m - national mathcounts

1994 target wrong 2 ( teacher sharing cost $78, two no money, others each add $1.30), 4, 5,6,8; 8 is AM-GM inequality; 6 exterior angle; 5 list out first and last 5 terms, others cancel out; 4 k, n, m int, 2 3 7;

1995 target wrong 5 7; inch took as feet; rectangle counting missing 1; use combination for counting, not baby counting!

1996 target 3 4 5; 7/10< 5/7<8/11, 5/7 goes in between!! 5/8 right; sprint wrong 3 6 8 10 13 15 18 20 22;

right 21 24 23 1/(1+sqrt2) = sqrt2-1; 25-30 not done; fixed all of it

1997 target 3 5 7 8 wrong; inch not feet; hollow cylinder has inside as area also! 0 and 1 are both power of 4 numbers, don't miss 0; sprint wrong 4, 8, 9, 12, 13, 14, 16, 18, 21; 25-30 not done; 27 got x-y coordinates but asking sum of them 9+7=16!

1998 sprint 2 11 13 21 wrong; 26/30 right; target 8/8 right

1999 sprint wrong 1 9 10 16 21 22 24-30; 23 right; target wrong 2 (8.5 month goes to September! not august) 8 semicircle counted only half of chord, should be 2x;

2000 national target wrong 4 5 8; 4 is 9 dots counting isoceles; 5 digits sieving; 9 marble blue and green easy; sprint worked backwards 1 ok; 2-11 not done; wrong 16, 17 19 23 24 30; total 14 right. hosting timed team work

2001 target 2 3 wrong; total 6 right; sprint 1-10 right; wrong 11 12 14 24 25 27 29 30; right 26 28; 23 answer -14 looks like -19!!

2002 national sprint 14 16 23 24 25 wrong; 28-29 not done; 30 right; target 8 wrong; 1-7 right

2002 target 8 plot x^(a/x) for any constant integer a, solution is e=2.718, so maximize number of 3's in!

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

2003 national target 3 4 wrong; 7 8 right

2004 national sprint 12 17 18 wrong; 21 24 25 right; total 20 right; 22 region is a triangle plus a circle segment - 2+135/360*pi*16=2+6*pi; 23 8!=40320=32*35*36, just factor it out; 26. 5^3*3^3*5^4*2^2*1, a=50, b=15,c=1 answer is 750-1; target 2 4 8 wrong; TA8 doesn't have a simple way, need to loop through all the 2x3^n numbers 2 6 18 54 162 486 1458, subtract from 2004, and mod 7; TA 4 convert from real numbers to shapes on x-y coordinate system, probability to geometry, connection!

2004 sprint 12 17 18 ; 22 23 not done; 24 25 right; 26-30 not done

2005 sprint total 18 right; wrong 8, 12 17 18 23 25 26 30 gecko; target wrong 3 4 6 7; right 8; hosting team timed work

2006 sprint wrong 4 13 14 15 16 17 22-30 not done; target 6 8 wrong;

2007 sprint wrong 5 7 10 16 19 21; 23-30 not done; 22 right; total 16 right; target 1 6 8 wrong

2008 target national 3 5 6 8 wrong; 4 crossed out; sprint wrong 17 18 21 22 23 25-30 not done; 24 right

2009 target 3 5 7 wrong; sprint wrong 6 8 14 15 18 21 25-26 not done but looked up solutions

2010 target 6 8 wrong 1-7 right; sprint 3 crossed out, 7 9 12 13 16 18 23 wrong; 19 20 right

2011 sprint 15 16 wrong, 21 crossed out; did up to 18 in school, figured out remaining later to #30; target 6 8 not done; 5 7 1-4 all right! 5 is semicircle one

2012 national sprint wrong 1 6 11 20-24 not done 25 right 26-30 not done; target 7 8 not done; 1-6 right?

