1. 3D plane problem using mass point geometry.
1 A handy fact to know for MATHCOUNTS is that the volume of a regular tetrahedron with edge length ๐ ๐ is given by V = โ2/12๐ ^3. In general, the volume of a tetrahedron with a triangular face of area๐ด and perpendicular height with respect to that baseโ is given by V = 1/3โ๐ด. Recall for a regular tetrahedron that each face is an equilateral triangle satisfying with area ๐ด=โ3/4๐ ^2
1 odd 3 or 5 multiples - must separate odd 3 multiples vs odd 5 multiples first!
3 9 15 21... gaps of 6
5 15 25 35 45.. gaps of 10
then overlaping 15 multiples 15, 45, 75...
2. dot product vs cross product, differentiation
- need to understand your own strength and weaknesses; fix past mistakes
1. 2019s29 state same as national 2019 target 4
https://iu.box.com/s/eugb0b38ty7ffgkx33doyi55fel0uhc4
better solution of 2019s29-
16C4-(5x11C4-10x6C4); the 5x11C4 includes twice of the two >=5H cases, so need to take away one;
https://artofproblemsolving.com/community/c3h1950938p13466408
https://artofproblemsolving.com/community/c3h1976740p13719538
https://artofproblemsolving.com/community/c3h1950938p13466408
We have tails, so 5 regions:
Now, we need to place heads in these regions, as such:
Now, we suppose there are in the first region, in the second, and so on. Then, we have , so this can be done in ways.
Now, we have to consider when we have one area with more than dots. There are ways to choose the area. WLOG, assume it is . Then, we have that , which can be done in ways. Thus, we need to subtract ways.
However, we overcomunted when we had numbers . Then, there are ways to choose them. WLOG, assume it is and . Then, , which can be done in ways. Thus, we need to add ways.
Thus our answer should be . Calculations give , , and . Thus our answer is ways.
no use for 2019 s29
1.
50 posts
#1 Jan 7, 2020, 5:04 pm
How many distinct circles of radius 2 units are in the coordinate place and pass through exactly two of the labeled points on this graph?
The points are (-1,1) (1,1) (3,1) (-1,-1) (1,-1) (3,-1)
my question -
https://artofproblemsolving.com/community/c3h1993131_past_mathcounts_problems_19901999
- How many circles in the 2-D coordinate plane go through at least 3 lattice points (x, y) such that |x|, |y| โค 1?
- (A) 39 (B) 63 (C) 72 (D) 78 (E) 84
past notes
#13 Jan 14, 2020, 8:35 pm
jlee28 wrote:
did anyone here get to nats countdown
too pr0
which state are you from? some states are much easier than others... speaking of free resources, you should do Alcumus (its organized by topic which is nice) and For The Win! (it helps to improve speed)
Use mathcounts trainer, do past mathcounts problems, and sleep a lot the day before. If you get a type of problem wrong often, like C&P, practice those problems and do the AoPS book.
I would recommend these applications:
a) Mathcounts Trainer
b) For the Win!
c) Alcumus
but before you try these out, please do some practice papers... it will tell you what you are good at and what you need to improve on....
Practice. Review missed problems, know ur weakness, practice in ur weaknesses, make them stronger, avoid making sillies by doing a lot of problems, and use your resources!
I was also first time MATHCOUNTS last year and I donโt know how many hours I spent on MATHCOUNTS trainer. It has a ton of old problems from other MATHCOUNTS competitions and lots of problems are similar to ones I did on the trainer. You can find it here: https://artofproblemsolving.com/mathcounts_trainer/play
When you get the question, don't pick up your pencil and start writing. Instead, read the whole question, and think about the steps you would take to solve it. Just get a rough plan in your head. In some cases, you may decide that the numbers given are good enough to do the question mentally. If so, then do it mentally. If you decide that you have a plan and the numbers are too big/messy for mental math, then use your pencil.
I used to just start writing without any idea of what I was doing. I was writing but had no aim. This got me faster. Think about it as if you're going somewhere. By thinking about it, you're giving yourself a map. If you had no map and you started writing, then it would be like you're driving without a map. If the distance was long, then which would be easier. Using the map, right?
