2010 MAΘ National Convention. Answer Key:

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1 MAΘ National Convention Answer Key: Section I Trivia. 79 (THE WHO). 86 (HIGHLANDER) (THE OFFICE). 37 (THE LION, THE WITCH AND THE WARDROBE) 5. 6 (ROLLING STONE) 6. (THE ONION) (WES ANDERSON) 8. 5 (THE CRITERION COLLECTION) (CHUCK PALAHNIUK). 39 (YOSHIMI BATTLES THE PINK ROBOTS) Section II What Number am I? Section III Unscramble Me. OCTAHEDRON (TETRAHEDRON, HEXAHEDRON, DODECAHEDRON, ICOSAHEDRON). RATIONAL or WHOLE (NATURAL, INTEGER, REAL, COMPLEX) 3. TWO (ONE, THREE, FOUR). CALCULATOR (ABACUS, SLIDE RULE) 5. GEOMETRIC (ARITHMETIC, HARMONIC) 6. MODE (MEAN, MEDIAN) 7. MU (ALPHA, THETA) 8. ZERO or NEUTRAL (POSITIVE, NEGATIVE) 9. RING (GROUP, INTEGRAL DOMAIN, FIELD). CALCULUS (ALGEBRA, GEOMETRY, STATISTICS)

2 MAΘ National Convention Section IV Somewhat Difficult Problems. 9 (RSS, SRS, SSR, PRR,RPR, RRP, SPP, PSP, PPS). 979 (Any number from a perfect square n to nn+ ( ), inclusive, will suffice for c. Since = 936 and 5 = 5, all numbers from 936 to *5 = 979 will suffice for c, and the next number that has a solution will be 5. The answer, FYI, is x = 979 / ) 3. 5 (<,6>; <,,6>; <,,6>; <,,,8,6>; <,,, 8,6>). /3 (Since A+ B= B/ A, A + AB+ B=, and solving for B gives B= A /( A+ ). Since AB = 6 / (7A), A /( A + ) = 6/7. The only rational solution to this is A = /3. Plugging back in gives B = / 3) 5. (The first term must be, and then the remaining terms are 3, 5, 7, 9,, n,. Therefore, the only even term is the first one.) 6. (This sequence looks like this:, 6,, 7,, 8,, 8, 8,,,, and then it repeats beginning with the first. So there are terms initially, with a repeating cycle of 8 terms after that. Since = + 5*8, the th term is the last in the cycle, meaning the th term is the two terms before it.) 7. 5 (Since the sum of the first n positive integers is nn+ ( )/, we are looking for the smallest value of n such that nn ( + ) = *3* kfor some positive integer k. Since 6 =, and 5 is divisible by 3, the smallest value of n is 5.) 8. (By squaring both sides, a+ a 8 = b+ 5. Since a and b are positive integers, a 8 = 5, or a =. Since a= b, b=.) ( X = + + ( )( ) + ( 3)( )( )..., so x =, x 5 = 9, and x 6 = =3995) x =, x =, x 3 =,. 9/65 (There are 5*5*5*9 = 697 different sequences of four cards, but when dealing to two people, the first person s cards switched in the order doesn t matter. Thus there are only / of these that are possible, so the denominator is 997/ = 635. For the numerator, if both cards are the same, there are 3 different ranks, along with 6 different combinations of two suits for the first person (the second person s suits are determined automatically once the first person s suits are determined), so there are 78 ways for both cards to have the same rank. If the two cards in each hand 3 have different ranks, there are = 78 different choices of two cards, and then there are *3= different combinations of one card, and *3= combinations of the other, so there are a total of 78** = 3 total ways to distribute the cards in this way. Thus, the total numerator is 78+3=3, and the fraction is 3/635, which reduces to 9/65.) (This is the sequence,,, 3, 3, 3,,,,,, where the positive integer n is repeated n times. Thus, the series is, s, 3 3s, s,, 6 6s, and 57 63s. The sum is thus *63 = 6*63*5 / = 8966.)

