Final Presentation V 1dot 0

Round 1 Marathon (Theoretical Mathematics) Buzzer Round 3 Questions 2

Question 1 1 0

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How many elements are there in the rotation group of a soccer ball (having 20 hexagonal faces and 12 pentagonal faces) ?

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Solution

60

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Question 2 1 5

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If you have N envelopes (N being very large) and each has its unique recipient . You don’t know which one is to be sent to whom. Sending all them randomly what is the probability that more than person receives a correct envelope. 7

Solution

1 n 1 − (1 − ) n

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Question 3 2 0

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• Assume that the random variable X1 and X2 are normally distributed . Mean Standard Deviation Mean Standard X1 u1 s1 deviation u2 X2 s2 X1 : u1 s1 X2 : u2 s2 • The co-relation between X1 and X2 is -1 . • How can you choose constants 'a' and 'b' such that (a*X! + b*X2) has minimum variance 10

Solution

a+b = 1 , 0
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Round 2 Shot Put (Audio/Visual Round)

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Question 1 1 5

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Draw the relation between the three sets Complete Metric Spaces

Universal Set

Compact Metric Spaces

Totally Bounded Metric Spaces

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The solution Compact Complete

Totally Bounded

Universal Set

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Question 2 1 5

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Join these with 4 straight lines without lifting your pen

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The solution

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Question 3 1 0

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Connect the three houses to water, electricity and gas suppliers without any lines crossing-

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Solution It is not possible because K3,3 is not planar. If it were, m<= 2n-4 (for bipartite) 9 <= 2*6 – 4 9 <= 8 Contradiction

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Question 4 1 0

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Insert operators to make the following statement true

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1

1 =6

Hint: Don’t constrain yourself to basic operators 23

Solution

(1 + 1 + 1 )! = 6 24

Cross Round 3 Country (History/Trivia) Buzzer Round 10 Questions 25

Question 1 Identify X 3 Hints 2 5 2 0 1 5

1 0 26

Ques 2 5 Hint 1 2 0 Hint 21 5 Hint 31 0

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Solution

Paul Erdos

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Question 2 2 Hints 2 5 2 0 1 5

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Ques 2 5

Hint 1 2 0

Hint 2 1 5

PTO 30

Hint 2 1 5

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Solution

Alan Turing

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Question 3 NO HINTS 1 0

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X (December 7, 1823 – December 29, 1891) argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the integers; all else is the work of man".

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Solution

Leopold Kronecker

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Question 4 NO HINTS 1 0

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" Is R a normal set? If it is normal, then it is a member of R, since R contains all normal sets. But if that is the case, then R contains itself as a member, and therefore is abnormal. On the other hand, if R is abnormal, then it is not a member of R, since R contains only normal sets. But if that is the case, then R does not contain itself as a member, and therefore is normal. Clearly, this is a paradox: if we suppose R is normal we can prove it is abnormal, and if we suppose R is abnormal we can prove it is normal. Hence, R is both normal and abnormal, which is a contradiction." What is this an explanation of?

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Solution

Russell's paradox

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Question 5 1 0

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In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence a^n-1= 1 mod n for all integers a which are relatively prime to n. What is the smallest Carmichael number?

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Solution

561

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Question 6

1 0

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X (17 June 1898 – 27 March 1972) was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs and mezzotints. These feature impossible constructions, explorations of infinity, architecture and tessellations.

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Solution

M.C. Escher

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Question 7

1 5

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The X is one of the earliest ________ curves to have been described. It can be created by starting with an equilateral triangle. At each step, each side is altered recursively as follows: 1. Divide the line segment into three segments of equal length. 2. Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. 3. Remove the line segment that is the base of the triangle from step 2.

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Solution

Koch snowflake

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Question 8 1 5

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The universe of X is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, A or B. Every cell interacts with its eight neighbours, which are the cells that are directly horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur: 1. Any A cell with fewer than two A neighbours or more than three A neighbours goes to state B. 2. Any A cell with two or three A neighbours stays in state A. 3. Any B cell with exactly three A neighbours comes to state A.

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Solution

The Game of Life

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Question 9 1 5

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attack, so named because of its relation to the ___________ problem in probability theory. Given a function f, the goal of the attack is to find two inputs x1,x2 such that f(x1) = f(x2). Inputs are chosen randomly until such a pair is obtained. This method can be rather efficient. Say a function f(x) yields any of H different outputs with equal probability. Then it is expected that the required pair will be obtained after testing 1.25*sqrt(H) arguments on average.

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Solution

Birthday attack

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Question 10 1 0

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In classical logic, _______ ___________ (Latin: mode that affirms by affirming) is the name given to an form of argument sometimes referred to as affirming the antecedent or the law of detachment. An example of an argument that fits the form for _____ _________: If today is Tuesday, then I will go to work. Today is Tuesday. Therefore, I will go to work.

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Solution

Modus ponens

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Sprint Rapid FIRE !!! 3 questions 120 seconds

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