Then, just start thinking through the question before touching your pencil. That should help.
I found a very simple solution using mass points. It is solvable in seconds now.
first geometry not able to solve, mock mathcounts 2017 target 8
https://www.mathopenref.com/trianglecircumcircle.html circle drawing
https://math.stackexchange.com/questions/3523202/triangle-circumcirle-geometry-question
https://math.stackexchange.com/questions/3523202/triangle-circumcircle-geometry-question
need tan, triangle cosine law, triangle area in sine, complementary angle sin=cos, sin+cos=1
1. Right Triangles: Median to the Hypotenuse is Equal to Half the Hypotenuse
external angle equals sum of two internal angles opposite; use always; don't do 2 steps if can be done in one step
1. circle angle jumping technique; two radius form isosceles triangle!
3. %07t7 - use 1024^odd power ends in 24! don't use 2 for calculation
\question When the expression $(2^1)(2^2)(2^3)...(2^{99})(2^{100})$ is written as an integer, what is the product of the tens digit and the ones digit?
https://web2.0calc.com/questions/mathcounts-state
there's not point to work on harder problems if speed of easier problems can't be improved - can't even reach there; must improve easy problem speed first!
%07t4 - print on paper wrong
\question In how many ways can 36 be written as the product $a \times b \times c \times d$, where a, b, c and d are positive integers such that $a \le b \le c \le d$?
use big number as category/sub-casing base: 36, 18, 12 down to 4 3 2
better than using number of 1's: 1x1x1x36; 1x1x2x18...
work on most recent year first 2019 backwards to 1990
practice makes perfect; don't over do it; don't overthink- past mistakes from real mathcount state problems
- 2019 handbook has 250 problems, did 27 so far 1/22/20
- team problems
- national problems no solutions only answer key
- AOPS mock mathcounts
chicken mcnugget theorem-
https://artofproblemsolving.com/wiki/index.php/Chicken_McNugget_Theorem
potatoes counting with 4-letter words:
https://math.stackexchange.com/questions/20238/6-letter-permutations-in-mississippi?rq=1
avoid over-counting, avoid under-counting; her sense of space/geometry is not quite well. reflect/rotation/translation, folding, 3D coordinate plane, not done well; drawing class?, cubing? make sketches; space imagination lacking
flawed notation system, use x for multiplication, curved x for x, . for decimal!
counting good - methodically counting
tormoehlen #22 honey creek middle school 24+5 better than most past results
2017 state target and sprint results
https://iu.box.com/s/pcwsgbk4qx5gjub6abwp2qql6c5uldh0
https://iu.box.com/s/cj4a39crt5t99p28zejj1opqc7rb486k
purdue math contest 2 and 1
https://iu.box.com/s/6r4ijbipq2gk6v3ak1tt527t7ni259fr
https://iu.box.com/s/gunh1l1n5adnom0io4hqhss7hqql27ao
leave factors multiplication at the end - try to cancel them before doing multiplication; saves time and avoid mistakes!
keep it simple, as simple as possible; don't over complicate things! she tend to think too complicately about a simple problem.
off by 1 - missing 0 - off by 2 (divided by 2; multiply by 2; D=2r; triangle/trapezoid/rhombus area all have 1/2 in front)
geometry especially 3D, percents and ratios, radius/diameter, circumference/perimeter weak - weakness; triangles and parallel lines ok;
1988-89 state 40/40 sprint round: wrong 9 11 23 28 29 30 34 35-40 not done;
1990 state target 5 7 wrong, 5 7 are both number theory; 8 not done; 5 right; 8 is very easy triangle altitude problem; late night work; 90 sprint 15, 17 18 30 wrong; other 26 right, easy year; 17 list out 4 values, -2 -1 0 1; not total number of 4; 30 semicircle perimeter include diameter itself!, she calculated only the circle part, not the diameter
94-95 wrong 2 is 3/4*1/4 shaded triangles with geometric sequence 3/4 ratio, sum of GP; 8 is right, 130.8 don't write as 136.8! 7 two dice need to list out with table for values.
95-96 didn't time 1 wrong; 2-8 right; 1 is how many literal '5' appears in 1-500? 20x5+1 in 500=101; 20 '5's in 1-100 not 19!
