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2018 BmMT Team Round - Berkley mini Math Tournament Fall

Source:

January 18, 2022
algebrageometrycombinatoricsnumber theorybmmt

Problem Statement

p1. What is the sum of the first 1212 positive integers?
p2. How many positive integers less than or equal to 100100 are multiples of both 22 and 55?
p3. Alex has a bag with 44 white marbles and 44 black marbles. She takes 22 marbles from the bag without replacement. What is the probability that both marbles she took are black? Express your answer as a decimal or a fraction in lowest terms.
p4. How many 55-digit numbers are there where each digit is either 11 or 22?
p5. An integer aa with 1≤a≤101\le a \le 10 is randomly selected. What is the probability that 100a\frac{100}{a} is an integer? Express your answer as decimal or a fraction in lowest terms.
p6. Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let PP be the number of points of intersection between any two circles. How many possible values of PP are there?
p7. Let x,y,zx, y, z be nonzero real numbers such that x+y+z=xyzx + y + z = xyz. Compute 1+yzyz+1+xzxz+1+xyxy.\frac{1 + yz}{yz}+\frac{1 + xz}{xz}+\frac{1 + xy}{xy}.
p8. How many positive integers less than 106106 are simultaneously perfect squares, cubes, and fourth powers?
p9. Let C1C_1 and C2C_2 be two circles centered at point OO of radii 11 and 22, respectively. Let AA be a point on C2C_2. We draw the two lines tangent to C1C_1 that pass through AA, and label their other intersections with C2C_2 as BB and CC. Let x be the length of minor arc BCBC, as shown. Compute xx. https://cdn.artofproblemsolving.com/attachments/7/5/915216d4b7eba0650d63b26715113e79daa176.png
p10. A circle of area π\pi is inscribed in an equilateral triangle. Find the area of the triangle.
p11. Julie runs a 22 mile route every morning. She notices that if she jogs the route 22 miles per hour faster than normal, then she will finish the route 55 minutes faster. How fast (in miles per hour) does she normally jog?
p12. Let ABCDABCD be a square of side length 1010. Let EFGHEFGH be a square of side length 1515 such that EE is the center of ABCDABCD, EFEF intersects BCBC at XX, and EHEH intersects CDCD at YY (shown below). If BX=7BX = 7, what is the area of quadrilateral EXCYEXCY ? https://cdn.artofproblemsolving.com/attachments/d/b/2b2d6de789310036bc42d1e8bcf3931316c922.png
p13. How many solutions are there to the system of equations a2+b2=c2a^2 + b^2 = c^2 (a+1)2+(b+1)2=(c+1)2(a + 1)^2 + (b + 1)^2 = (c + 1)^2 if a,ba, b, and cc are positive integers?
p14. A square of side length s s is inscribed in a semicircle of radius r r as shown. Compute sr\frac{s}{r}. https://cdn.artofproblemsolving.com/attachments/5/f/22d7516efa240d00d6a9743a4dc204d23d190d.png
p15. SS is a collection of integers n with 1≤n≤501 \le n \le 50 so that each integer in SS is composite and relatively prime to every other integer in SS. What is the largest possible number of integers in SS?
p16. Let ABCDABCD be a regular tetrahedron and let W,X,Y,ZW, X, Y, Z denote the centers of faces ABCABC, BCDBCD, CDACDA, and DABDAB, respectively. What is the ratio of the volumes of tetrahedrons WXYZWXYZ and WAYZWAYZ? Express your answer as a decimal or a fraction in lowest terms.
p17. Consider a random permutation {s1,s2,...,s8}\{s_1, s_2, ... , s_8\} of {1,1,1,1,−1,−1,−1,−1}\{1, 1, 1, 1, -1, -1, -1, -1\}. Let SS be the largest of the numbers s1s_1, s1+s2s_1 + s_2, s1+s2+s3s_1 + s_2 + s_3, ...... , s1+s2+...+s8s_1 + s_2 + ... + s_8. What is the probability that SS is exactly 33? Express your answer as a decimal or a fraction in lowest terms.
p18. A positive integer is called almost-kinda-semi-prime if it has a prime number of positive integer divisors. Given that there are168are 168 primes less than 10001000, how many almost-kinda-semi-prime numbers are there less than 10001000?
p19. Let ABCDABCD be a unit square and let X,Y,ZX, Y, Z be points on sides ABAB, BCBC, CDCD, respectively, such that AX=BY=CZAX = BY = CZ. If the area of triangle XYZXYZ is 13\frac13 , what is the maximum value of the ratio XB/AXXB/AX? https://cdn.artofproblemsolving.com/attachments/5/6/cf77e40f8e9bb03dea8e7e728b21e7fb899d3e.png
p20. Positive integers a≤b≤ca \le b \le c have the property that each of a+ba + b, b+cb + c, and c+ac + a are prime. If a+b+ca + b + c has exactly 44 positive divisors, find the fourth smallest possible value of the product c(c+b)(c+b+a)c(c + b)(c + b + a).
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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