MathDB

3

Part of 2013 Stanford Mathematics Tournament

Problems(9)

2013 General Problem 3

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2/4/2013
Queen Jack likes a 5-card hand if and only if the hand contains only queens and jacks. Considering all possible 5-card hands that can come from a standard 52-card deck, how many hands does Queen Jack like?
SMT 2013 Calculus #3

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2/3/2013
Suppose aa and bb are real numbers such that lim⁡x→0sin⁡2xeax−bx−1=12.\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}. Determine all possible ordered pairs (a,b)(a, b).
calculuslimittrigonometryanalytic geometry
SMT 2013 Advanced Topics Tiebreaker #3

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11/2/2014
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability 12\frac{1}{2}. What is the expected value of the number of games they will play?
probabilityexpected value
SMT 2013 Algebra Tiebreaker #3

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11/2/2014
Find the minimum of f(x,y,z)=x3+12yzx+16(1yz)32f(x,y,z)=x^3+12\frac{yz}{x}+16(\frac{1}{yz})^{\frac{3}{2}} where x,yx,y, and zz are all positive. Note: The problem as given in the tiebreaker did not specify that each of x,yx,y, and zz had to be positive. Without this constraint, the answer is −∞-\infty, as x3x^3 can be an arbitrarily large negative value and dominate the expression.
SMT 2013 Geometry Tiebreaker #3

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11/2/2014
In quadrilateral ABCDABCD, diagonals ACAC and BDBD intersect at EE. If AB=BE=5AB=BE=5, EC=CD=7EC=CD=7, and BC=11BC=11, compute AEAE.
geometry
2013 SMT Geometry #3

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11/2/2014
Robin has obtained a circular pizza with radius 22. However, being rebellious, instead of slicing the pizza radially, he decides to slice the pizza into 44 strips of equal width both vertically and horizontally. What is the area of the smallest piece of pizza?
geometryAMC 10AMC 10 B
SMT 2013 Algebra #3

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2/3/2013
Karl likes the number 1717 his favorite polynomials are monic quadratics with integer coefficients such that 1717 is a root of the quadratic and the roots differ by no more than 1717. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose x2x^2 term has a coefficient of 11.)
algebrapolynomialquadratics
SMT 2013 Team #3

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2/4/2013
Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?
SMT 2013 Advanced Topics #3

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12/31/2016
Nick has a terrible sleep schedule. He randomly picks a time between 4 AM and 6 AM to fall asleep, and wakes up at a random time between 11 AM and 1 PM of the same day. What is the probability that Nick gets between 6 and 7 hours of sleep?
2013Advanced Topics