96-97 target 1-8 right; 4 is over time, divisible by 72 means divisible by 8 and 9! sprint wrong 1 3 16 17 21 22 27 28 29 30; total 20 right
97-98 target 1-8 all right; s4 11 18 27 wrong; total 26 right 1-30 done; 27 end of six day completed 1/3, so next day will double that to get 2/3, adding up will finish the day; so answer is 7 days; no need to do lengthy calculation!
98-99 target 1-6 right; wrong 7 is a 3D cube stacking thing; 8 right
1998-99 state sprint wrong 10 11 13 17 20 30 not done; 24 right? 10 LCM wrong; 11 counting wrong; 13 percent;17 semicircle missed 1/2; 20 pigeon hole principle; money-barrel principle wrong;
99 second problem wrong - How many pairs of vertical angles are formed 2.by five distinct linesthat have a common point of intersection? answer is 20 not 10!!!
1999 answer key: 89/91, 20 not 10, 6, 1.25*10^27; 34
16 80 67 666,666, 2030 not 2060
26, 4 25% 7/16, 39/2 or 19.50 (15)
8/3, 8pi 646 -1, 21
31 16 9 8 77
226 12 3/14, 27 2.5pi (30)
LCM calculator https://www.calculatorsoup.com/calculators/math/lcm.php
2000 state sprint 14 25 27 28 29-30 missing 50 minutes; target 5 and 7 both crossed out; 2000 state sprint she did #13 counting triangles right - very good! she checked answer on 29-30; 2000 24 right for sprint; 6 right on target
2000 national target wrong 4 5 8; 4 is 9 dots counting isoceles; 5 digits sieving; 9 marble blue and green easy
2001 state target 1 wrong! 3 wrong decimal point miss 1; 5 not done; 2001 sprint 5, 8 13 15 18 23 24 wrong; 26-30 not done in 30 min; 2001 target 2nd time: 1 3 right; 5 tried to use chord multiplication - wrong; 5 may not be worth it? don't over think!
2001 solve for wrong variable; didn't read the question
2001 state sprint 8 4x4x4 cubes has how many cubes? correct way is 1^3 (size 4)+2^3 (size 3)+3^3 (size 2)+4^3 (size 1) =100! Total number of boxes of any size is 4C2x4C2x4C2=6^3=216 not 1000; out of 216 boxes, 100/216 of them are cubes!
If it is a 3x4x5 box, total number of cubes would be 3x4x5(size 1)+ 2x3x4(size 2) + 1x2x3 (size 3) = 90; total number of boxes would be 3C2x4C2*5C2=3*6*10=180; half of them are cubes!
s5 dumb copy number error; s8 is WRONG METHOD - 5C2 x 5C2, cube choosing two planes determines the third plane, no need to multiply a third 5C2; but is this correct?s13 ratio of rectangle easy; s15 counting pentagons and hexagons; divide by 2; sprint18 should be distance, not rate as she thought! 23 need to calculate hexagon and equilateral area;
24 https://www.whitman.edu/mathematics/cgt_online/book/section03.03.html
partition of integers using generating function; no forumula, just count: 1 numbers, 2 numbers, ... 7 numbers; 26 telescoping 27 just try each number; 28 595628 29 simple linear equation with one unknown; 30 heron forumula
2002 target 7 and 8 did again right 2/1/20
2002 state target 1? did again and fixed; sprint wrong 9 19 23 27 29 30; 8 right need pm after 2:05; 19 off by 1 day; 28 right; 24 right all; 9 circle partial angle area easy; 23 tricky 3 multiples which are even; 5 multiples which are even, 5 multiples which are 3 multiples; 27 extremely easy after drawing pictures; 29 count and try; 30 try counting, tricky
2003 state target 4 crossed out; sprint wrong 10 11 14 15 16 17 24 25 29; 27 crossed out but right; 30 ok; 21 right in 40 mins late night work; 10 is segment reflection over x-axis, sign wrong forget negative sign; 11 greater than 5 not include 5! 14 24/96 make numerator as small as possible, denominator as large as possible; 15 division slow; add 6*4, not babycounting; 16 mislabeled A and C, mixed up; 17 next palindrome needs to be 404xx area, not 41xxx area!! 24 case work: 136, 145, 262 253 244 334; get permutation numbers for each of them; 25 1-10 7; 11-20 6; 21-30 5; 31-40 1; (19 total) 41-50 6; (25) 51-60 5; 61-70 7 (37) 71 74 76 answer; 29 50*60/4 minutes later = 12 hours 30 minutes; 4:50
