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2017 Mathcounts State Prep: Some Counting and Probability Questions on Dot
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#5 1993 Mathcounts National Target : Find the probability that four randomly selected points on the
geoboard below will be the vertices of a square? Express your answer as a common fraction.
21 Mathcounts Competition Preparation
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2011 Mathcounts Chapter Sprint solutions
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#5 2004 AMC 10A: A set of three points is chosen randomly from the grid shown. Each three-point
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(same image as the below question) set has the same probability of being chosen. What is the
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probability that the points lie on the same straight line?
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2007 Mathcounts Chapter Sprint #29 : The points of this 3-by-3 grid are equally spaced
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horizontally and vertically. How many different sets of three points of this grid can be the three
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vertices of an isosceles triangle?
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Solution:
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mathcounts notes: counting problems
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#5 National Target: There are 16C4 =
= 1820 ways to select 4 points on the geoboard.
Advanced beginning level (1)
AIME (1)
There are 3 x 3 = 9 one by one squares and 2 x 2 = 4 two by two squares and 1 x 1 = 1 three by three squares. (Do you
AIME I (1)
AIME II (1)
see the pattern?)
AIME prep (1)
algebra (5)
algebra manipulation (2)
Algebra problems (1)
There are 4 other squares that have side length of √ 2
AMC (3)
and 2 other larger squares that have side length of √ 5.
AMC 10 prep (1)
AMC 12 prep (1)
9 + 4 + 1 + 4 + 2 = 20 and
AMC 8 (2)
AMC 8 prep (2)
AMC 8 problems (2)
AMC hardest problems (1)
AMC prep (5)
AMC problems (2)
AMC test (1)
#5: Solution:
AMC-10A: There are 9C3 =
AMC-10 (1)
= 84 ways to chose the three dots and 8 of the lines connecting the three dots
will form straight lines. (Three verticals, three horizontals and two diagonals.) so
AMC-12 (1)
AMC-8 (3)
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AMC8 (1)
#29: Solution:
Use the length of the two congruent legs to solve this problem systematically.
AMCs (4)
analytical geometry (1)
angle bisector (1)
Angle trisect (1)
There are 16 1 - 1 -
isosceles triangles.
There are 8
by 2 isosceles triangles. (See that ?)
by
Angles (2)
Area (1)
area of a regular hexagon (1)
area of an equilateral triangle (1)
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articles on math education (1)
There are 4
2-2-
There are 4
by
Finally, there are 4
isosceles triangles.
by 2 isosceles triangles.
by
by
isosceles triangles.
Add them up and the answer is 36.
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2013 Mathcounts State Prep: Counting Problems
constructive counting (2)
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counting (6)
Counting and Probability (9)
#1: 2006 Mathcounts state : My three-digit code is 023. Reckha can’t choose a code that is the same as
counting backwards (1)
mine in two or more of the three digit-positions, nor that is the same as mine except for switching the
counting coins (1)
positions of two digits (so 320 and 203, for example, are forbidden, but 302 is fine). Reckha can
counting problems (5)
otherwise choose any three-digit code where each digit is in the set {0, 1, 2, ..., 9}. How many codes
counting with restrictions (1)
are available for Reckha?
cube (1)
difference of squares (1)
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mathcounts notes: counting problems
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Solution:
digit sum (1)
Do complementary counting. Use total possible ways minus those that are not allowed.
dimension change (1)
dimensional change (5)
You can't use two or more of the numbers that are at the same position (given) as 203, which means that you can't
have 0 __ 3, __ 23, or 02__.
dimensional change I (1)
dimensional change II. Mathcounts (1)
For each of the __, you can use 10 digits (from 0, 1, 2 ... to 9) so 10 + 10 + 10 = 30.
