

I'll send you detailed solutions once we finish all the problems.

Practice more "at least" problems :

2014 AMC-10 B problem 16 

Solution

From Mathcounts Mini : Video tutorials on counting and probability for Mathcounts state/national prep concepts are in order the of difficulty.

Counting the Number of Subsets of a Set

Constructive Counting

More Constructive Counting 

Probability and Counting

This Week's Work : Week 43 - for Inquisitive Young Mathletes

From Mathcounts Mini : Using Similarity to Solve Geometry Problems

Also try the warm-up, retry the problems in the video on your own and spend some time
pondering on those follow-up, harder problems.

#4, 5, 6 are state level problems. #7 is challenging.

From Mathcounts Mini : Tangent Segments and Similar Triangles

#8 is state/national level.

Have fun problem solving !! Mrs. Lin

Find the area of the petal, or the football shape.

Find the area of the football shape, or the petal shape.
The below Mathcounts mini presents two methods.

Circle and area revisited from Mathcounts mini

The first question is exactly the same as this one.
Besides the two methods on the videos, you can also use the following methods.

Solution III:
You can also look at this as a Venn Diagram question.
One quarter circle is A and the other is B, and both are congruent. (center at opposite corner vertexes)

The overlapping part is C.

A + B - C = 6^2 so C = A + B - 36 or 18pi - 36





                                                                                               

Solution IV:

If you use the area of the rectangle,
which is 6 x 12 minus, the area of the half circle with a radius 6, you get the two white spots that are un-shaded.

Use the area of the square minus that will again give you the answer.
\(6^{2}-\left( 6*12-\dfrac {6^{2}\pi } {2}\right)\)
= 18pi - 36




Similar triangles and triangles that share the same vertexes/or/and trapezoid

Another link from my blog

Similar triangles, dimensional change questions are all over the place so make sure you really
understand them.

Take care and happy problem solving !!

This Week's Work : Week 15 - for Inquisitive Young Mathletes

Part I work for this week:

See if you can write proof to show the exterior angle of any regular convex polygon is \(\frac{360}{n}\).
I'll include that in my blog for better proof.

Polygons Part I : interior angle, exterior angle, sum of all the interior angles in a polygon, how many diagonals
in a polygon

Polygons Part II : reviews and applicable word problems

Interior angles of polygons from "Math Is Fun"

Exterior angles of polygons from "Math Is Fun"

Supplementary angles

Complementary angles

Get an account from Alcumus and choose focus topics on "Polygon Angles" to practice.
Instant feedback is provided. This is by far the best place to learn problem solving, so make the best use of
these wonderful features.

This week's video on math or science : Moebius Transformations Revealed

Part II work online timed test word problems and link to key in the answers will be sent out through e-mail.

Have fun problem solving !!

This Week's Work : Week 14 - for Inquisitive Young Mathletes

Venn Diagram explained from AoPS : learn the two set and three set examples

Principle of Inclusion-Exclusion from AoPS: again, two set and three set examples

Video on Venn Diagrams with Two categories from AoPS

Video on Venn Diagrams with Three categories from AoPS

Word problems on Venn diagrams medium level (instant feedback)

Word problems on Venn diagrams medium hard level (instant feedback)

The common error for most students (especially boys, oh oh) is arithmetic error.

Make sure you line the category with specific number given right and check your math.

Chicken, rabbit algebra speed test : aim for less than 5 minutes. No calculator, please !!

Motivation, Not IQ, Matters Most For Learning New Math Skills from Time Magazine

Power of Ten Video

This Week's Work : Week 13 - for Inquisitive Young Mathletes

First, review target level questions you got wrong last week and practice counting systematically.

Right triangle inscribed in a circle proof from Khan Academy

Review 30-60-90 special right triangle from AOPS and practice getting the two legs/hypotenuse fast.

Review 45-45-90 special right triangle from AOPS

The harder questions are the ones when given the length to the 60 degree of the 30-60-90 special right triangle or
the length to the 90 degree of the 45-45-90 special right triangle so make sure you can get them fast and right.

practice here  (instant feedback)

more practices (instant feedback)

See if you can solve all these questions as they are countdown problems.
(hint : lots of Pythagorean triples or applicable special right triangle ratio concept)

from Regents prep (high school level)

Videos/articles on math for this week:

Fractals: Hunting the Hidden Dimension

 

Nature by numbers

 

What Is It About Bees And Hexagons?

