MathDB

2014 BMT Spring

Part of BMT Problems

Subcontests

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2014 BMT Team 18

Monty wants to play a game with you. He shows you five boxes, one of which contains a prize and four of which contain nothing. He allows you to choose one box but not to open it. He then opens one of the other four boxes that he knows to contain nothing. Then, he makes you switch and choose a different, unopened box. However, Monty sketchily reveals the contents of another (empty) box, selected uniformly at random from the two or three closed boxes (that you do not currently have chosen) that he knows to contain no prize. He then offers you the chance to switch again. Assuming you seek to maximize your return, determine the probability you get a prize.

BMT 2014 Spring - Individual 18

Suppose the polynomial f(x)=x2014f(x) = x^{2014} is equal to f(x)=∑k=02014ak(xk)f(x) =\sum^{2014}_{k=0} a_k {x \choose k} for some real numbers a0,...,a2014a_0,... , a_{2014}. Find the largest integer mm such that 2m2^m divides a2013a_{2013}.
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2014 BMT Team 15

Suppose a box contains 2828 balls: 11 red, 22 blue, 33 yellow, 44 orange, 55 purple, 66 green, and 77 pink. One by one, each ball is removed uniformly at random and without replacement until all 2828 balls have been removed. Determine the probability that the most likely “scenario of exhaustion” occurs; that is, determine the probability that the first color to have all such balls removed from the box is red, that the second is blue, the third is yellow, the fourth is orange, the fifth is purple, the sixth is green, and the seventh is pink.

BMT 2014 Spring - Individual 15

Albert and Kevin are playing a game. Kevin has a 10%10\% chance of winning any given round in the match. If Kevin wins the first game, he wins the match. If not, he requests that the match be extended to a best of 33. If he wins the best of 33, he wins the match. If not, then he requests the match be extended to a best of 55, and so forth. What is the probability that Kevin eventually wins the match? (A best of 2n+12n+ 1 match consists of a series of rounds. The first person to reach n+1n + 1 winning games wins the match)
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2014 BMT Team 14

Let (x,y)(x, y) be an intersection of the equations y=4x2−28x+41y = 4x^2 - 28x + 41 and x2+25y2−7x+100y+3494=0x^2 + 25y^2 - 7x + 100y +\frac{349}{4}= 0. Find the sum of all possible values of xx.

BMT 2014 Spring - Individual 14

Suppose that f(x)=xx2−2x+2f(x) = \frac{x}{x^2 - 2x + 2} and g(x1,x2,...,x7)=f(x1)+f(x2)+...+f(x7)g(x_1, x_2, ... , x_7) = f(x_1) + f(x_2) +... + f(x_7). If x1,x2,...,x7x_1, x_2,..., x_7 are non-negative real numbers with sum 55, determine for how many tuples (x1,x2,...,x7)(x_1, x_2, ... , x_7) does g(x1,x2,...,x7)g(x_1, x_2, ... , x_7) obtain its maximal value.
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P1
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BMT 2014 Spring - Geometry P1

Let ABCABC be a triangle. Let r r denote the inradius of △ABC\vartriangle ABC. Let rar_a denote the AA-exradius of △ABC\vartriangle ABC. Note that the AA-excircle of △ABC\vartriangle ABC is the circle that is tangent to segment BCBC, the extension of ray ABAB beyond B B and the extension of ACAC beyond CC. The AA-exradius is the radius of the AA-excircle. Define rb r_b and rc r_c analogously. Prove that 1r=1ra+1rb+1rc\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}

BMT 2014 Spring - Analysis P1

Suppose that a,b,c,da,b,c,d are non-negative real numbers such that a2+b2+c2+d2=2a^2+b^2+c^2+d^2=2 and ab+bc+cd+da=1ab+bc+cd+da=1. Find the maximum value of a+b+c+da+b+c+d and determine all equality cases.

BMT 2014 Spring - Discrete P1

Let a simple polygon be defined as a polygon in which no consecutive sides are parallel and no two non-consecutive sides share a common point. Given that all vertices of a simple polygon PP are lattice points (in a Cartesian coordinate system, each vertex has integer coordinates), and each side of PP has integer length, prove that the perimeter must be even.
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BMT 2014 Spring - Geometry P2

Let ABCABC be a fixed scalene triangle. Suppose that X,YX, Y are variable points on segments ABAB, ACAC, respectively such that BX=CYBX = CY . Prove that the circumcircle of △AXY\vartriangle AXY passes through a fixed point other than AA.

BMT 2014 Spring - Analysis P2

Define η(f)\eta(f) to be the number of roots that are repeated of the complex-valued polynomial ff (e.g. η((x−1)3⋅(x+1)4)=5\eta((x-1)^3\cdot(x+1)^4)=5). Prove that for nonconstant, relatively prime f,g∈C[x]f,g\in\mathbb C[x], η(f)+η(g)+η(f+g)<deg⁡f+deg⁡g\eta(f)+\eta(g)+\eta(f+g)<\deg f+\deg g

BMT 2014 Spring - Discrete P2

Given an integer n≥2n\ge2, the graph GG is defined by:
- Vertices of GG are represented by binary strings of length nn - Two vertices a,ba,b are connected by an edge if and only if they differ in exactly 22 places
Let SS be a subset of the vertices of GG, and let S′S' be the set of edges between vertices in SS and vertices not in SS. Show that if ∣S∣|S| (the size of SS) ≤2n−2\le2^{n-2}, then ∣S′∣≥∣S∣|S'|\ge|S|.
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