However, you repeat 023 three times in each case so you need to minus 2 back so not to over count.
dimensional change III (1)
distance (1)
elementary school level (1)
equilateral triangle (1)
30-2 = 28
essay (1)
Also, you can't just switch two digits, which means 320, 203 and 032 are not allowed. { but 302 and 230 are allowed
since you are switching all the digits }
etc... (1)
exterior angle of a regular polygon (1)
face diagonal; space diagonal (1)
There are 10 x 10 x 10 = 1000 digits total and 1000 - 28 - 3 = 969 The answer
factors (1)
Fibonacci Numbers (1)
#2: 2011 AMC-8 # 23: How many 4-digit positive integers have four different digits where the leading
digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?
folding paper questions (1)
for young mathletes (1)
frog puzzle (1)
Solution:
For the integer to be a multiple of 5, there are two cases:
games (2)
geometric mean of a right triangle (1)
Case I: The unit digit is 5 : __ __ __ 5
geometric sequence (1)
There are 4 numbers to choose for the thousandth digit [since 5 is the largest digit and you can't have "0" for the
geometry (29)
leading digit so there are 4 numbers 1, 2, 3, 4 that you can use], 4 numbers to choose for the hundredth
grid (2)
digit (0 and one of the remaining 3 numbers that are not the same number as the one in the thousandth digit) and 3
harder Mathcounts problem (4)
numbers to choose for the tenth digit (the remaining 3 numbers) so total 4 x 4 x 3 = 48 ways
harder Mathcounts problems (1)
height to the hypotenuse (1)
Case II: The unit digit is 0: __ __ __ 0
One of the remaining three numbers has to be 5, and for the remaining 2 numbers, there are 4C2 = 6 ways
to choose the 2 numbers from the numbers 1, 2, 3 or 4.
There are 3! arrangements for the three numbers so 6 x 3! = 36
Heron's formula (2)
how many animals (1)
how many diagonals in a polygon (1)
how many ways (2)
how many ways from point A to B (1)
48 + 36 = 84 ways
How many ways to make 25 cents (1)
How many zeros? (2)
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2013 Mathcounts State Prep: Partition Questions
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#24 2001 Mathcounts Sate Sprint Round: The number 4 can be written as a sum of one or more
inverse and direct proportion (1)
natural numbers in exactly five ways: 4, 3+1, 2 + 1 + 1, 2 +2 and 1 + 1 + 1 + 1; and so 4 is said to have
inverse and direct relation (1)
five partitions. What is the number of partitions for the number 7?
largest area (1)
#2: Extra: Try partition the number 5 and the number 8.
learn how to learn; how to prepare for
Mathcounts (1)
learning (1)
Solution:
#24: You can solve this problem using the same technique as counting coins:
Counting coins questions
7
math (1)
math cartoons (1)
math concept (1)
6 5 4 3 2 1
1
Mass point geometry (2)
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1 way
Math Jams (1)
1 way
math word problems (1)
2 ways ( 5 + 2 or 5 + 1 + 1)
math word problems of the day (2)
1 way
Mathconts (1)
1
2 ways ( 4 + 2 + 1 or 4 + 1 + 1 + 1)
Mathcounts (86)
2
0 1
1 ways
Mathcounts chapter level (1)
1
2
3 ways ( 3 + 2 + 2, 3 + 2 + 1 + 1 and 3 + 1 + 1 + 1 + 1)
3
4 ways (2 + 2 + 2 + 1, 2 + 2 + 3 ones, 2 + 5 ones and 7 ones.)
1
1
1
1 1
1
Total 15 ways.
Mathcounts coach (1)
Mathcounts competition problem solutions
(1)
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mathcounts notes: counting problems
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The partitions of 5 are listed below (There are 7 ways total.):
Mathcounts concept (1)
5 4 3 2 1
1
1
1 1
2
Mathcounts geometry (1)
1 way
1 way
2 ways (3 + 2 and 3 + 1 + 1)
3 ways (2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1)
Mathcounts harder problems (1)
Mathcounts National (3)
Mathcounts national prep (2)
There are 22 ways to partition the number 8.
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Mathcounts geometry questions (1)
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S A T U R D AY, M AY 1 9 , 2 0 1 2
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How Many Numbers (Terms)? Space, Inclusive and Exclusive Notes
Please write a comment and give me feedback. Thanks a lot!!
Mathcounts preparation (1)
Mathcounts problem (3)
Mathcounts problems (49)
Quite a lot of my students have problems figuring out this type of problems so here are the notes.
Mathcounts problems. competition math
(1)
#1: How many consecutive numbers from 1 to 5 inclusive?
