MathDB

2

Part of 2013 Stanford Mathematics Tournament

Problems(9)

2013 General Problem 2

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2/4/2013
Jimmy runs a successful pizza shop. In the middle of a busy day, he realizes that he is running low on ingredients. Each pizza must have 1 lb of dough, 14\frac14 lb of cheese, 16\frac16 lb of sauce, and 13\frac13 lb of toppings, which include pepperonis, mushrooms, olives, and sausages. Given that Jimmy currently has 200 lbs of dough, 20 lbs of cheese, 20 lbs of sauce, 15 lbs of pepperonis, 5 lbs of mushrooms, 5 lbs of olives, and 10 lbs of sausages, what is the maximum number of pizzas that JImmy can make?
SMT 2013 Calculus #2

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2/3/2013
Compute all real values of bb such that, for f(x)=x2+bx−17,f(4)=f′(4)f(x) = x^2+bx-17, f(4)=f'(4).
calculus
SMT 2013 Algebra Tiebreaker #2

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11/2/2014
If ff is a monic cubic polynomial with f(0)=−64f(0)=-64, and all roots of ff are non-negative real numbers, what is the largest possible value of f(−1)f(-1)? (A polynomial is monic if it has a leading coefficient of 11.)
algebrapolynomial
SMT 2013 Geometry Tiebreaker #2

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11/2/2014
Points AA, BB, and CC lie on a circle of radius 55 such that AB=6AB=6 and AC=8AC=8. Find the smaller of the two possible values of BCBC.
geometryquadraticstrigonometryarea of a triangletrig identitiesLaw of Sines
SMT 2013 Advanced Topics Tiebreaker #2

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11/2/2014
How many alphabetic sequences (that is, sequences containing only letters from a⋯za\cdots z) of length 20132013 have letters in alphabetic order?
2013 SMT Geometry #2

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11/2/2014
What is the perimeter of a rectangle of area 3232 inscribed in a circle of radius 44?
geometryperimeterrectangle
2013 SMT Algebra #2

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11/2/2014
A tree has 1010 pounds of apples at dawn. Every afternoon, a bird comes and eats xx pounds of apples. Overnight, the amount of food on the tree increases by 10%10\%. What is the maximum value of xx such that the bird can sustain itself indefinitely on the tree without the tree running out of food?
SMT 2013 Team #2

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2/4/2013
In unit square ABCDABCD, diagonals AC‾\overline{AC} and BD‾\overline{BD} intersect at EE. Let MM be the midpoint of CD‾\overline{CD}, with AM‾\overline{AM} intersecting BD‾\overline{BD} at FF and BM‾\overline{BM} intersecting AC‾\overline{AC} at GG. Find the area of quadrilateral MFEGMFEG.
geometryanalytic geometry
SMT 2013 Advanced Topics #2

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12/31/2016
Consider the numbers {24,27,55,64,x}\{24,27,55,64,x\}. Given that the mean of these five numbers is prime and the median is a multiple of 33, compute the sum of all possible positive integral values of xx.
2013Advanced Topics