This Week's Work : Week 12 -- for Inquisitive Young Mathletes

Link to the online timed test on questions you mostly got wrong or not fast enough. 
(sent through e-mail)

Common Pythagorean Triples: 
3, 4, 5 and its derivatives 
5, 12, 13 
8, 15, 17 
7, 24, 25 (at least these for SAT I and II) 

9, 40, 41, (the rest for state and Nationals, so we'll learn them later)
11, 60, 61
12, 35, 37
13, 84, 85
20, 21, 29

Shoe string method in finding the area of any polygon

Heron's formula in finding the area of a triangle.

Don't mix up the "s" with the other "S" of finding

the area of an equilateral triangle -- proof and formula (You can also use 30-60-90 special right
triangle to get that.)
or
the area of a regular hexagon

In Heron's case, "s" stands for half of the perimeter.

Besides, I've noticed most of the questions, when given the sides, are best solved by using Pythagorean triples, especially in sprint round questions, so make sure to actively evaluate the question(s) at hand and use the most efficient strategy.

Here is the link to 2003 chapter #29 that most of you got wrong:

You don't need to use complementary counting for that specific question since it's equal cases either way. Make sure you understand why you need to times 3. (AA_, A_A, and _AA for team A to be chosen two out of three days).

From Mathcounts Mini: Area of irregular polygon

See if you can use shoestring method to get the same answer.
Second half is again on similar triangles, dimensional change and sometimes
Pythagorean triples.

From NOVA : Fractals - Hunting the Hidden Dimension

This Week's Work : Week 11 -- for Inquisitive Young Mathletes

Area of an equilateral triangle

You can also use angle ratio 30-60-90 to prove that; besides, if you times that by 6, you'll get the area of a regular hexagon.

More links on Pascal's Triangle from Art of Problem Solving:

Introducing Pascal's Triangle

Patterns in Pascal's Triangle

Other review links:

From Math is Fun

Pascal's Triangle and Its Patterns

Triangular numbers

Science link :

From NOVA : How Smart Can We Get ?

This Week's Work : Week 10 -- for Inquisitive Young Mathletes

Learn or review : How many zeros?

Learn or review : Special Right Trianlges

From Art of Problem Solving of some triangles:

Isosceles and Equilateral Triangles

Isosceles Right Triangles (45-45-90 degree special right triangles)

30-60-90 Degree Special Right Triangles

Math and Science go hand in hand so watch this fascinating video from NOVA just for inspiration. 

From NOVA: What Will the Future Be Like? 

Have fun problem soling and exploring !! 

This Week's Work : Week 9 -- for Inquisitive Young Mathletes

Part 1:
See below for links:
They are all related to dimensional change and similar polygons.

Dimensional change questions I 

Dimensional change questions II 
 
Dimensional change questions III : Similar Triangles 

Par II:
Tangent Segments and Similar Triangles from Mathcounts Mini 

If you have more time, download the extra word problems to see if you can solve them at
reasonable speed and accuracy.

Online timed test and problem of the week will be sent out through e-mail.
Time: 40  minutes without a calculator.

The Monty Hall Problem explained

This Week's Work : Week 8 -- for Inquisitive Young Mathletes

Assignment 1:
Using Algebra and Number Sense as Shortcuts from Mathcounts Mini

Watch the video and work on the activity sheet below the video on the same link for more practices.

 Also, review the following:
\(x^{2}-y^{2}=\left( x+y\right) \left( x-y\right)\) \(\left( x+y\right) ^{2}=x^{2}+2xy+y^{2}\) \(\left( x-y\right) ^{2}=x^{2}-2xy+y^{2}\) \(\left( x+y\right) ^{2}-2xy =x^{2}+ y^{2}\)
\(\left( x-y\right) ^{2}+ 2xy =x^{2}+y^{2}\)
\(\left( x+y\right) ^{2}-4xy =\left( x-y\right) ^{2}\)

Assignment 2:

Pascal's Triangle  from Math is Fun

Pascal's Triangle and Its Patterns

Assignment 3:
It'll be sent through e-mail.
Happy problem solving !!

This Week's Work : Week 6 and 7 Review -- for Inquisitive Young Mathletes

Watch Joint Proportion from Art of Problem Solving 

Spend some time pondering on "Work" word problems from Purple Math. 
These are some very standard word problems you'll encounter in competition math.  

