MathDB

2021 MMATHS

Part of MMATHS problems

Subcontests

(14)
Mixer Round
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2021 MMATHS Mixer Round - Math Majors of America Tournament for High Schools

p1. Prair takes some set SS of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from 11 to 1010 is on her list! What is the smallest possible value of ∣S∣|S|, the number of elements in her set SS?
p2. Jake has 20212021 balls that he wants to separate into some number of bags, such that if he wants any number of balls, he can just pick up some bags and take all the balls out of them. What is the least number of bags Jake needs?
p3. Claire has stolen Cat’s scooter once again! She is currently at (0; 0) in the coordinate plane, and wants to ride to (2,2)(2, 2), but she doesn’t know how to get there. So each second, she rides one unit in the positive xx or yy-direction, each with probability 12\frac12 . If the probability that she makes it to (2,2)(2, 2) during her ride can be expressed as ab\frac{a}{b} for positive integers a,ba, b with gcd(a,b)=1gcd(a, b) = 1, then find a+ba + b.
p4. Triangle ABCABC with AB=BC=6AB = BC = 6 and ∠ABC=120o\angle ABC = 120^o is rotated about AA, and suppose that the images of points BB and CC under this rotation are B′B' and C′C', respectively. Suppose that AA, B′B' and CC are collinear in that order. If the area of triangle B′CC′B'CC' can be expressed as a−bca - b\sqrt{c} for positive integers a,b,ca, b, c with csquarefree, find a+b+ca + b + c.
p5. Find the sum of all possible values of a+b+c+da + b + c + d if a,b,c,a, b, c, d are positive integers satisfying ab+cd=100,ab + cd = 100, ac+bd=500.ac + bd = 500.
p6. Alex lives in Chutes and Ladders land, which is set in the coordinate plane. Each step they take brings them one unit to the right or one unit up. However, there’s a chute-ladder between points (1,2)(1, 2) and (2,0)(2, 0) and a chute-ladder between points (1,3)(1, 3) and (4,0)(4, 0), whenever Alex visits an endpoint on a chute-ladder, they immediately appear at the other endpoint of that chute-ladder! How many ways are there for Alex to go from (0,0)(0, 0) to (4,4)(4, 4)?
p7. There are 88 identical cubes that each belong to 88 different people. Each person randomly picks a cube. The probability that exactly 33 people picked their own cube can be written as ab\frac{a}{b} , where aa and bb are positive integers with gcd(a,b)=1gcd(a, b) = 1. Find a+ba + b.
p8. Suppose that p(R)=Rx2+4xp(R) = Rx^2 + 4x for all RR. There exist finitely many integer values of RR such that p(R)p(R) intersects the graph of x3+2021x2+2x+1x^3 + 2021x^2 + 2x + 1 at some point (j,k)(j, k) for integers jj and kk. Find the sum of all possible values of RR.
p9. Let a,b,ca, b, c be the roots of the polynomial x3−20x2+22x^3 - 20x^2 + 22. Find bca2+acb2+abc2\frac{bc}{a^2} +\frac{ac}{b^2} +\frac{ab}{c^2}.
p10. In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it, this grid’s score is the sum of all numbers recorded this way. Deyuan shades each square in a blank n×nn \times n grid with probability kk; he notices that the expected value of the score of the resulting grid is equal to kk, too! Given that k>0.9999k > 0.9999, find the minimum possible value of nn.
p11. Find the sum of all xx from 22 to 10001000 inclusive such that ∏n=2xlog⁡nn(n+1)n+2\prod^x_{n=2} \log_{n^n}(n + 1)^{n+2} is an integer.
p12. Let triangle ABCABC with incenter II and circumcircle Γ\Gamma satisfy AB=63AB = 6\sqrt3, BC=14BC = 14, and CA=22CA = 22. Construct points PP and QQ on rays BABA and CACA such that BP=CQ=14BP = CQ = 14. Lines PIPI and QIQI meet the tangents from BB and CC to Γ\Gamma, respectively, at points XX and YY . If XYXY can be expressed as ab−ca\sqrt{b}-c for positive integers a,b,ca, b, c with cc squarefree, find a+b+ca + b + c.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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2021 MMATHS Mathathon Rounds 4-5 Math Majors of America Tournament for HS