3 MAΘ National Convention., 5, and 9 (The Pythagorean triples are in the form a b, ab, a + b. The first such triple is 3,,5 ( a=, b= ). Succeeding such triples come from a recursive definition where a = n a + + n bnand b = n+ an. Thus, the next triples are,,9 ( a= 5, b= ) ; 9,,69 ( a=, b= 5) ; 697,696,985 ( a= 9, b= ) ; and 59,6,57 ( a 7, b= 9). The last digits of the hypotenuse lengths form the sequence 5,9,9,5,,,5,9,9,5,,,5,9,9,5,, ) (Let n be the common integer root. Then n An B= = n An+ B. Therefore, 3 ( ) B= n n /, and the smallest odd value of B is 9, which occurs when n = 3.) 3. 5 (Powerful numbers can be written in the form ab, where a and b are positive integers. Of the first positive integers, the ones that are powerful are,, 8, 9, 6, 5, 7, 3, 36, 9, 6, 7, 8,, 8,, 5, 8,, 69, 96,, 6, 5, 3, 56, 88, 89, 3, 33, 36, 39,, 3,, 8, 5, 5, 59, 576, 65, 68, 675, 676, 79, 78, 8, 8, 86, 9, 96, 968, 97,, and there are 5 of these.) n ( n 3 3 n n n = 3, where n is an odd multiple of 3,, and 3 (hence, n = 9 ); n is an even multiple of and 3, and one more than a multiple of 3 (hence, n = 86 ); n 3 is an even multiple of 3 and 3, and one more than a multiple of (hence, n 3 = 78 ); and n is an even multiple of 3 and, and one more than a multiple of 3 (hence, n = 66 ). The sum is thus ( 9) + ( 3 86) ( 78) + ( 3 66) = 58.) Section V Minority Game Answers to be determined after test is given. Section VI Game Theory. 98 coins for head pirate, coin for third pirate, coin for fifth pirate. Both prisoners should defect from the other 3.. The bidding will increase indefinitely 5. $99 for first person, $ for second person

4 MAΘ National Convention Section VII Sum of Four Perfect Squares For example, there are others: = + = = = 9+9+ = ++ 5 = = 6+6 = = = 6+ 9 = +5+ = = + 7 = = 9+ = = ++ 8 = = = = = = 6+6+ = 7 = +5+8 = = = + 8 = = 3 = = = 9+ = + 35 = = = = = = + = = = = 9+ = = = = = = +5+ Section VIII The Game For example, there are others (probably): = =! 35 =!!!!!!!!!! + = =!! 36 =!!!!!!!!! + + 3= = + 37 =!!!!!!!!! +! + = + +! + = =!!!!!!!!! +! +! 5= + +! +!! 6 = (+ )! = +! +! = = (( + ) + + ) 7 ( )!! = + + = ( + + )!!!!!!!! 39!!!!!!!!!!! =!!!!!! + =!!!!!! + 8 = ( + )! +! +! 5 = ( +!)!!!!!!!!!!!! = ( +! +!)!!!!!!!!!!! 9= 6 = ( +! + )!!!!!!!!!!! 3 = (!)!!!!!!!! = ( ) = + 8 (!!)!!!!!!!!!!!! 7 =!!!!!! = (! + )!!!!!!! = + + = ( ) 5!!!! = =!!!!!!! +! 6 = ( +! +!)!!!!!!!!!!!!!!!!!!!!! 3 = +! + 3 = + 7 =!!!!!!!!! + = +! +! 3 =!!!!!!! +! 8 =!!!!!!!! (!!)!! = ( ) 6 (( )!!! )!!!!!! = ( )!!!!!!!! = + + = (( + ) + ) 7 (( )! )!!!! 3 = ( + )! +! +!!!!! 9 =!!!!!!!! +! + = + 5 =!!!!!!!! +! +! 3 ( )!!!!!!!!!!!!!!!!!!!!

5 MAΘ National Convention Section IX Cryptograms. There is nothing strange in the circle being the origin of any and every marvel. Aristotle. I don t believe in mathematics. Albert Einstein 3. Mathematics is a game played according to certain rules with meaningless marks on paper. David Hilbert. A mathematician is a blind man in a dark room looking for a black cat which isn t there. Charles Darwin 5. Mathematics is the only science where one never knows what one is talking about nor whether what is said is true. Bertrand Russell Section X Wild Card Question Answer to be determined after test is given.

Notes: Pythagorean Triples

Math 5330 Spring 2018 Notes: Pythagorean Triples Many people know that 3 2 + 4 2 = 5 2. Less commonly known are 5 2 + 12 2 = 13 2 and 7 2 + 24 2 = 25 2. Such a set of integers is called a Pythagorean Triple.