03s15 - need to use multiplication not addition, no baby counting;
03s25 - baby counting, too time consuming
2003 national sprint wrong 2 4 13 up to 23; 20 right ; 21 is 1/4 not 1/9! finished 24 -30; wrong 27 4 vs 8 digits; 29 horseshoe shape with 3 lines cut region - crooked get 10; 30 - don't count >6 digits! https://math.stackexchange.com/questions/758466/probability-of-number-formed-from-dice-rolls-being-multiple-of-8
2004 state target 8 crossed out; sprint wrong 4, 23 24 29; right 26 in 40 min; easy year; 3/24 not simplified to 1/8 dumb mistakes; 23 (12, 35) is above, missed one point; 24 29 quite hard; 24 needs to know odd number cubed, 5^3=125, 7^3=343, 9^3=729; 8^3=512 6^3=216; 125+343+729=1197 how they add up, negative and positive cubes cancelled out; 29 combination with circle center points, methodically need to go through each combination with restrictions
2005 target 5 and 6 done right 2/1/20
2005 state target 8 not done; 4 crossed out, 5 crossed out; 6 not done; 7 right; state sprint wrong 7 18 21 28-30 not done in 40 min;24 right; more notes below;
2005chapter sprint wrong 18 20 21 22 23; 24 25 right; 26-30 not done; 20 right total for chapter 2015 sprint
2006 state target 7 8 wrong; others right; sprint 1-12 ok; wrong 16 18 22 28; 29-30 not done; 24 right in 40 mins; 2/1/20 target 7 8 correct
2007 state target 4 5 marked out; 7 not done; 8 right; sprint 1 8 13 17 18 19 23 26 wrong; 25 right; 27-30 not done. total 18 right 40 min; 2/3/20 s26 wrong again
2008 state target 2 4 6 8 wrong; 1357 right; 3 marked out should know; no decimal, only fraction! 2008 sprint 3 11 12 15 22 23 24 wrong; 26-30 not done; 25-7=18 right in 40 min; 3 needs equation; 11 need to include 0! 12 dumb copy numbers and only do calculate in the end, no middle calculation except to simplify; 15 1-15 in a circle, start with a number, eliminate the 3rd; when only 11 remains, which number was first erased? Note the starting integer and first erased integer is different! try one specific number, like 1 and count until the end, 5 is the last number, so if you start at x, the last left number is x+4. That's true for 11 left, so 11-4=7 is the starting integer, and the first erased integer is 9 answer. very genuine idea from sandra, much better solution than provided by mathcounts. 22 is essentially figuring out 90 has how many integer factors? prime factorization=3^2x2x5, so it has 3x2x2=12 which is the answer; 23 cut 1/4 of the cylinder and unroll it, get 4-3 right triangle, 5 is the hypotenuse, times 4 gives 20 as answer; 24 is asking for perimeter of the small square, not the big square! don't mix up; 25 sandy is right, list and eliminate; 26 need to set up w x y z 4 variables and sum up to w+z+3x+3y, only 4C2 combinations, list out and calculate to 24 22 20 18 16 so answer is 5; 27 mr+2=(m+1)(r-2), r=2(m+2), r must be even, try 98, mr=96=16*6, r=16 m=6 satisfied the condition, so first try give answer 98; 28 cylinder and sphere volume, make the circle center and sphere center connection to form a right triangle, connect to intersecting points not other points; 29 must recognize perfect square form x^4-10x^3+25x^2 30. list out the 10 songs: 1/2 1 1.5 2 2.5 3 3.5 4 4.5 5; complementary counting, consider within 4.5 minutes she hears all of her favorite 3.5 minute song, it could be first 1/10, or second with two shortest songs first, 2/10*1/9 total is 11/90, and 1-11/90=79/90 answer. this is sandra's idea very good
2009 target 2 shouldn't wrong! 2009 sprint 2 5 6wrong! 23 24 not done; 25 26 done right; 27-30 not done; total 21 right in 40 min late night work; redid all
2010 state target 2? 6? 8?
