Subcontests
(6)2019 MMATHS Mixer Round - Math Majors of America Tournament for High Schools
p1. An ant starts at the top vertex of a triangular pyramid (tetrahedron). Each day, the ant randomly chooses an adjacent vertex to move to. What is the probability that it is back at the top vertex after three days?
p2. A square “rolls” inside a circle of area π in the obvious way. That is, when the square has one corner on the circumference of the circle, it is rotated clockwise around that corner until a new corner touches the circumference, then it is rotated around that corner, and so on. The square goes all the way around the circle and returns to its starting position after rotating exactly 720o. What is the area of the square?
p3. How many ways are there to fill a 3×3 grid with the integers 1 through 9 such that every row is increasing left-to-right and every column is increasing top-to-bottom?
p4. Noah has an old-style M&M machine. Each time he puts a coin into the machine, he is equally likely to get 1 M&M or 2 M&M’s. He continues putting coins into the machine and collecting M&M’s until he has at least 6 M&M’s. What is the probability that he actually ends up with 7 M&M’s?
p5. Erik wants to divide the integers 1 through 6 into nonempty sets A and B such that no (nonempty) sum of elements in A is a multiple of 7 and no (nonempty) sum of elements in B is a multiple of 7. How many ways can he do this? (Interchanging A and B counts as a different solution.)
p6. A subset of {1,2,3,4,5,6,7,8} of size 3 is called special if whenever a and b are in the set, the remainder when a+b is divided by 8 is not in the set. (a and b can be the same.) How many special subsets exist?
p7. Let F1=F2=1, and let Fn=Fn−1+Fn−2 for all n≥3. For each positive integer n, let g(n) be the minimum possible value of ∣a1F1+a2F2+...+anFn∣, where each ai is either 1 or −1. Find g(1)+g(2)+...+g(100).
p8. Find the smallest positive integer n with base-10 representation 1a1a2...ak such that 3n = \overline{a_1a_2 a_k1}.
p9. How many ways are there to tile a 4×6 grid with L-shaped triominoes? (A triomino consists of three connected 1×1 squares not all in a line.)
p10. Three friends want to share five (identical) muffins so that each friend ends up with the same total amount of muffin. Nobody likes small pieces of muffin, so the friends cut up and distribute the muffins in such a way that they maximize the size of the smallest muffin piece. What is the size of this smallest piece?
Numerical tiebreaker problems:
p11. S is a set of positive integers with the following properties:
(a) There are exactly 3 positive integers missing from S.
(b) If a and b are elements of S, then a+b is an element of S. (We allow a and b to be the same.)
How many possibilities are there for the set S?
p12. In the trapezoid ABCD, both ∠B and ∠C are right angles, and all four sides of the trapezoid are tangent to the same circle. If AB=13 and CD=33, find the area of ABCD.
p13. Alice wishes to walk from the point (0,0) to the point (6,4) in increments of (1,0) and (0,1), and Bob wishes to walk from the point (0,1) to the point (6,5) in increments of (1,0) and (0,1). How many ways are there for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times)?
p14. The continuous function f(x) satisfies 9f(x+y)=f(x)f(y) for all real numbers x and y. If f(1)=3, what is f(−3)?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. 2019 MMATHS Tiebreaker p3 - from (0, 0) to the (m,n)
Let m and n be positive integers. Alice wishes to walk from the point (0,0) to the point (m,n) in increments of (1,0) and (0,1), and Bob wishes to walk from the point (0,1) to the point (m,n+1) in increments of(1,0) and (0,1). Find (with proof) the number of ways for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times). 2019 MMATHS Mathathon Rounds 1-4 Math Majors of America Tournament for HS
Round 1
p1. A small pizza costs $4 and has 6 slices. A large pizza costs $9 and has 14 slices. If the MMATHS organizers got at least 400 slices of pizza (having extra is okay) as cheaply as possible, how many large pizzas did they buy?
p2. Rachel flips a fair coin until she gets a tails. What is the probability that she gets an even number of heads before the tails?
p3. Find the unique positive integer n that satisfies n!⋅(n+1)!=(n+4)!.
Round 2
p4. The Portland Malt Shoppe stocks 10 ice cream flavors and 8 mix-ins. A milkshake consists of exactly 1 flavor of ice cream and between 1 and 3 mix-ins. (Mix-ins can be repeated, the number of each mix-in matters, and the order of the mix-ins doesn’t matter.) How many different milkshakes can be ordered?
p5. Find the minimum possible value of the expression (x)2+(x+3)4+(x+4)4+(x+7)2, where x is a real number.
p6. Ralph has a cylinder with height 15 and volume π960 . What is the longest distance (staying on the surface) between two points of the cylinder?
Round 3
p7. If there are exactly 3 pairs (x,y) satisfying x2+y2=8 and x+y=(x−y)2+a, what is the value of a?
p8. If n is an integer between 4 and 1000, what is the largest possible power of 2 that n4−13n2+36 could be divisible by?
(Your answer should be this power of 2, not just the exponent.)
p9. Find the sum of all positive integers n≥2 for which the following statement is true: “for any arrangement of n points in three-dimensional space where the points are not all collinear, you can always find one of the points such that the n−1 rays from this point through the other points are all distinct.”