Mathcounts problems. Special factoring
technique (1)
1 _ 2 _ 3 _ 4 _ 5 There are 5 numbers if you just list them and count them out; however, what if
Mathcounts questions (4)
the question is:
Mathcounts state (15)
#2: How many consecutive numbers from 34 to 200 (SAT type problem)?
Most students would think it's 200 - 34 = 166, but it's not.
Using #1 case, if you do 5 - 1 = 4, you are only getting how many spaces between those consecutive numbers.
Mathcounts state/national prep (1)
Mathcounts state/national prep (1)
Mathcounts strategies (1)
Mathcounts strategy (1)
Mathcounts word problems (1)
Mathcouts prep (1)
Thus for question #2, the correct answer is 200 - 34 + 1 or 200 - 33 = 167
mathleague (1)
medians (1)
#3: What about how many consecutive numbers from 5 to 100 exclusive?
mental math (1)
mental math trick I (1)
Inclusive means including the first and the last numbers; exclusive means not including the first and the last
numbers, so for this question, you do 100 - 5 - 1 = 94.
Use # 1 case to help you figure out and really understand the concepts involved.
Here are other questions to help you practice the skills.
monument Valley (1)
number 11 (1)
number theory (2)
numbers (1)
painted cube (1)
painted cube problems (1)
partition (2)
Word problems: Answers below.
pascal's triangle (3)
#1: How many numbers from 45 to 100 inclusive?
permutations (1)
#2: How many numbers from 17 to 127 inclusive?
polygons (1)
#3: How many numbers from 12 to 34 exclusive?
prime factorization (1)
#4: How many multiples of 9 from 1 to 200 inclusive?
prime number chart (1)
prime numbers (1)
probability (4)
problem solving (25)
#5: The dimension of the square on the left is 20 feet by 20 feet. If you place a
post every four feet, starting at one corner, how many posts will be placed?
problem solving strategies (2)
problem solving strategy (1)
puzzles (2)
Pythagorean Triples (2)
rabbit questions (1)
Rate (3)
rate time distance questions (1)
#6: The distance from exit 13 to 21 is 216 miles. How many miles is the distance between two exits if all exits are
equally spaced?
#7: How many multiples of 5 from 120 to 218 exclusive?
rectangle inscribed in a triangle (1)
rectangular prism (1)
recursion (1)
Richard Rusczyk (1)
right triangle (1)
#8: Who is right? The teacher or the student? Try this question.
right triangle in a circle (1)
rt = d (1)
#9: How many numbers from -12, -11, -10.........56 inclusive?
SAT harder math (1)
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SAT II harder math (2)
What is their sum?
SAT math (1)
#10: How many numbers are in the list: 17.25, 18.25, 19, 25...111.25?
SAT math harder problem strategies. (1)
set (2)
shoelace method (1)
Shoestring method (3)
similar polygons (1)
similar triangle (1)
similar triangles (14)
similar triangles II (1)
Simon's Favorite Factoring Trick (2)
slope (1)
special right triangles (4)
square based pyramid (1)
square numbers (1)
Solutions: To excel at Mathcounts state/national, you need to practice all these questions mentally.
Stars and bars (1)
sticks and stones (2)
student's original problem (1)
#1: 100 - 45 + 1 = 100 - 44 = 56
study habits (1)
#2: 127-17 + 1 = 127 - 16 = 111
#3: Exclusive: 34 -12 -1 = 34 - 13 = 21
sum (1)
#4: Multiples of 9 from 1 to 200 starts with 9 and ends in 198.
sum and product of roots. (1)
Sum of all the possible arrangements (1)
Solution I: 9 , 18, 27...198 = 9 (1, 2, 3, ...22) The answer is 22.
sum of the angles of a polygon (1)
Solution II:
Sunday nights' problem solving group
lessons (1)
tangent segments (1)
tenth digit (1)
test taking strategies (1)
testemonials (1)
#5: Just observe one side first. Exclude the 4 corners, the other posts
are similar to those exclusive type problems.