2013 national target - 3 4 5 6 8 wrong; sprint 15 16 22 wrong; 24 right; 23 25-30 not done; need review

2014 targe 4 6 7 8 wrong; sprint 2 10 13 wrong; only did 1-13

2015 target 3 4 6 8 wrong; 2015 sprint explained 25-30; 2nd time doing it 26 wrong even though it was right before; 29-30 not done? sprint 1-14 16 18 20-23 right; 15 17 19 24wrong; 25-30 not done; team did 7 binary tree combination 14C7x6C3^2x2C1^4; 10 did trichotomy with 3 weather types

2016 national sprint did only one 27 right; 2016 national target 2 should know

2017 national sprint, target and team round all done - not sure how many correct

2018 national target 2 3 7 8 wrong; total 4 right; sprint hard only did 1-15 in time; 26-30 not figured out

2018 sprint wrong 9 10 12-15; right 11 1-8; team has solutions from yongcheng, didn't do

2019 national sprint 1-12 right; 13 14 15 16 18 wrong; 17 19 20 right; 21-30 not done; target wrong 2 4 6 8; total 4 correct; team did 5 magic square

self solutions below -

2010 national team round -

1. 4 multiples 15, 5 multiples 12, with 3 overlapping: 24

4*3's 12 24 36 48 60

5*6's 30

24 take out 6 is 18;

2. linear equations - sandra knows

3. mean median mode and range of 8 intergers are each 10, what is larget integer?

https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_15

7 7 9 10 10 10 10 17

4. linear equations - sandra knows

5. recursively solve down on 5 to 0

6. cylinder volume - sandra knows

7. total is 3^6; 6 x 9 - 1; 4x9, 1x10,8 - 30; 2x9, 2x8, 10 - 90; 3x8,10 - 20; add up to

141/3^6 = 47/243

8. chicken nugget theorem shift right to 1; 7*11-7-11+1

9. use similar shape ratio 1/9 for small segments - on stack exchange math!

10. simple geometry with middle point coordinates - easy

http://mathteamnyc.weebly.com/mathcounts.html has most past years mathcounts

https://www.sycamoreschool.org/academics/academic-teams

http://www.aquatutoring.org/miscmath.php chicago

https://mathprize.atfoundation.org/resources/past-tests/2019/mathprize2019problems.pdf

math for girls

https://ivyleaguecenter.org/2017/09/30/35-sets-of-previous-official-amc-10-tests-with-answer-keys/

https://ivyleaguecenter.files.wordpress.com/2016/02/2016_amc10b.pdf washington DC

AMC 10 https://www.cut-the-knot.org/proofs/ptolemy.shtml

2016 national sprint

4. 7/2>14/5>1/10

biggest is (7/2+14/5)/(1/10) smallest is 1/10 / (7/2+14/5)!

10. biggest square number <43 is 6^2, so A=6

11. look for 1/a near 1/3, a=3, b=4, c=6

12. pick any point first, then 3 subcases:

case 1) picking 2 next points, then 2 opposite points: 2/5 * 2/4 = 1/5;

case 2) picking 2 longer points, then each has 2 points of choice: 2/5 * 2/4 = 1/5;

case 3) picking 1 opposite point, any third point will for a right triangle: 1/5

sums up to 3/5 answer

19 https://artofproblemsolving.com/wiki/index.php/1985_AHSME_Problems/Problem_14

24 use logrithm, y+y^2=3/2, solve for y positive, -1+sqrt7/2

z^2=(1+2y)^2=7

20 can only be 785 + 785 + 785 = 02355, THREE sums to 15

21 4/5 x 2/3 = 8/15

22 b=2+3a, solve 3 a^2 - 2a -225=0, a=9, b=29 answer 38

27. total number ways of moves are 4^4=256

face 1 to 2 as one example (there are 4 of them): to go back to face 1, there are total 12 ways:

face 3 first: 12341, 12351, 12361, 12321 - 4 of them

face1 next: 12121, 12151, 12161, 12151 - 4 of them

face 5 next: 12521, 12541 - 2 of them

face 6 next: 12621, 12641 - 2 of them

so each of the 4 face 1 goes to, there are 12 ways to go back to face 1, that's 12*4=48 ways

48/256=3/16 answer

markov chain - https://math.stackexchange.com/questions/2542476/random-walk-over-a-cubeprobability-of-returning-back?rq=1

28 programming - n28.cc

programming seal meeting problem? how to program combinatorics? and probability?

https://math.stackexchange.com/questions/3493755/regular-octagon-with-inscribed-square-area-ratio-question

not understanding it first - why? regular octagon has different squares inscribed????

https://web2.0calc.com/questions/regular-octagon-abcdefgh-has-a-side-length-of-2-cm

https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_23

30. easy with pythagorean theorem, drop altitude to BC

x/y=9/5

h^2+(9+5/2)^2 =x^2

h^2+2.5^2=y^2

solve y=15/2, x=27/2, perimeter =27+23=50 answer

2016 national team

6. 11*7*19-17*7*12 = 35

7. let small side to be x, 1+x+sqrt(x^2+1)=6, solve x=2.4, area = 36-2.4*2=31.2 answer

