MathDB

2022

Part of MBMT Guts Rounds

Problems(3)

2022 MBMT Guts Round D1-15/ Z1-8 Montgomery Blair Math Tournament

Source:

9/1/2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
Set 1
D1 / Z1. What is 1+2⋅31 + 2 \cdot 3?
D2. What is the average of the first 99 positive integers?
D3 / Z2. A square of side length 22 is cut into 44 congruent squares. What is the perimeter of one of the 44 squares?
D4. Find the ratio of a circle’s circumference squared to the area of the circle.
D5 / Z3. 66 people split a bag of cookies such that they each get 2121 cookies. Kyle comes and demands his share of cookies. If the 77 people then re-split the cookies equally, how many cookies does Kyle get?
Set 2
D6. How many prime numbers are perfect squares?
D7. Josh has an unfair 44-sided die numbered 11 through 44. The probability it lands on an even number is twice the probability it lands on an odd number. What is the probability it lands on either 11 or 33?
D8. If Alice consumes 10001000 calories every day and burns 500500 every night, how many days will it take for her to first reach a net gain of 50005000 calories?
D9 / Z4. Blobby flips 44 coins. What is the probability he sees at least one heads and one tails?
D10. Lillian has nn jars and 4848 marbles. If George steals one jar from Lillian, she can fill each jar with 88 marbles. If George steals 33 jars, Lillian can fill each jar to maximum capacity. How many marbles can each jar fill?
Set 3
D11 / Z6. How many perfect squares less than 100100 are odd?
D12. Jash and Nash wash cars for cash. Jash gets $6\$6 for each car, while Nash gets $11\$11 per car. If Nash has earned $1\$1 more than Jash, what is the least amount of money that Nash could have earned?
D13 / Z5. The product of 1010 consecutive positive integers ends in 33 zeros. What is the minimum possible value of the smallest of the 1010 integers?
D14 / Z7. Guuce continually rolls a fair 66-sided dice until he rolls a 11 or a 66. He wins if he rolls a 66, and loses if he rolls a 11. What is the probability that Guuce wins?
D15 / Z8. The perimeter and area of a square with integer side lengths are both three digit integers. How many possible values are there for the side length of the square?
PS. You should use hide for answers. D.16-30/Z.9-14, 17, 26-30 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMTalgebrageometrycombinatoricsnumber theory
2022 MBMT Guts Round Z15-25 Montgomery Blair Math Tournament

Source:

9/1/2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
Z15. Let AOBAOB be a quarter circle with center OO and radius 44. Let ω1\omega_1 and ω2\omega_2 be semicircles inside AOBAOB with diameters OAOA and OBOB, respectively. Find the area of the region within AOBAOB but outside of ω1\omega_1 and ω2\omega_2.
Set 4
Z16. Integers a,b,ca, b, c form a geometric sequence with an integer common ratio. If c=a+56c = a + 56, find bb.
Z17 / D24. In parallelogram ABCDABCD, ∠A⋅∠C−∠B⋅∠D=720o\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o where all angles are in degrees. Find the value of ∠C\angle C.
Z18. Steven likes arranging his rocks. A mountain formation is where the sequence of rocks to the left of the tallest rock increase in height while the sequence of rocks to the right of the tallest rock decrease in height. If his rocks are 1,2,...,101, 2, . . . , 10 inches in height, how many mountain formations are possible? For example: the sequences (1−3−5−6−10−9−8−7−4−2)(1-3-5-6-10-9-8-7-4-2) and (1−2−3−4−5−6−7−8−9−10)(1-2-3-4-5-6-7-8-9-10) are considered mountain formations.
Z19. Find the smallest 55-digit multiple of 1111 whose sum of digits is 1515.
Z20. Two circles, ω1\omega_1 and ω2\omega_2, have radii of 22 and 88, respectively, and are externally tangent at point PP. Line ℓ\ell is tangent to the two circles, intersecting ω1\omega_1 at AA and ω2\omega_2 at BB. Line mm passes through PP and is tangent to both circles. If line mm intersects line ℓ\ell at point QQ, calculate the length of PQP Q.
Set 5
Z21. Sen picks a random 11 million digit integer. Each digit of the integer is placed into a list. The probability that the last digit of the integer is strictly greater than twice the median of the digit list is closest to 1a\frac{1}{a}, for some integer aa. What is aa?
Z22. Let 66 points be evenly spaced on a circle with center OO, and let SS be a set of 77 points: the 66 points on the circle and OO. How many equilateral polygons (not self-intersecting and not necessarily convex) can be formed using some subset of SS as vertices?
Z23. For a positive integer nn, define rnr_n recursively as follows: rn=rn−12+rn−22+...+r02r_n = r^2_{n-1} + r^2_{n-2} + ... + r^2_0,where r0=1r_0 = 1. Find the greatest integer less than r2r12+r3r22+...+r2023r20222.\frac{r_2}{r^2_1}+\frac{r_3}{r^2_2}+ ...+\frac{r_{2023}}{r^2_{2022}}.
Z24. Arnav starts at 2121 on the number line. Every minute, if he was at nn, he randomly teleports to 2n22n^2, n2n^2, or n24\frac{n^2}{4} with equal chance. What is the probability that Arnav only ever steps on integers?
Z25. Let ABCDABCD be a rectangle inscribed in circle ω\omega with AB=10AB = 10. If PP is the intersection of the tangents to ω\omega at CC and DD, what is the minimum distance from PP to ABAB?
PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here and D.16-30/Z.9-14, 17, 26-30 [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMTalgebrageometrycombinatoricsnumber theory
2022 MBMT Guts Round D16-30/ Z9-14,17,26-30 Montgomery Blair Math Tournament

