MathDB

2021 BmMT

Part of BmMT problems

Subcontests

(4)
Pacer Round
1

2021 BmMT Pacer Round - Berkley mini Math Tournament

p1. 17.5%17.5\% of what number is 4.5%4.5\% of 2800028000?
p2. Let xx and yy be two randomly selected real numbers between −4-4 and 44. The probability that (x−1)(y−1)(x - 1)(y - 1) is positive can be written in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm + n.
p3. In the xyxy-plane, Mallen is at (−12,7)(-12, 7) and Anthony is at (3,−14)(3,-14). Mallen runs in a straight line towards Anthony, and stops when she has traveled 23\frac23 of the distance to Anthony. What is the sum of the xx and yy coordinates of the point that Mallen stops at?
p4. What are the last two digits of the sum of the first 20212021 positive integers?
p5. A bag has 1919 blue and 1111 red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the 88th ball he draws is red can be written in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm + n.
p6. How many terms are in the arithmetic sequence 33, 1111, ......, 779779?
p7. Ochama has 2121 socks and 44 drawers. She puts all of the socks into drawers randomly, making sure there is at least 11 sock in each drawer. If xx is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of xx?
p8. What is the least positive integer nn such that n+1−n<120\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}?
p9. Triangle △ABC\vartriangle ABC is an obtuse triangle such that ∠ABC>90o\angle ABC > 90^o, AB=10AB = 10, BC=9BC = 9, and the area of △ABC\vartriangle ABC is 3636. Compute the length of ACAC. https://cdn.artofproblemsolving.com/attachments/a/c/b648d0d60c186d01493fcb4e21b5260c46606e.png
p10. If x+y−xy=4x + y - xy = 4, and xx and yy are integers, compute the sum of all possible values ofx+y x + y.
p11. What is the largest number of circles of radius 11 that can be drawn inside a circle of radius 22 such that no two circles of radius 11 overlap?
p12. 22.5%22.5\% of a positive integer NN is a positive integer ending in 77. Compute the smallest possible value of NN.
p13. Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years?
p14. Say there is 11 rabbit on day 11. After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day 55?
15. Ajit draws a picture of a regular 6363-sided polygon, a regular 9191-sided polygon, and a regular 105105-sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have?
p16. Grace, a problem-writer, writes 99 out of 1515 questions on a test. A tester randomly selects 33 of the 1515 questions, without replacement, to solve. The probability that all 33 of the questions were written by Grace can be written in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm + n.
p17. Compute the number of anagrams of the letters in BMMTBMMTBMMTBMMT with no two MM's adjacent.
p18. From a 1515 inch by 1515 inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form a−bπa-b\pi, where aa and bb are positive integers. Compute a+ba + b.
p19. Bayus has 20212021 marbles in a bag. He wants to place them one by one into 99 different buckets numbered 11 through 99. He starts by putting the first marble in bucket 11, the second marble in bucket 22, the third marble in bucket 33, etc. After placing a marble in bucket 99, he starts back from bucket 11 again and repeats the process. In which bucket will Bayus place the last marble in the bag? https://cdn.artofproblemsolving.com/attachments/9/8/4c6b1bd07367101233385b3ffebc5e0abba596.png
p20. What is the remainder when 15+25+35+...+202151^5 + 2^5 + 3^5 +...+ 2021^5 is divided by 55?
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Ind. Round
1