8. 3 30 300 12 and 75! sums to 420 answer

https://thestarman.pcministry.com/math/rec/RepeatDec.htm

9. http://www.aquatutoring.org/miscmath.php has solution as 92 not 90 as in the answer sheet

10. let BC/AB=y; larger square side a, smaller square side b

triangle area is BC*AB/2 = 1/2 y AB^2

a/AB = ay/(ay+1) = y/(y+1), so a = y/(y+1) AB

solve for y by the small triangle area 9/289

(a-b) /b = y, so b = a/(y+1)=y/(y+1)^2 AB,

1/2 b*(a-b) = 9/289 1/2 y AB^2

b/AB=3/17= y/(y+1)^2

so y = 3/17 * (y+1)^2, can solve for y

a^2/ (1/2*y*AB^2) = 2y/(y+1)^2 = 2y/(17/3*y) = 6/17 answer

2017 national team

7. can solve hot dog = 2.15, patties=4.15

.15 .30 .45 .60 .75 .9 .05 .2 .35 .5

.9 = 6, .05=7 x2=14; 14*2.15+6*4.15=55 answer

8. 1.9=19/10>1, so must have 1, then 9/10>1/2, must have 2; 4/10=2/5>1/3, must have 3,

2/5-1/3=1/15, so have 15, 1+2+3+15=21 answer

9. there are two geometric sequences, first one: 210 *3/5 (1+1/15+1/15^2+....)

second: 42*2/3(1+1/15+1/15^2+....), sum both up to get 135+30=165

zeno paradox?? went to 164.2 after 3 rounds, not to the end

https://math.stackexchange.com/questions/3495487/bird-going-between-two-moving-objects-with-different-speed?noredirect=1#comment7188298_3495487

10. 7 cars to 5 children

https://math.stackexchange.com/questions/3493423/7-different-cars-distributed-to-three-daughters-and-two-sons-with-restrictions

https://math.stackexchange.com/questions/2673819/in-how-many-ways-can-six-different-gifts-be-given-to-five-different-children-wit/2673833?noredirect=1#comment5522567_2673833

2017 national target

3. https://www.mathhomeworkanswers.org/64588/permutation

As there are two senators for each state then 100 senators can be considered as 50 pairs covering 50 states. Either senator could be picked to serve on the committee. Instead of thinking in terms of senators think in terms of state representation. There are 50C5 ways of combining the representation into a committee of 5, equal to 2,118,760 (50*49*48*47*46 divided by the number of ways of arranging 5 objects, because we are only interested in combinations, not permutations). We could pick either senator for each of the five states represented, so we need to multiply this number by 2^5=32: 67,800,320

100*98*96*94*92/5!

x k m z. - 4

wxyz - 4

total 16 ways

but x and z can repeat, wxxy wxxz, wxyz, xxyz, 24+12*3=60, 60*2=120

k and m each has 24*4=96, 96*2=192

192+120=312,

wxyz repeats between x and z, 24 counted twice, 312-24=288 Answer

2017 national sprint

15. 1000 times both sides, subtract

999n/814=D75, divisible by 9, D must be 6, then n = 550

22. case work, subcase

11 12

10 12 - 9

9 12

12 11 - 45 total

10 9. -

8 7 - 5

8 6 - 1

8 5 - 1

7 6 - 5

7 5 - 1

6 5 -1

24 03 30 - 1/8 two heads 1/2;

1-2 2-1 3/8 each 2 head 1/4; 1/8+3/8*1/2=5/16;

25. drop an altitude and use pythagorean theorem to solve; or use existing median theorem

26. 25 tiles A1 A2...D5, random select 3 without replacement, prob of at least one pair having same letter or number?

method 2 - two cases

case 1: first tile doesn't matter A1; second tile has common with first C1; and third tile common with either first or second(7 of A1 then 4 of C1): 8/24*(7+4)/23=11/69

case 2: first tile doesn't matter; second tile differ from first; third tile common match with both first and second: 16/24*2/23=4/69

total = 15/69=5/23

method 1 - complementary - consider no same letter or number cases,

16/24 x 9 /23 * 3 = 18/23

1 - 18/23 = 5/23

27. similar trapezoids - ratio of sides

28. 1.75+sqrt3=(1+sqrt3/2)^2

29. x+y+z=36

x=5+y

x=2y or z=2x; or z=2y, or x=2z; y=2z

five solutions for longest length: 21 20.5 15.5 16.4 17.4

30.