2010 sprint 5 wrong missing 1/2; 23 not done; 25 27 right; 26 28-30 not done
2010 chapter sprint 15 palindrome; 20 subset; 29 wrong
2011 state target 5? 22 not 120 7? 0.36 not .36!
2011 sprint 1-11 ok; 12 wrong20 23 wrong; 24 25 26 right; 27-30 not done in 40min
2012 state target all correct appears;
2012 state sprint 23? 25 wrong; 20-22, 24 right
2013 state target 7 wrong; (3, 6 marked out);
2013 sprint 22 crossed out, 28-30 not done; 19-21, 23 24 25-27 right
2014 state target 5, 6 marked out others correct; 2/1/20 did t5-6 again right ok
2014 sprint 25-30 not done; 23 crossed out but correct so 21-24 right
2015 state target 7 wrong, logic 3 out of 4 conditions true, 6 marked out
2015 sprint 1-17 right; total 21 right in 40min; 18 wrong 22 23 25 wrong ; 26 right; 27-30 not done; 23 square both sides, not just one! finished 27-30 with looking at solutions
2015 chapter sprint wrong 18 22 23; 20-21 not done; 24 25 right; 26-30 not done;
2016 19-30 only; sprint 20 21 did wrong again; dumb mistakes 2^2+2=8; 3/12+4/12=7/12 not 5/12; 29-30 not done; others 19-30 right. 29 done right; 30 still have difficulties; need to test 20, 21 30 again
2016 state target 7, 8 right; was it genuine; 6 has marks; 3 has marks;
2016 state sprint 1-6 right; 16 crossed out but right; 40 minutes to 21; 19 right; 20 not done; 21 did wrong;
worked through 25-30, 30 is hexagonal bipyramid, 3D geometry is hard for sandy to imagine
https://youtu.be/dzBkIPEi_zE?t=77 has hexagonal bipyramid https://youtu.be/bq2cae4VMH8
2016 national target 2 should know
2017 target 7 done right, there is a shortcut to get the area: rotate DAE and it lines up with ABC, same base, same height, so same area! same for ICH has same area as ABC; target 8 need to do again - it should be easy and done in one minute
2017 target 7-8 not done, (4 was wrong before but correct later); 7 was correct once before, but wrong now; target 6 right! quite good; 2017 state sprint wrong ones are 3, 5, 14, 16, 17, 19 22 23 24 25 26-30 2017 state sprint 15 right;redo 3 5 14 16 17 ok; 19-30 not done 19 is pole with snake 60-degree angle; just do one line/turn to get to top, snake lenght = 9pi, h=9pi*sqrt3/2; no need to do more than 1 turn to reach the top, problem solved very easily; 27 is standard star and bars problem 15C3; 29 set up xy coordinate systems and solve linear equations to get x part of intersecting points; easily solved; 2017 explained all the problems to sandy, she'll need to do it again. aops guy didn't explain 19, 27 and 29. cut the crap out
2018 state target 8 not done; other correct
2018 state sprint 1-10 correct, 11-18, 20 correct; 19 wrong; path counting both forward and backward works, just avoid the black areas completely; complementary counting won't work!
2018 sprint 40 minutes did 18-19 problems right; did 20-30 but many hours. there are two 3D problems ax+by+cz=d, cube xyz coordinates
2019 state target 8 marked, 3 crossed out, others correct;
2019 state sprint 1-10 right; 11 wrong; 12-18 right,19 not done; 20 right; 21-24 right; 25 wrong; 26-30 not done; got 22 total right for 2019 state sprint
2019 team 8 wrong;hard`10 wrong hard; 8 lucky correct
purdue sprint got 22 right; target 3 right?
2018 AMC10 A 7 wrong - 0 is int!
2019 AMC10 A 4 6 10; B 3
1617 handbook 237 238 angles and arcs
1415 handbook mass point geometry stretch 299 300 not done?
do 2019 state sprint 11 19 21-30