2021 BmMT Individual Round - Berkley mini Math Tournament

p1. What is the largest number of five dollar footlongs Jimmy can buy with 88 dollars?
p2. Austin, Derwin, and Sylvia are deciding on roles for BMT 20212021. There must be a single Tournament Director and a single Head Problem Writer, but one person cannot take on both roles. In how many ways can the roles be assigned to Austin, Derwin, and Sylvia?
p3. Sofia has7 7 unique shirts. How many ways can she place 22 shirts into a suitcase, where the order in which Sofia places the shirts into the suitcase does not matter?
p4. Compute the sum of the prime factors of 20212021.
p5. A sphere has volume 36π36\pi cubic feet. If its radius increases by 100%100\%, then its volume increases by aπa\pi cubic feet. Compute aa.
p6. The full price of a movie ticket is $10\$10, but a matinee ticket to the same movie costs only 70%70\% of the full price. If 30%30\% of the tickets sold for the movie are matinee tickets, and the total revenue from movie tickets is $1001\$1001, compute the total number of tickets sold.
p7. Anisa rolls a fair six-sided die twice. The probability that the value Anisa rolls the second time is greater than or equal to the value Anisa rolls the first time can be expressed as mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n.
p8. Square ABCDABCD has side length AB=6AB = 6. Let point EE be the midpoint of BC‾\overline{BC}. Line segments AC‾\overline{AC} and DE‾\overline{DE} intersect at point FF. Compute the area of quadrilateral ABEF.
p9. Justine has a large bag of candy. She splits the candy equally between herself and her 44 friends, but she needs to discard three candies before dividing so that everyone gets an equal number of candies. Justine then splits her share of the candy between herself and her two siblings, but she needs to discard one candy before dividing so that she and her siblings get an equal number of candies. If Justine had instead split all of the candy that was originally in the large bag between herself and 1414 of her classmates, what is the fewest number of candies that she would need to discard before dividing so that Justine and her 1414 classmates get an equal number of candies?
p10. For some positive integers aa and bb, a2−b2=400a^2 - b^2 = 400. If aa is even, compute aa.
p11. Let ABCDEFGHIJKLABCDEFGHIJKL be the equilateral dodecagon shown below, and each angle is either 90o90^o or 270o270^o. Let MM be the midpoint of CD‾\overline{CD}, and suppose HM‾\overline{HM} splits the dodecagon into two regions. The ratio of the area of the larger region to the area of the smaller region can be expressed as mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n. https://cdn.artofproblemsolving.com/attachments/3/e/387bcdf2a6c39fcada4f21f24ceebd18a7f887.png
p12. Nelson, who never studies for tests, takes several tests in his math class. Each test has a passing score of 60/10060/100. Since Nelson's test average is at least 60/10060/100, he manages to pass the class. If only nonnegative integer scores are attainable on each test, and Nelson gets a di erent score on every test, compute the largest possible ratio of tests failed to tests passed. Assume that for each test, Nelson either passes it or fails it, and the maximum possible score for each test is 100.
p13. For each positive integer nn, let f(n)=nn+1+n+1nf(n) = \frac{n}{n+1} + \frac{n+1}{n} . Then f(1)+f(2)+f(3)+...+f(10)f(1)+f(2)+f(3)+...+f(10) can be expressed as mn\frac{m}{n} where mm and nn are relatively prime positive integers. Compute m+nm + n.
p14. Triangle △ABC\vartriangle ABC has point DD lying on line segment BC‾\overline{BC} between BB and CC such that triangle △ABD\vartriangle ABD is equilateral. If the area of triangle △ADC\vartriangle ADC is 14\frac14 the area of triangle △ABC\vartriangle ABC, then (ACAB)2\left( \frac{AC}{AB}\right)^2 can be expressed as mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n.
p15. In hexagon ABCDEFABCDEF, AB=60AB = 60, AF=40AF = 40, EF=20EF = 20, DE=20DE = 20, and each pair of adjacent edges are perpendicular to each other, as shown in the below diagram. The probability that a random point inside hexagon ABCDEFABCDEF is at least 20220\sqrt2 units away from point DD can be expressed in the form a−bπc\frac{a-b\pi}{c} , where aa, bb, cc are positive integers such that gcd(a,b,c)=1(a, b, c) = 1. Compute a+b+ca + b + c. https://cdn.artofproblemsolving.com/attachments/3/c/1b45470265d10a73de7b83eff1d3e3087d6456.png
p16. The equation x+20−x=20+20x−x2\sqrt{x} +\sqrt{20-x} =\sqrt{20 + 20x - x^2} has 44 distinct real solutions, x1x_1, x2x_2, x3x_3, and x4x_4. Compute x1+x2+x3+x4x_1 + x_2 + x_3 + x_4.
p17. How many distinct words with letters chosen from B,M,TB, M, T have exactly 1212 distinct permutations, given that the words can be of any length, and not all the letters need to be used? For example, the word BMMTBMMT has 1212 permutations. Two words are still distinct even if one is a permutation of the other. For example, BMMTBMMT is distinct from TMMBTMMB.
p18. We call a positive integer binary-okay if at least half of the digits in its binary (base 22) representation are 11's, but no two 11s are consecutive. For example, 1010=1010210_{10} = 1010_2 and 510=10125_{10} = 101_2 are both binary-okay, but 1610=10000216_{10} = 10000_2 and 1110=1011211_{10} = 1011_2 are not. Compute the number of binary-okay positive integers less than or equal to 20202020 (in base 1010).
p19. A regular octahedron (a polyhedron with 88 equilateral triangles) has side length 22. An ant starts on the center of one face, and walks on the surface of the octahedron to the center of the opposite face in as short a path as possible. The square of the distance the ant travels can be expressed as mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n. https://cdn.artofproblemsolving.com/attachments/f/8/3aa6abe02e813095e6991f63fbcf22f2e0431a.png
p20. The sum of 1a\frac{1}{a} over all positive factors aa of the number 360360 can be expressed in the form mn\frac{m}{n} ,where mm and nn are relatively prime positive integers. Compute m+nm + n.
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Ind. Tie
1