Review --
dimensional change and probability links from previous weeks.  

More practices on inverse and direct relation 

From Regents Exam Prep 
Link I 

Link II

New concepts:

Height to the hypotenuse
How many ways to arrange the word "banana"? (with elements repeating)
Probability that two of the 3 friends were born on the same week day.

Question : If you can earn 0, 1, 3, 7 or 10 points with each shot and each person has three chances, how many scores can't be made? 

Note:
Percentage increase (don't forget to minus the original 100% or 1) is very different from at what percent will it return to the original size or what is the size compared to the original.

This Week's Work : Week 5 -- for Inquisitive Young Mathletes

Evan (a 5th grader in PA) 's Problem of the Week: 
A man notices a sign in Shop-a-Lot that says: "All prices are marked 25% off today only!" He decides to buy a shirt that costs $65.12 before the discount. He then uses a $16 gift certificate and the clerk applies 12% sales tax. What is the final cost of the shirt after all the steps are applied? Express your answer to the nearest hundredth. 
 
Solution: 
When there is a discount that is 25% off, the original price is going to be 75% times the original price. Therefore, 75% of $65.12 is $48.84. Next, subtracting $16, we obtain $32.84. Finally, applying 12% sales tax, we get $32.84 x 112% (since original price is added to the sales tax "price") which is $36.7808. This rounded to the nearest hundredth is $36.78.

Assignment 1: 

Painted Cube Problems

Visualization of the Painted Cube Problems 

It's more fun if you dig out Legos or Unit Cubes and just build some cubes (and later rectangular prism) 
and spend some time observing how it works. 

Assignment 2:

Review special right triangle, Pythagorean triples, theorem :

30-60-90 , 45-45-90 special right triangle angle ratios

Dimensional Change

Inscribed and Circumscribed Circle Radius of an Equilateral Triangle

This Week's Work : Week 4 -- for Inquisitive Young Mathleges

First of all, problem of the week from Evan, a 5th grader:
A man notices a sign in Shop-a-Lot that says: "All prices are marked 25% off today only!" He decides to buy a shirt that costs $65.12 before the discount. He then uses a $16 gift certificate and the clerk applies 12% sales tax. What is the final cost of the shirt after all the steps are applied? Express your answer to the nearest hundredth. 

This week, we'll learn two very common sequences : arithmetic and geometric sequences.
There are quite a few similarities between these two types and they are closely linked to ratio, proportion
so just watch the videos and play around/generate a few/ponder on those sequences. I don't expect you to learn them in just one week.

Notes from Regents Exam Prep: Arithmetic and Geometric Sequences and Series

From Mthcounts Mini:

Easier concepts:

Sequences

Arithmetic sequence/determine the nth term

Arithmetic and geometric sequences

Harder concepts:

Relationship between arithmetic sequences, mean and median

Sequences, series and patterns

From my blog : 
Some special arithmetic sequences and the easier way to find their sum

Write some notes of the most important features of the arithmetic sequence.
The best note will be posted here to share with other students.
Be creative !!

This Week's Work : week 3 -- for Inquisitive Young Mathletes

Assignment 1: 
Watch and learn Simon's Favorite Factoring Trick.

Work on some of the problems as well to check your understanding.

Assignment 2: 
Just learn as much as you can.
We'll keep practicing counting and probability.

Counting Permutations : from Art of Problem Solving

With or Without Replacement : from Art of Problem Solving

Notes on Permutations from the Math Page

Permutations with Some Identical Elements

This Week's Work : Week 2 -- for Inquisitve Young Mathletes

Assignment 1:
Mathcounts Mini related to the "Set" concept
Download the word problems below the video and work on them for this week.

Pascal's Triangle from Math is Fun.

Below is problem of the week, which continues with Evan's problem from last week so read it carefully.
Two players play a game starting with a pile of 26 sticks. The players alternate turns, each taking 1, 2, or 3 sticks on his or her turn.The player who takes the last stick wins.Who has the winning strategy in this game, the first player or the second player? How many sticks he/she needs to take? Why? 

Assignment 2:
Review special right triangles: Notes from my blog

30-60-90 Triangles from Art of Problem Solving

Powers of Pythagorean Triples from Art of Problem Solving

Working together rate  problems from Art of Problem Solving.