Round 4
p10. How many divisors of 101110^{11} have at least half as many divisors that 101110^{11} has?
p11. Let f(x,y)=xy+yxf(x, y) = \frac{x}{y}+\frac{y}{x} and g(x,y)=xy−yxg(x, y) = \frac{x}{y}-\frac{y}{x} . Then, if f(f(...f(f(⏟2021fsf(f(1,2),g(2,1)),2),2)...,2),2)\underbrace{f(f(... f(f(}_{2021 fs} f(f(1, 2), g(2,1)), 2), 2)... , 2), 2) can be expressed in the form a+bca + \frac{b}{c}, where aa, bb,cc are nonnegative integers such that b<cb < c and gcd(b,c)=1gcd(b,c) = 1, find a+b+⌈(log⁡2(log⁡2c)⌉a + b + \lceil (\log_2 (\log_2 c)\rceil
p12. Let ABCABC be an equilateral triangle, and letDEF DEF be an equilateral triangle such that DD, EE, and FF lie on ABAB, BCBC, and CACA, respectively. Suppose that ADAD and BDBD are positive integers, and that [DEF][ABC]=97196\frac{[DEF]}{[ABC]}=\frac{97}{196}. The circumcircle of triangle DEFDEF meets ABAB, BCBC, and CACA again at GG, HH, and II, respectively. Find the side length of an equilateral triangle that has the same area as the hexagon with vertices D,E,F,G,HD, E, F, G, H, and II.
Round 5
p13. Point XX is on line segment ABAB such that AX=25AX = \frac25 and XB=52XB = \frac52. Circle Ω\Omega has diameter ABAB and circle ω\omega has diameter XBXB. A ray perpendicular to ABAB begins at XX and intersects Ω\Omega at a point YY. Let ZZ be a point on ω\omega such that ∠YZX=90o\angle YZX = 90^o. If the area of triangle XYZXYZ can be expressed as ab\frac{a}{b} for positive integers a,ba, b with gcd(a,b)=1gcd(a, b) = 1, find a+ba + b.
p14. Andrew, Ben, and Clayton are discussing four different songs; for each song, each person either likes or dislikes that song, and each person likes at least one song and dislikes at least one song. As it turns out, Andrew and Ben don't like any of the same songs, but Clayton likes at least one song that Andrew likes and at least one song that Ben likes! How many possible ways could this have happened?
p15. Let triangle ABCABC with circumcircle Ω\Omega satisfy AB=39AB = 39, BC=40BC = 40, and CA=25CA = 25. Let PP be a point on arc BCBC not containing AA, and let QQ and RR be the reflections of PP in ABAB and ACAC, respectively. Let AQAQ and ARAR meet Ω\Omega again at SS and TT, respectively. Given that the reflection of QRQR over BCBC is tangent to Ω\Omega , STST can be expressed as ab\frac{a}{b} for positive integers a,ba, b with gcd(a,b)=1gcd(a,b)= 1. Find a+ba + b.
PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here ,Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2021 MMATHS Mathathon Rounds 6-7 Math Majors of America Tournament for HS

Round 6
p16. Let ABCABC be a triangle with AB=3AB = 3, BC=4BC = 4, and CA=5CA = 5. There exist two possible points XX on CACA such that if YY and ZZ are the feet of the perpendiculars from XX to ABAB and BC,BC, respectively, then the area of triangle XYZXY Z is 11. If the distance between those two possible points can be expressed as abc\frac{a\sqrt{b}}{c} for positive integers aa, bb, and cc with bb squarefree and gcd(a,c)=1gcd(a, c) = 1, then find a+b+ca +b+ c.
p17. Let f(n)f(n) be the number of orderings of 1,2,...,n1,2, ... ,n such that each number is as most twice the number preceding it. Find the number of integers kk between 11 and 5050, inclusive, such that f(k)f (k) is a perfect square.
p18. Suppose that ff is a function on the positive integers such that f(p)=pf(p) = p for any prime p, and that f(xy)=f(x)+f(y)f (xy) = f(x) + f(y) for any positive integers xx and yy. Define g(n)=∑k∣nf(k)g(n) = \sum_{k|n} f (k); that is, g(n)g(n) is the sum of all f(k)f(k) such that kk is a factor of nn. For example, g(6)=f(1)+1(2)+f(3)+f(6)g(6) = f(1) + 1(2) + f(3) + f(6). Find the sum of all composite nn between 5050 and 100100, inclusive, such that g(n)=ng(n) = n.
Round 7
p19. AJ is standing in the center of an equilateral triangle with vertices labelled AA, BB, and CC. They begin by moving to one of the vertices and recording its label; afterwards, each minute, they move to a different vertex and record its label. Suppose that they record 2121 labels in total, including the initial one. Find the number of distinct possible ordered triples (a,b,c)(a, b, c), where a is the number of AA's they recorded, b is the number of BB's they recorded, and c is the number of CC's they recorded.
p20. Let S=∑n=1∞(1−{(2+3)n})S = \sum_{n=1}^{\infty} (1- \{(2 + \sqrt3)^n\}), where {x}=x−⌊x⌋\{x\} = x - \lfloor x\rfloor , the fractional part of xx. If S=a−bcS =\frac{\sqrt{a} -b}{c} for positive integers a,b,ca, b, c with aa squarefree, find a+b+ca + b + c.
p21. Misaka likes coloring. For each square of a 1×81\times 8 grid, she flips a fair coin and colors in the square if it lands on heads. Afterwards, Misaka places as many 1×21 \times 2 dominos on the grid as possible such that both parts of each domino lie on uncolored squares and no dominos overlap. Given that the expected number of dominos that she places can be written as ab\frac{a}{b}, for positive integers aa and bb with gcd(a,b)=1gcd(a, b) = 1, find a+ba + b.
PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here and 4-5 [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2021 MMATHS Mathathon Rounds 1-3 Math Majors of America Tournament for HS

