Subcontests
(6)2016 MMATHS Mixer Round - Math Majors of America Tournament for High Schools
p1. Give a fake proof that 0=1 on the back of this page. The most convincing answer to this question at this test site will receive a point.
p2. It is often said that once you assume something false, anything can be derived from it. You may assume for this question that 0=1, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number.
p3. Suppose you write out all integers between 1 and 1000 inclusive. (The list would look something like 1, 2, 3, ... , 10, 11, ... , 999, 1000.) Which digit occurs least frequently?
p4. Pick a real number between 0 and 1 inclusive. If your response is r and the standard deviation of all responses at this site to this question is σ, you will receive r(1−(r−σ)2) points.
p5. Find the sum of all possible values of x that satisfy 243x+1=81x2+2x.
p6. How many times during the day are the hour and minute hands of a clock aligned?
p7. A group of N+1 students are at a math competition. All of them are wearing a single hat on their head. N of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime N such that nobody will ever steal the blue hat.
p8. On the back of this page, prove there is no function f(x) for which there exists a (finite degree) polynomial p(x) such that f(x)=p(x)(x+3)+8 and f(3x)=2f(x).
p9. Given a cyclic quadrilateral YALE with YA=2, AL=10, LE=11, EY=5, what is the area of YALE?
p10. About how many pencils are made in the U.S. every year? If your answer to this question is p, and our (good) estimate is ρ, then you will receive max(0,1−21∣log10(p)−log10(ρ)∣) points.
p11. The largest prime factor of 520,302,325 has 5 digits. What is this prime factor?
p12. The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide 520,302,325 by an appropriate integer. Unfortunately, when solving the problem many people divide it by n instead, and then they fail to see the pattern. What is n?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. 2016 MMATHS Mathathon Rounds 5-7 Math Majors of America Tournament for HS
Round 5
p13. Let {a}n≥1 be an arithmetic sequence, with a1=0, such that for some positive integers k and x we have ak+1=(xk), a2k+1=(x+1k) , and a3k+1=(x+2k). Let {b}n≥1 be an arithmetic sequence of integers with b1=0. Given that there is some integer m such that bm=(xk), what is the number of possible values of b2?
p14. Let A=arcsin(41) and B=arcsin(71). Find sin(A+B)sin(A−B).
p15. Let {fi}i=19 be a sequence of continuous functions such that fi:R→Z is continuous (i.e. each fi maps from the real numbers to the integers). Also, for all i, fi(i)=3i. Compute ∑k=19fk∘fk−1∘...∘f1(3−k).
Round 6
p16. If x and y are integers for which 20310x3+10x2y+xy3+y4=1134341 and x−y=1, then compute x+y.
p17. Let Tn be the number of ways that n letters from the set {a,b,c,d} can be arranged in a line (some letters may be repeated, and not every letter must be used) so that the letter a occurs an odd number of times. Compute the sum T5+T6.
p18. McDonald plays a game with a standard deck of 52 cards and a collection of chips numbered 1 to 52. He picks 1 card from a fully shuffled deck and 1 chip from a bucket, and his score is the product of the numbers on card and on the chip. In order to win, McDonald must obtain a score that is a positive multiple of 6. If he wins, the game ends; if he loses, he eats a burger, replaces the card and chip, shuffles the deck, mixes the chips, and replays his turn. The probability that he wins on his third turn can be written in the form z3x2⋅y such that x,y, and z are relatively prime positive integers. What is x+y+z?(NOTE: Use Ace as 1, Jack as 11, Queen as 12, and King as 13)
Round 7
p19. Let fn(x)=ln(1+x2n+x2n+1+x3⋅2n). Compute ∑k=0∞f2k(21).
p20. ABCD is a quadrilateral with AB=183, BC=300, CD=55, DA=244, and BD=305. Find AC.
