MathDB

5

Part of 2016 Harvard-MIT Mathematics Tournament

Problems(7)

2016 Algebra #5

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12/24/2016
An infinite sequence of real numbers a1,a2,…a_1, a_2, \dots satisfies the recurrence an+3=an+2−2an+1+an a_{n+3} = a_{n+2} - 2a_{n+1} + a_n for every positive integer nn. Given that a1=a3=1a_1 = a_3 = 1 and a98=a99a_{98} = a_{99}, compute a1+a2+⋯+a100a_1 + a_2 + \dots + a_{100}.
2016 Geo #5

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12/30/2016
Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of (92)=36\binom 92 = 36 lines. Assume that all 3636 lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?
2016 Combo #5

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12/30/2016
Let aa, bb, cc, dd, ee, ff be integers selected from the set {1,2,…,100}\{1,2,\dots,100\}, uniformly and at random with replacement. Set M=a+2b+4c+8d+16e+32f. M = a + 2b + 4c + 8d + 16e + 32f. What is the expected value of the remainder when MM is divided by 6464?
2016 Guts #5

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12/24/2016
Patrick and Anderson are having a snowball fight. Patrick throws a snowball at Anderson which is shaped like a sphere with a radius of 1010 centimeters. Anderson catches the snowball and uses the snow from the snowball to construct snowballs with radii of 44 centimeters. Given that the total volume of the snowballs that Anderson constructs cannot exceed the volume of the snowball that Patrick threw, how many snowballs can Anderson construct?
2016 Team #5

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12/30/2016
Find all prime numbers pp such that y2=x3+4xy^2 = x^3+4x has exactly pp solutions in integers modulo pp.
In other words, determine all prime numbers pp with the following property: there exist exactly pp ordered pairs of integers (x,y)(x,y) such that x,y∈{0,1,…,p−1}x,y \in \{0,1,\dots,p-1\} and p divides y2−x3−4x. p \text{ divides } y^2 - x^3 - 4x.
2016 General #5: Sequences

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11/15/2016
Let the sequence {ai}i=0∞\{a_i\}^\infty_{i=0} be defined by a0=12a_0 =\frac12 and an=1+(an−1−1)2a_n = 1 + (a_{n-1} - 1)^2. Find the product ∏i=0∞ai=a0a1a2…\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots
HMMT
2016 Theme #5: More basketball points

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11/22/2016
Steph Curry is playing the following game and he wins if he has exactly 55 points at some time. Flip a fair coin. If heads, shoot a 33-point shot which is worth 33 points. If tails, shoot a free throw which is worth 11 point. He makes 12\frac12 of his 33-point shots and all of his free throws. Find the probability he will win the game. (Note he keeps flipping the coin until he has exactly 55 or goes over 55 points)
HMMT