MathDB

2022 BmMT

Part of BmMT problems

Subcontests

(3)
Ind. Round
1

2022 BmMT Individual Round - Berkley mini Math Tournament

p1. Nikhil computes the sum of the first 1010 positive integers, starting from 11. He then divides that sum by 5. What remainder does he get?
p2. In class, starting at 8:008:00, Ava claps her hands once every 44 minutes, while Ella claps her hands once every 66 minutes. What is the smallest number of minutes after 8:008:00 such that both Ava and Ella clap their hands at the same time?
p3. A triangle has side lengths 33, 44, and 55. If all of the side lengths of the triangle are doubled, how many times larger is the area?
p4. There are 5050 students in a room. Every student is wearing either 00, 11, or 22 shoes. An even number of the students are wearing exactly 11 shoe. Of the remaining students, exactly half of them have 22 shoes and half of them have 00 shoes. How many shoes are worn in total by the 5050 students?
p5. What is the value of −2+4−6+8−...+8088-2 + 4 - 6 + 8 - ... + 8088?
p6. Suppose Lauren has 22 cats and 22 dogs. If she chooses 22 of the 44 pets uniformly at random, what is the probability that the 2 chosen pets are either both cats or both dogs?
p7. Let triangle △ABC\vartriangle ABC be equilateral with side length 66. Points EE and FF lie on BCBC such that EE is closer to BB than it is to CC and FF is closer to CC than it is to BB. If BE=EF=FCBE = EF = FC, what is the area of triangle △AFE\vartriangle AFE?
p8. The two equations x2+ax−4=0x^2 + ax - 4 = 0 and x2−4x+a=0x^2 - 4x + a = 0 share exactly one common solution for xx. Compute the value of aa.
p9. At Shreymart, Shreyas sells apples at a price cc. A customer who buys nn apples pays ncnc dollars, rounded to the nearest integer, where we always round up if the cost ends in .5.5. For example, if the cost of the apples is 4.24.2 dollars, a customer pays 44 dollars. Similarly, if the cost of the apples is 4.54.5 dollars, a customer pays 55 dollars. If Justin buys 77 apples for 33 dollars and 44 apples for 11 dollar, how many dollars should he pay for 2020 apples?
p10. In triangle △ABC\vartriangle ABC, the angle trisector of ∠BAC\angle BAC closer to AC‾\overline{AC} than AB‾\overline{AB} intersects BC‾\overline{BC} at DD. Given that triangle △ABD\vartriangle ABD is equilateral with area 11, compute the area of triangle △ABC\vartriangle ABC.
p11. Wanda lists out all the primes less than 100100 for which the last digit of that prime equals the last digit of that prime's square. For instance, 7171 is in Wanda's list because its square, 50415041, also has 11 as its last digit. What is the product of the last digits of all the primes in Wanda's list?
p12. How many ways are there to arrange the letters of SUSBUSSUSBUS such that SUSSUS appears as a contiguous substring? For example, SUSBUSSUSBUS and USSUSBUSSUSB are both valid arrangements, but SUBSSUSUBSSU is not.
p13. Suppose that xx and yy are integers such that x≥5x \ge 5, y≥3y \ge 3, and x−5+y−3=x+y\sqrt{x - 5} +\sqrt{y - 3} = \sqrt{x + y}. Compute the maximum possible value of xyxy.
p14. What is the largest integer kk divisible by 1414 such that x2−100x+k=0x^2-100x+k = 0 has two distinct integer roots?
p15. What is the sum of the first 1616 positive integers whose digits consist of only 00s and 11s?
p16. Jonathan and Ajit are flipping two unfair coins. Jonathan's coin lands on heads with probability 120\frac{1}{20} while Ajit's coin lands on heads with probability 122\frac{1}{22} . Each year, they flip their coins at thesame time, independently of their previous flips. Compute the probability that Jonathan's coin lands on heads strictly before Ajit's coin does.
p17. A point is chosen uniformly at random in square ABCDABCD. What is the probability that it is closer to one of the 44 sides than to one of the 22 diagonals?
p18. Two integers are coprime if they share no common positive factors other than 11. For example, 33 and 55 are coprime because their only common factor is 11. Compute the sum of all positive integers that are coprime to 198198 and less than 198198.
p19. Sumith lists out the positive integer factors of 1212 in a line, writing them out in increasing order as 11, 22, 33, 44, 66, 1212. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as a1a_1, a2a_2, a3a_3, a4a_4, a5a_5, a6a_6. Luke then calculates gcd(a1,2a2,3a3,4a4,6a5,12a6).gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6). Given that Luke's result is greater than 11, how many possible permutations could he have written?
p20. Tetrahedron ABCDABCD is drawn such that DA=DB=DC=2DA = DB = DC = 2, ∠ADB=∠ADC=90o\angle ADB = \angle ADC = 90^o, and ∠BDC=120o\angle BDC = 120^o. Compute the radius of the sphere that passes through AA, BB, CC, and DD.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Pacer Round
1