Round 4
p10. Donald writes the number 12121213131415 on a piece of paper. How many ways can he rearrange these fourteen digits to make another number where the digit in every place value is different from what was there before?
p11. A question on Joe’s math test asked him to compute ba+43 , where a and b were both integers. Because he didn’t know how to add fractions, he submitted b+4a+3 as his answer. But it turns out that he was right for these particular values of a and b! What is the largest possible value that a could have been?
p12. Christopher has a globe with radius r inches. He puts his finger on a point on the equator. He moves his finger 5π inches North, then π inches East, then 5π inches South, then 2π inches West. If he ended where he started, what is the largest possible value of r?
PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2789002p24519497]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. 2019 MMATHS Mathathon Rounds 5-7 Math Majors of America Tournament for HS
Round 5
p13. Suppose △ABC is an isosceles triangle with AB=BC, and X is a point in the interior of △ABC. If m∠ABC=94o, m∠ABX=17o, and m∠BAX=13o, then what is m∠BXC (in degrees)?
p14. Find the remainder when ∑n=120191+2n+4n2+8n3 is divided by 2019.
p15. How many ways can you assign the integers 1 through 10 to the variables a,b,c,d,e,f,g,h,i, and j in some order such that a<b<c<d<e,f<g<h<i, a<g,b<h,c<i, f<b,g<c, and h<d?
Round 6
p16. Call an integer n equi-powerful if n and n2 leave the same remainder when divided by 1320. How many integers between 1 and 1320 (inclusive) are equi-powerful?
p17. There exists a unique positive integer j≤10 and unique positive integers nj , nj+1, ..., n10 such that j≤nj<nj+1<...<n10 and (10n10)+(9n9)+...+(jnj)=2019. Find nj+nj+1+...+n10.
p18. If n is a randomly chosen integer between 1 and 390 (inclusive), what is the probability that 26n has more positive factors than 6n?
Round 7
p19. Suppose S is an n-element subset of {1,2,3,...,2019}. What is the largest possible value of n such that for every 2≤k≤n, k divides exactly n−1 of the elements of S?
p20. For each positive integer n, let f(n) be the fewest number of terms needed to write n as a sum of factorials. For example, f(28)=3 because 4!+2!+2!=28 and 28 cannot be written as the sum of fewer than 3 factorials. Evaluate f(1)+f(2)+...+f(720).
p21. Evaluate ∑n=1∞101n−1ϕ(n) , where ϕ(n) is the number of positive integers less than or equal to n that are relatively prime to n.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2788993p24519281]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. 2019 MMATHS Individual Round - Math Majors of America Tournament for High School
p1. When Charles traveled from Hawaii to Chicago, he moved his watch 5 hours backwards instead of 5 hours forwards. He plans to wake up at 7:00 the next morning (Chicago time). When he wakes up during the night and sees that his watch says 6:00, how many more hours should he sleep? (He has a 12-hour watch, not a 24-hour watch.)
p2. Rover’s dog house in the middle of a large grassy yard is a regular hexagon with side length 10. His leash, which has length 20, connects him to one vertex on the outside of the dog house. His leash cannot pass through the interior of the dog house. What is the total area of the yard (i.e., outside the doghouse) that he can roam? (Give your answer in units squared.)
p3. Daniel rolls three fair six-sided dice. Given that the sum of the three numbers he rolled was 6, what is the probability that all of the dice showed different numbers?
p4. The points A, B, and C lie on a circle centered at the point O. Given that m∠AOB=110o and m∠CBO=36o, there are two possible values of m∠CAO. Give the (positive) difference of these two possibilities (in degrees).
p5. Joanne has four piles of sand, which weigh 1, 2, 3, and 4 pounds, respectively. She randomly chooses a pile and distributes its sand evenly among the other three piles. She then chooses one of the remaining piles and distributes its sand evenly among the other two. What is the expected weight (in pounds) of the larger of these two final piles?
p6. When 15! is converted to base 8, it is expressed as 230167356abc00 for some digits a, b, and c. Find the missing string abc.
p7. Construct triangles △ABC and △A′B′C′ such that AB=10, BC=11, AC=12, C lies on segment A′A, B lies on C′C, A lies on B′B, and A′C=C′B=B′A=1. Find the ratio of the area of △A′B′C′ to the area of △ABC.
p8. Given that x4+y4+z4=1, let a be the maximum possible value of x+y+z, let b be the minimum possible value of x+y+z, let c be the maximum possible value of x−y−z, and let d be the minimum possible value of x−y−z. What is the value of abcd?
p9. How many (possibly empty) subsets of {1,2,3,4,5,6,7,8,9,10,11} do not contain any pair of elements with difference 2?
p10. The positive real numbers x and y satisfy x2=y2+72. If x2, y2, and (x+y)2 are all integers, what is the largest possible value of x2+y2?
p11. There are N ways to decompose a regular 2019-gon into triangles (by drawing diagonals between the vertices of the 2019-gon) such that each triangle shares at least one side with the 2019-gon. What is the largest integer a such that 2a divides N?
p12. Anna has a 5×5 grid of pennies. How many ways can she arrange them so that exactly two pennies show heads in each row and in each column?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.