There are
\frac{20}{4} - 1 = 4
posts on each side so 4 * 4 + 4 (corner posts)
= 20
The Grid technique (1)
The Hockey Stick Identity (1)
this week's work (19)
this week's work 10 (1)
this week's work 11 (1)
this week's work 12 (1)
#6: There are 21 - 13 = 8 space so
\frac{216}{8} = 27
miles. The answer is 27 miles.
this week's work 13 (1)
this week's work 14 (1)
#7: Multiples of 5 from 120 to 218 start with 120 and end in 215.
this week's work 15 (1)
Since it's asking exclusive, 120, 125, ...215 = 5(24, 25, ...43)
this week's work 2 (1)
43 - 24 - 1 = 43 - 25 = 18
this week's work 3 (1)
#8: You only need two cuts to get 3 pieces so 2 * 10 = 20 minutes. The student is right.
this week's work 35 (1)
#9: 56 - ( -12) + 1 = 56 + 13 = 69
The sum is from 13 to 56 since up to 12 it got cancelled with the negative equivalent numbers.
Use average * the term you got
\frac{(13 + 56)}{2}* (56 - 13 +1) = 1518
. The sum is 1518.
#10:111.25 - 117.25 + 1 = 111.25 - 116.25 = 95
PO ST ED BY S OME ON E O PPO SI TE O F PI ER RE AT 10 :1 4 AM
this week's work 38 (1)
this week's work 4 (1)
this week's work 43 (1)
this week's work 44 (1)
1 CO MMEN T:
LABELS : BEGI NN IN G LEVE L , COU NTI NG PROBLE MS , MAT HCOU NT S
this week's work 45 (1)
this week's work 46 (1)
this week's work 5 (1)
this week's work 6 (1)
F R I D AY, M AY 1 1 , 2 0 1 2
this week's work 7 (1)
Problem Solving Strategy: Counting Coins
this week's work 8 (1)
Q: How many ways can you make 25 cents if you can use quarters, dimes, nickels or pennies?
this week's work 9 (1)
Start with the largest value:
three pole problems (1)
time (1)
1 Quarter
1 way
2 Dimes, 1 Nickel You can stop here since it implies 2 ways.
2 Dimes, 0 Nickel ( which implies 5 pennies)
1 Dime, 3 Nickels , which implies 4 ways.
1 Dime, 2 Nickels ( 5 pennies)
trapezoid (3)
triangles share the same vertex (3)
triangles shares the same base (1)
Triangular numbers (1)
triangular numbers. chapter level (1)
unit digit (1)
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mathcounts notes: counting problems
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1 Dime, 1 Nickel (10 pennies)
Venn Diagram (1)
1 Dime, 0 Nickel ( 15 pennies)
Vieta (1)
0 Dime, 5 N, which implies 6 ways.
Weird question (1)
Vinjai's notes/solutions (1)
: , 4 N (5 pennies)
with or without replacement (1)
: , 3 N (10 pennies)
wolf sheep cabbage (1)
: , 2 N (15 pennies)
word problems (9)
: , 1 N (20 pennies)
working together (1)
: , 0 N (25 pennies)
So altogether 13 ways.
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▼ 2021 (2)
Practice questions: (answers below)
▼ 12/19 - 12/26 (2)
pennies?
2021 Mathcounts Competition
Preparation Strategies
Q 2 : How many different combination of coins could a person have if she has exactly 21 cents?
Face Diagonal and Space Diagonal of
a Rectangular ...
Q 1 : How many ways can you make a. 15cents, b. 20cents, c. 30 cents if you can use quarters, dimes, nickels or
► 2019 (5)
Q 3 :Using nickels, dimes, quarters and/or half-dollars, how many ways can you make 75 cents?
► 2018 (11)
Q 4: 20 coins of quarters and nickels add up to 4 dollars. How many nickels are there?
► 2017 (5)
Q5: What is the least number of US coins to make changes possible from 1 to 99 cents inclusive? (half dollar is
► 2016 (16)
► 2015 (12)
allowed)
► 2014 (24)
► 2013 (28)
► 2012 (26)
► 2008 (1)
Answers:
#1: a. 6 ways ; b. 9 ways; c 18 ways ;
#2: 9 ways
#3: 22 ways
#4: 5 nickels and 15 quarters
#5: 9 coins (1 half dollar, 1 quarter, 2 dimes, 1 nickel and 4 pennies)
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NO COMMEN TS :
LABELS : BEGI NN IN G LEVE L , COU NTI NG COIN S, COUN TING P ROBLEMS, E TC. .., H O W M A N Y WAY S TO MAKE 2 5 CENTS
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