2021 BmMT Individual Tiebreaker Round - Berkley mini Math Tournament

p1. Isosceles trapezoid ABCDABCD has AB=2AB = 2, BC=DA=17BC = DA =\sqrt{17}, and CD=4CD = 4. Point EE lies on CD‾\overline{CD} such that AE‾\overline{AE} splits ABCDABCD into two polygons of equal area. What is DEDE?
p2. At the Berkeley Sandwich Parlor, the famous BMT sandwich consists of up to five ingredients between the bread slices. These ingredients can be either bacon, mayo, or tomato, and ingredients of the same type are indistiguishable. If there must be at least one of each ingredient in the sandwich, and the order in which the ingredients are placed in the sandwich matters, how many possible ways are there to prepare a BMT sandwich?
p3. Three mutually externally tangent circles have radii 22, 33, and 33. A fourth circle, distinct from the other three circles, is tangent to all three other circles. The sum of all possible radii of the fourth circle can be expressed as mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Team Round
1

2021 BmMT Team Round - Berkley mini Math Tournament Spring

p1. What is the area of a triangle with side lengths 6 6, 8 8, and 1010?
p2. Let f(n)=nf(n) = \sqrt{n}. If f(f(f(n)))=2f(f(f(n))) = 2, compute nn.
p3. Anton is buying AguaFina water bottles. Each bottle costs 1414 dollars, and Anton buys at least one water bottle. The number of dollars that Anton spends on AguaFina water bottles is a multiple of 1010. What is the least number of water bottles he can buy?
p4. Alex flips 33 fair coins in a row. The probability that the first and last flips are the same can be expressed in the form m/nm/n for relatively prime positive integers mm and nn. Compute m+nm + n.
p5. How many prime numbers pp satisfy the property that p2−1p^2 - 1 is not a multiple of 66?
p6. In right triangle △ABC\vartriangle ABC with AB=5AB = 5, BC=12BC = 12, and CA=13CA = 13, point DD lies on CA‾\overline{CA} such that AD=BDAD = BD. The length of CDCD can then be expressed in the form m/nm/n for relatively prime positive integers mm and nn. Compute m+nm + n.
p7. Vivienne is deciding on what courses to take for Spring 20212021, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Vivienne have?
p8. Square ABCDABCD has side length 22. Square ACEFACEF is drawn such that BB lies inside square ACEFACEF. Compute the area of pentagon AFECDAFECD.
p9. At the Boba Math Tournament, the Blackberry Milk Team has answered 44 out of the first 1010 questions on the Boba Round correctly. If they answer all pp remaining questions correctly, they will have answered exactly 9p5%\frac{9p}{5}\% of the questions correctly in total. How many questions are on the Boba Round?
p10. The sum of two positive integers is 20212021 less than their product. If one of them is a perfect square, compute the sum of the two numbers.