Round 1
p1. Ben the bear has an algorithm he runs on positive integers- each second, if the integer is even, he divides it by 22, and if the integer is odd, he adds 11. The algorithm terminates after he reaches 11. What is the least positive integer n such that Ben's algorithm performed on n will terminate after seven seconds? (For example, if Ben performed his algorithm on 33, the algorithm would terminate after 33 seconds: 3→4→2→13 \to 4 \to 2 \to 1.)
p2. Suppose that a rectangle RR has length pp and width qq, for prime integers pp and qq. Rectangle SS has length p+1p + 1 and width q+1q + 1. The absolute difference in area between SS and RR is 2121. Find the sum of all possible values of pp.
p3. Owen the origamian takes a rectangular 12×1612 \times 16 sheet of paper and folds it in half, along the diagonal, to form a shape. Find the area of this shape.
Round 2
p4. How many subsets of the set {G,O,Y,A,L,E}\{G, O, Y, A, L, E\} contain the same number of consonants as vowels? (Assume that YY is a consonant and not a vowel.)
p5. Suppose that trapezoid ABCDABCD satisfies AB=BC=5AB = BC = 5, CD=12CD = 12, and ∠ABC=∠BCD=90o\angle ABC = \angle BCD = 90^o. Let ACAC and BDBD intersect at EE. The area of triangle BECBEC can be expressed as ab\frac{a}{b}, for positive integers aa and bb with gcd(a,b)=1gcd(a, b) = 1. Find a+ba + b.
p6. Find the largest integer nn for which 101n+103n101n−1+103n−1\frac{101^n + 103^n}{101^{n-1} + 103^{n-1}} is an integer.
Round 3
p7. For each positive integer n between 11 and 10001000 (inclusive), Ben writes down a list of nn's factors, and then computes the median of that list. He notices that for some nn, that median is actually a factor of nn. Find the largest nn for which this is true.
p8. ([color=#f00]voided) Suppose triangle ABCABC has AB=9AB = 9, BC=10BC = 10, and CA=17CA = 17. Let xx be the maximal possible area of a rectangle inscribed in ABCABC, such that two of its vertices lie on one side and the other two vertices lie on the other two sides, respectively. There exist three rectangles R1R_1, R2R_2, and R3R_3 such that each has an area of xx. Find the area of the smallest region containing the set of points that lie in at least two of the rectangles R1R_1, R2R_2, and R3R_3.
p9. Let a,b,a, b, and cc be the three smallest distinct positive values of θ\theta satisfying cos⁡θ+cos⁡3θ+...+cos⁡2021θ=sin⁡θ+sin⁡3θ+...+sin⁡2021θ.\cos \theta + \cos 3\theta + ... + \cos 2021\theta = \sin \theta+ \sin 3 \theta+ ... + \sin 2021\theta. What is 4044π(a+b+c)\frac{4044}{\pi}(a + b + c)?
[color=#f00]Problem 8 is voided.
PS. You should use hide for answers.Rounds 4-5 have been posted [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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MMATHS 2021, Problem 4: The Return of Cat and Claire

Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"
Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
Proposed by Andrew Wu

MMATHS 2021, TB Problem 4: Initially I Asked For XY^2 But I'm Dumb

Let triangle ABCABC with incenter II and circumcircle Γ\Gamma satisfy AB=63,BC=14,AB = 6\sqrt{3}, BC = 14, and CA=22CA = 22. Construct points PP and QQ on rays BABA and CACA such that BP=CQ=14BP = CQ = 14. Lines PIPI and QIQI meet the tangents from BB and CC to Γ\Gamma, respectively, at points XX and YY. If XYXY can be expressed as ab−ca\sqrt{b} - c for positive integers a,b,ca,b,c with cc squarefree, find a+b+ca + b + c.
Proposed by Andrew Wu
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MMATHS 2021, Problem 2: Digital Reductions

Define the digital reduction of a two-digit positive integer AB‾\underline{AB} to be the quantity AB‾−A−B\underline{AB} - A - B. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of 6262 is 62−6−2=54.62 - 6 - 2 = 54.)
Proposed by Andrew Wu

MMATHS 2021, TB Problem 2: Expected Value of a Grid

In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's score is the sum of all numbers recorded this way. Deyuan shades each square in a blank n×nn\times n grid with probability kk; he notices that the expected value of the score of the resulting grid is equal to kk, too! Given that k>0.9999k > 0.9999, find the minimum possible value of nn.
Proposed by Andrew Wu
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