p21. Define xyz(t+1)=1000x+100y+10z+t+1 as the decimal representation of a four digit integer. You are given that 3x5y7z2t=xyz(t+1) where x,y,z, and t are non-negative integers such that t is odd and 0≤x,y,z,(t+1)≤9. Compute3x5y7z
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782822p24445934]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. 2016 MMATHS Mathathon Rounds 1-4 Math Majors of America Tournament for HS
Round 1
p1. This year, the Mathathon consists of 7 rounds, each with 3 problems. Another math test, Aspartaime, consists of 3 rounds, each with 5 problems. How many more problems are on the Mathathon than on Aspartaime?
p2. Let the solutions to x3+7x2−242x−2016=0be a,b, and c. Find a2+b2+c2. (You might find it helpful to know that the roots are all rational.)
p3. For triangle ABC, you are given AB=8 and ∠A=30o . You are told that BC will be chosen from amongst the integers from 1 to 10, inclusive, each with equal probability. What is the probability that once the side length BC is chosen there is exactly one possible triangle ABC?
Round 2
p4. It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat?
p5. Let a be the square root of the least positive multiple of 2016 that is a square. Let b be the cube root of the least positive multiple of 2016 that is a cube. What is a−b?
p6. Hypersomnia Cookies sells cookies in boxes of 6,9 or 10. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy 8 cookies.)
Round 3
p7. There is a store that sells each of the 26 letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out 1 cost $1001, the letters to spell out 2 cost $1010, and the letters to spell out 11 cost $2015, how much do the letters to spell out 12 cost?
p8. There is a square ABCD with a point P inside. Given that PA=6, PB=9, PC=8. Calculate PD.
p9. How many ordered pairs of positive integers (x,y) are solutions to x2−y2=2016?
Round 4
p10. Given a triangle with side lengths 5,6 and 7, calculate the sum of the three heights of the triangle.
p11. There are 6 people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them?
p12. Find all x such that ∑i=0∞ixi=43.
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. 2016 MMATHS Individual Round - Math Majors of America Tournament for High School
p1. For what value of x is the function f(x)=(x−2)2 minimized?
p2. Two spheres A and B have centers (0,0,0) and (2016,2016,1008). Sphere A has a radius of 2017. If A and B are externally tangent, what is the radius of sphere B?
p3. Consider a white, solid cube of side length 5 made of 5×5×5=125 identical unit cubes with faces parallel to the faces of the larger cube. The cube is submerged in blue paint until the entire exterior of the cube is painted blue, so that a face of a smaller cube is blue if and only if it is part of a face of the larger cube. A random smaller cube is selected and rolled. What is the probability that the up-facing side is blue?p4. [This problem was thrown out.]
p5. Find the largest prime factor of 4003997, given that 4003997 is the product of two primes.
p6. Elaine is writing letters to six friends. She has six addressed letters and six addressed envelopes. She puts each letter randomly into an envelope without first checking the name on the envelope. What is the probability that exactly one envelope has the correct letter?
p7. Colin has written the numbers 1,2,...,n on a chalk board. He will erase at most 4 of the numbers (he might choose not to erase any of the numbers) and then circle n−4 of the remaining numbers. There are exactly 2016 possible ways to do this. Find n. (You should assume that circling the same set of numbers but erasing different numbers should count as different possible ways.)p8. Let Rn=r1,r2,r3,...,rn be a finite sequence of integers such that for all possible i, ri is either −1, 0, 1 or 2. Furthermore, for all i such that 1≤i<n, ri and ri+1 have opposite parity (i.e. one is odd and the other is even). Finally, −1 and 2 do not occur adjacently in the sequence. Given that r1 must be even (i.e. either r1=0 or r1=2), S(n) is the number of possible sequences Rn could be. For example, S(1)=2. For what k is S(k)=122?
p9. What is the largest prime p for which the numbers p2−8, p2−2, and p2+10 are all prime as well?
p10. Express 25+21⋅22⋅23⋅24⋅25 as an integer.
p11. Let ABCD be a quadrilateral where AC bisects ∠A, AB=AD, BC=CD=7, and AC⋅BD=36. Find AB+AD.
p12. Find the largest integer x for which there is an integer y such that x4+12x3+39x2+17x−57=y3.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.