AMS 303 SECOND TEST REVIEW Solutions to Past Tests


Spring 2013 AMS 303 Second Test
1. Test A: x sub 9, Test B: x sub 10.
2. Test A: a (-,oo), b(a+,1), c(j-,1), d(h-,1), e(i-,1), f(k-,1), g(e+ or f+,1) h(b+,1), i(c+,1), j(b+,1), k(d+,1), z(g+,1). One new solution b-j, c-i, d-h, e-g, f-k.
Test B: a (-,oo), b(a+,1), c(j-,1), d(h-,1), e(i-,1), f(k-,1), g(c+ or d+,1), h (e+,1), i(b+,1), j(f+,1), k(b+,1), z(g+,1). One new solution b-i, c-j, d-g, e-h, f-k
3. Test A: a) digital sum of piles is 011=3, b) remove 3 from pile 2 or 3, b) take 1 from pile 2, c) g(0)=0, g(1)=1, g(2)=2, g(3)=0, g(4)=1, g(5)=2, g(6)=0, g(7)=1, sum is 01, remove 1 from pile 3.
Test B: a) digital sum of piles is 010=2, b) remove 2 from pile 3 or 4, b) impossible, gr. # already 2, c) g(0)=0, g(1)=1, g(2)=2, g(3)=0, g(4)=1, g(5)=2, g(6)=0, g(7)=1, sum is 01, remove 2 from pile 1.
4.Test A: (1/4)[(b+w)^9 +(b+w)(b^2 + w^2)^4 +2(b+w)^3(b^2 + w^2)^3].
Test B: (1/4)[(b+w)^8 +(b^2 + w^2)^4 +2(b+w)^2(b^2 + w^2)^3].
5.Test A: Not possible,L & T play two games but each can win only 1 game.
Test B: possible-- e.g.,L beats T twice, L beats V twice, T beats L once, and V beats T once.
6. Test A: Initial solution: x11=40, x12=30, x22=10, x32=20, x33=40. Prices u1=$10, u2=$9, u3=$11, v1=v3=$15, v2=14. Select x13. Increase x13 and x32 by 30, reduce x12 and x33 by 30.
Test B: Initial solution: x11=10, x21=30, x22=30, x32=10, x33=60. Prices u1=$ 10, u2=$12, u3=$14, v1=v2=$17, v3=18. Select x12 or x13. For x12, increase x12 and x21 by 10, reduce x11 and x22 by 10.

Fall 2012 AMS 303 Second Test (Test A)
1. (4 pt) x sub 8.
2. (15 pt) a (-,00), b(g-,1), c(h-,1), d (i-,1), e (a+,1), f (k-,1), g(e+,1), h(e+,1), i(b+,1), j(f+,1) or (d+,1), z(j+,1), one new matching b-i,c-h,d-j,e-g,f-k.
3. (15 pt- 2,3,3,7) a) 5, b) remove 3 from pile 2, or 5 from pile 3 or 4, c) remove 4 from piles 2,3, or 4, d) g(0)=0, g(1)=1, g(2)=0, g(3)=1, g(4)=2,g(5)=3, g(6)=2, g(7)=3, digit sum is 011-- remove 1 from pile 2, or 5 from piles 3 or 4.
4. (15 pt) (1/12)[(x1)^18 + 2(x6)^3 + 2(x3)^6 + (x2)^9 + 3(x1)^2*(x2)^8 + 3(x1)^4*(x2)^7].
5. (8 pt) Not possible-- 6 games to play but only supply of 5 wins.

Spring 2011 AMS 303 Second Test
1. (x sub 3)^4.
2. a (-,oo), b(g-,1), c(a+,1), d(i-,1), e(h-,1), f(j-,1), g(c+,1), h(b+,1), i(f+,1), j(c+,1), k(e+,1 or d+,1), z(k+,1). One new solution b-h, c-g, d-i, e-k, f-j.
3. a) digital sum of initial pile is 111, remove 1 from pile 2, b) remove 4 from pile 3, c) g(0)=0, g(1)=1, g(2)=2, g(3)=0, g(4)=1, g(5)=2, g(6)=3, g(7)=0, sum is 010, remove 2 from pile 3.
4. (1/6)[(b+w)^9 + 2(b^3 + w^3)^3 + 3(b+w)(b^2 + w^2)^4].
5. Not possible, only 4 wins available for 5 games to be played.
6. Initial solution: x11=20, x12=30, x22=20, x32=20, x33=40. Prices u1=$10, u2=$8, u3=$9, v1=v2=v3=$14. Select x13. Increase x13 and x32 by 30, reduce x12 and x33 by 30.