Source:

9/1/2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
Set 4
D16. The cooking club at Blair creates 1414 croissants and 2121 danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes?
D17. Each digit in a 33 digit integer is either 1,21, 2, or 44 with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit?
D18 / Z11. How many two digit numbers are there such that the product of their digits is prime?
D19 / Z9. In the coordinate plane, a point is selected in the rectangle defined by −6≤x≤4-6 \le x \le 4 and −2≤y≤8-2 \le y \le 8. What is the largest possible distance between the point and the origin, (0,0)(0, 0)?
D20 / Z10. The sum of two numbers is 66 and the sum of their squares is 3232. Find the product of the two numbers.
Set 5
D21 / Z12. Triangle ABCABC has area 44 and AB‾=4\overline{AB} = 4. What is the maximum possible value of ∠ACB\angle ACB?
D22 / Z13. Let ABCDABCD be an iscoceles trapezoid with AB=CDAB = CD and M be the midpoint of ADAD. If △ABM\vartriangle ABM and △MCD\vartriangle MCD are equilateral, and BC=4BC = 4, find the area of trapezoid ABCDABCD.
D23 / Z14. Let xx and yy be positive real numbers that satisfy (x2+y2)2=y2(x^2 + y^2)^2 = y^2. Find the maximum possible value of xx.
D24 / Z17. In parallelogram ABCDABCD, ∠A⋅∠C−∠B⋅∠D=720o\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o where all angles are in degrees. Find the value of ∠C\angle C.
D25. The number 12ab987654312ab9876543 is divisible by 101101, where a,ba, b represent digits between 00 and 99. What is 10a+b10a + b?
Set 6
D26 / Z26. For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get nn. Estimate the greatest integer aa such that 2a2^a evenly divides nn.
D27 / Z27. Circles of radius 55 are centered at each corner of a square with side length 66. If a random point PP is chosen randomly inside the square, what is the probability that PP lies within all four circles?
D28 / Z28. Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s 44th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class?
D29 / Z29. Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are 1010 meters west from a roast turkey. Beard, can turn exactly 0.7o0.7^o and Bored can turn exactly 0.2o0.2^o degrees. Driving at a consistent 22 meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey. Suppose Beard gets to the Turkey in about 818.5818.5 seconds. Estimate the amount of time it will take Bored.
D30 / Z30. Let a be the probability that 44 randomly chosen positive integers have no common divisor except for 11. Estimate 300a300a. Note that the integers 1,2,3,41, 2, 3, 4 have no common divisor except for 11.
Remark. This problem is asking you to find 300lim⁡n→∞an300 \lim_{n\to \infty} a_n, if ana_n is defined to be the probability that 44 randomly chosen integers from {1,2,...,n}\{1, 2, ..., n\} have greatest common divisor 11.
PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MBMTalgebrageometrycombinatoricsnumber theory