2022 BmMT Pacer Round - Berkley mini Math Tournament

p1. Frankie the frog likes to hop. On his first hop, he hops 11 meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops 22 meters, and on his third hop, he hops 44 meters. How many meters, in total, has he travelled after 66 hops?
p2. Anton flips 55 fair coins. The probability that he gets an odd number of heads can be written in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm + n.
p3. April discovers that the quadratic polynomial x2+5x+3x^2 + 5x + 3 has distinct roots aa and bb. She also discovers that the quadratic polynomial x2+7x+4x^2 + 7x + 4 has distinct roots cc and dd. Compute ac+bc+bd+ad+a+b.ac + bc + bd + ad + a + b.
p4. A rectangular picture frame that has a 22 inch border can exactly fit a 1010 by 77 inch photo. What is the total area of the frame's border around the photo, in square inches?
p5. Compute the median of the positive divisors of 99999999.
p6. Kaity only eats bread, pizza, and salad for her meals. However, she will refuse to have salad if she had pizza for the meal right before. Given that she eats 33 meals a day (not necessarily distinct), in how many ways can we arrange her meals for the day?
p7. A triangle has side lengths 33, 44, and xx, and another triangle has side lengths 33, 44, and 2x2x. Assuming both triangles have positive area, compute the number of possible integer values for xx.
p8. In the diagram below, the largest circle has radius 3030 and the other two white circles each have a radius of 1515. Compute the radius of the shaded circle. https://cdn.artofproblemsolving.com/attachments/c/1/9eaf1064b2445edb15782278fc9c6efd1440b0.png
p9. What is the remainder when 20222022 is divided by 99?
p10. For how many positive integers xx less than 20222022 is x3−x2+x−1x^3 - x^2 + x - 1 prime?
p11. A sphere and cylinder have the same volume, and both have radius 1010. The height of the cylinder can be written in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm + n.
p12. Amanda, Brianna, Chad, and Derrick are playing a game where they pass around a red flag. Two players "interact" whenever one passes the flag to the other. How many different ways can the flag be passed among the players such that (1) each pair of players interacts exactly once, and (2) Amanda both starts and ends the game with the flag?
p13. Compute the value of 121+121+121+...\dfrac{12}{1 + \dfrac{12}{1+ \dfrac{12}{1+...}}}
p14. Compute the sum of all positive integers aa such that a2−505a^2 - 505 is a perfect square.
p15. Alissa, Billy, Charles, Donovan, Eli, Faith, and Gerry each ask Sara a question. Sara must answer exactly 55 of them, and must choose an order in which to answer the questions. Furthermore, Sara must answer Alissa and Billy's questions. In how many ways can Sara complete this task?
p16. The integers −x-x, x2−1x^2 - 1, and x3x3 form a non-decreasing arithmetic sequence (in that order). Compute the sum of all possible values of x3x^3.
p17. Moor and his 33 other friends are trying to split burgers equally, but they will have 22 left over. If they find another friend to split the burgers with, everyone can get an equal amount. What is the fewest number of burgers that Moor and his friends could have started with?
p18. Consider regular dodecagon ABCDEFGHIJKLABCDEFGHIJKL below. The ratio of the area of rectangle AFGLAFGL to the area of the dodecagon can be written in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm + n. https://cdn.artofproblemsolving.com/attachments/8/3/c38c10a9b2f445faae397d8a7bc4c8d3ed0290.png
p19. Compute the remainder when 34563^{4^{5^6}} is divided by 44.
p20. Fred is located at the middle of a 99 by 1111 lattice (diagram below). At every second, he randomly moves to a neighboring point (left, right, up, or down), each with probability 1/41/4. The probability that he is back at the middle after exactly 44 seconds can be written in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm + n.
https://cdn.artofproblemsolving.com/attachments/7/c/f8e092e60f568ab7b28964d23b2ee02cdba7ad.png
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Team Round
1

