MathDB

1

Part of 2013 Stanford Mathematics Tournament

Problems(8)

2013 General Problem 1

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2/4/2013
Robin goes birdwatching one day. he sees three types of birds: penguins, pigeons, and robins. 23\frac23 of the birds he sees are robins. 18\frac18 of the birds he sees are penguins. He sees exactly 55 pigeons. How many robins does Robin see?
SMT 2013 Advanced Topics Tiebreaker #1

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11/2/2014
Andrew flips a fair coin 55 times, and counts the number of heads that appear. Beth flips a fair coin 66 times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.
probability
SMT 2013 Algebra Tiebreaker #1

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11/2/2014
xx is a base-1010 number such that when the digits of xx are interpreted as a base-2020 number, the resulting number is twice the value as when they are interpreted as a base-1313 number. Find the sum of all possible values of xx.
SMT 2013 Geometry Tiebreaker #1

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11/2/2014
A circle of radius 22 is inscribed in equilateral triangle ABCABC. The altitude from AA to BCBC intersects the circle at a point DD not on BCBC. BDBD intersects the circle at a point EE distinct from DD. Find the length of BEBE.
geometrycircumcircleincenter
2013 SMT Geometry #1

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11/2/2014
In triangle ABCABC, AC=7AC=7. DD lies on ABAB such that AD=BD=CD=5AD=BD=CD=5. Find BCBC.
geometrycircumcircle
2013 SMT Algebra #1

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11/2/2014
Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of 1010 mph. If he completes the first three laps at a constant speed of only 99 mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?
SMT 2013 Team #1

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2/4/2013
Let f1(n)f_1(n) be the number of divisors that nn has, and define fk(n)=f1(fk−1(n))f_k(n)=f_1(f_{k-1}(n)). Compute the smallest integer kk such that fk(20132013)=2f_k(2013^{2013})=2.
SMT 2013 Advanced Topics #1

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12/31/2016
How many positive three-digit integers a‾ b‾ c‾\underline a\, \underline b\,\underline c can represent a valid date in 20132013, where either aa corresponds to a month and b‾ c‾\underline b\,\underline c corresponds to the day in that month, or a‾ b‾\underline a\, \underline b corresponds to a month and cc corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.
2013Advanced Topics