p11. Points EE and FF lie on edges BC‾\overline{BC} and DA‾\overline{DA} of unit square ABCDABCD, respectively, such that BE=13BE =\frac13 and DF=13DF =\frac13 . Line segments AE‾\overline{AE} and BF‾\overline{BF} intersect at point GG. The area of triangle EFGEFG can be written in the form m/nm/n , where mm and nn are relatively prime positive integers. Compute m+nm+n.
p12. Compute the number of positive integers n≤2020n \le 2020 for which nk+1n^{k+1} is a factor of (1+2+3+⋅⋅⋅+n)k(1+2+3+· · ·+n)^k for some positive integer kk.
p13. How many permutations of 123456123456 are divisible by their last digit? For instance, 123456123456 is divisible by 66, but 561234561234 is not divisible by 44.
p14. Compute the sum of all possible integer values for nn such that n2−2n−120n^2 - 2n - 120 is a positive prime number.
p15. Triangle △ABC\vartriangle ABC has AB=10AB =\sqrt{10}, BC=17BC =\sqrt{17}, and CA=41CA =\sqrt{41}. The area of △ABC\vartriangle ABC can be expressed in the form m/nm/n for relatively prime positive integers mm and nn. Compute m+nm + n.
p16. Let f(x)=1+x3+x101+x10f(x) = \frac{1 + x^3 + x^{10}}{1 + x^{10}} . Compute f(−20)+f(−19)+f(−18)+...+f(20)f(-20) + f(-19) + f(-18) + ...+ f(20).
p17. Leanne and Jing Jing are walking around the xyxy-plane. In one step, Leanne can move from any point (x,y)(x, y) to (x+1,y)(x + 1, y) or (x,y+1)(x, y + 1) and Jing Jing can move from (x,y)(x, y) to (x−2,y+5)(x - 2, y + 5) or (x+3,y−1)(x + 3, y - 1). The number of ways that Leanne can move from (0,0)(0, 0) to (20,20)(20, 20) is equal to the number of ways that Jing Jing can move from (0,0)(0, 0) to (a,b)(a, b), where a and b are positive integers. Compute the minimum possible value of a+ba + b.
p18. Compute the number positive integers 1<k<20211 < k < 2021 such that the equation x+kx=kx+xx +\sqrt{kx} = kx +\sqrt{x} has a positive rational solution for xx.
p19. In triangle △ABC\vartriangle ABC, point DD lies on BC‾\overline{BC} with AD‾⊥BC‾\overline{AD} \perp \overline{BC}. If BD=3ADBD = 3AD, and the area of △ABC\vartriangle ABC is 1515, then the minimum value of AC2AC^2 is of the form pq−rp\sqrt{q} - r, where p,qp, q, and rr are positive integers and qq is not divisible by the square of any prime number. Compute p+q+rp + q + r.
p20. Suppose the decimal representation of 1n\frac{1}{n} is in the form 0.p1p2...pjd1d2...dk‾0.p_1p_2...p_j\overline{d_1d_2...d_k}, where p1,...,pjp_1, ... , p_j , d1,...,dkd_1,... , d_k are decimal digits, and jj and kk are the smallest possible nonnegative integers (i.e. it’s possible for j=0j = 0 or k=0k = 0). We define the preperiod of 1n\frac{1}{n} to be jj and the period of 1n\frac{1}{n} to be kk. For example, 16=0.16666...\frac16 = 0.16666... has preperiod 11 and period 11, 17=0.142857‾\frac17 = 0.\overline{142857} has preperiod 00 and period 66, and 14=0.25\frac14 = 0.25 has preperiod 22 and period 00. What is the smallest positive integer nn such that the sum of the preperiod and period of 1n\frac{1}{n} is 8 8?
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