2022 BmMT Team Round - Berkley mini Math Tournament Spring

p1. If x2=7x^2 = 7, what is x4+x2+1x^4 + x^2 + 1?
p2. Richard and Alex are competing in a 150150-meter race. If Richard runs at a constant speed of 55 meters per second and Alex runs at a constant speed of 33 meters per second, how many more seconds does it take for Alex to finish the race?
p3. David and Emma are playing a game with a chest of 100100 gold coins. They alternate turns, taking one gold coin if the chest has an odd number of gold coins or taking exactly half of the gold coins if the chest has an even number of gold coins. The game ends when there are no more gold coins in the chest. If Emma goes first, how many gold coins does Emma have at the end?
p4. What is the only 33-digit perfect square whose digits are all different and whose units digit is 55?
p5. In regular pentagon ABCDEABCDE, let FF be the midpoint of AB‾\overline{AB}, GG be the midpoint of CD‾\overline{CD}, and HH be the midpoint of AE‾\overline{AE}. What is the measure of ∠FGH\angle FGH in degrees?
p6. Water enters at the left end of a pipe at a rate of 11 liter per 3535 seconds. Some of the water exits the pipe through a leak in the middle. The rest of the water exits from the right end of the pipe at a rate of 11 liter per 3636 seconds. How many minutes does it take for the pipe to leak a liter of water?
p7. Carson wants to create a wire frame model of a right rectangular prism with a volume of 20222022 cubic centimeters, where strands of wire form the edges of the prism. He wants to use as much wire as possible. If Carson also wants the length, width, and height in centimeters to be distinct whole numbers, how many centimeters of wire does he need to create the prism?
p8. How many ways are there to fill the unit squares of a 3×53 \times 5 grid with the digits 11, 22, and 33 such that every pair of squares that share a side differ by exactly 11?
p9. In pentagon ABCDE, AB=54AB = 54, AE=45AE = 45, DE=18DE = 18, ∠A=∠C=∠E\angle A = \angle C = \angle E, DD is on line segment BE‾\overline{BE}, and line BDBD bisects angle ∠ABC\angle ABC, as shown in the diagram below. What is the perimeter of pentagon ABCDEABCDE? https://cdn.artofproblemsolving.com/attachments/2/0/7c25837bb10b128a1c7a292f6ce8ce3e64b292.png
p10. If xx and yy are nonzero real numbers such that 7x+8y=91\frac{7}{x} + \frac{8}{y} = 91 and 6x+10y=89\frac{6}{x} + \frac{10}{y} = 89, what is the value of x+yx + y?
p11. Hilda and Marianne play a game with a shued deck of 1010 cards, numbered from 11 to 1010. Hilda draws five cards, and Marianne picks up the five remaining cards. Hilda observes that she does not have any pair of consecutive cards - that is, no two cards have numbers that differ by exactly 11. Additionally, the sum of the numbers on Hilda's cards is 11 less than the sum of the numbers on Marianne's cards. Marianne has exactly one pair of consecutive cards - what is the sum of this pair?
p12. Regular hexagon AUSTINAUSTIN has side length 22. Let MM be the midpoint of line segment ST‾\overline{ST}. What is the area of pentagon MINUSMINUS?
p13. At a collector's store, plushes are either small or large and cost a positive integer number of dollars. All small plushes cost the same price, and all large plushes cost the same price. Two small plushes cost exactly one dollar less than a large plush. During a shopping trip, Isaac buys some plushes from the store for 59 dollars. What is the smallest number of dollars that the small plush could not possibly cost?
p14. Four fair six-sided dice are rolled. What is the probability that the median of the four outcomes is 55?
p15. Suppose x1,x2,...,x2022x_1, x_2,..., x_{2022} is a sequence of real numbers such that: x1+x2=1x_1 + x_2 = 1 x2+x3=2x_2 + x_3 = 2 ...... x2021+x2022=2021x_{2021} + x_{2022} = 2021 If x1+x499+x999+x1501=222x_1 + x_{499} + x_{999} + x_{1501} = 222, then what is the value of x2022x_{2022}?
p16. A cone has radius 33 and height 44. An infinite number of spheres are placed in the cone in the following way: sphere C0C_0 is placed inside the cone such that it is tangent to the base of the cone and to the curved surface of the cone at more than one point, and for i≥1i \ge 1, sphere CiC_i is placed such that it is externally tangent to sphere Ci−1C_{i-1} and internally tangent to more than one point of the curved surface of the cone. If ViV_i is the volume of sphere CiC_i, compute V0+V1+V2+...V_0 + V_1 + V_2 + ... . https://cdn.artofproblemsolving.com/attachments/b/4/b43e40bb0a5974dd9d656691c14b4ae268b5b5.png
p17. Call an ordered pair, (x,y)(x, y), relatable if xx and yy are positive integers where yy divides 36003600, xx divides yy and yx\frac{y}{x} is a prime number. For every relatable ordered pair, Leanne wrote down the positive difference of the two terms of the pair. What is the sum of the numbers she wrote down?
p18. Let r,sr, s, and tt be the three roots of P(x)=x3−9x−9P(x) = x^3 - 9x - 9. Compute the value of (r3+r2−10r−8)(s3+s2−10s−8)(t3+t2−10t−8)(r^3 + r^2 - 10r - 8)(s^3 + s^2 - 10s - 8)(t^3 + t^2 - 10t - 8).
p19. Compute the number of ways to color the digits 0,1,2,3,4,5,6,7,80, 1, 2, 3, 4, 5, 6, 7, 8 and 99 red, blue, or green such that: (a) every prime integer has at least one digit that is not blue, and (b) every composite integer has at least one digit that is not green.
Note that 00 is not composite. For example, since 1212 is composite, either the digit 11, the digit 22, or both must be not green.
p20. Pentagon ABCDEABCDE has AB=DE=4AB = DE = 4 and BC=CD=9BC = CD = 9 with ∠ABC=∠CDE=90o\angle ABC = \angle CDE = 90^o, and there exists a circle tangent to all five sides of the pentagon. What is the length of segment AE‾\overline{AE}?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.