MathDB

MBMT Guts Rounds

Part of Montgomery Blair

Subcontests

(36)
2023
2

2023 MBMT Guts Round B1-15, G1-10 Montgomery Blair Math Tournament

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names
Set 1
B1 / G1 Find 203+22+3120^3 + 2^2 + 3^1.
B2 A piece of string of length 1010 is cut 44 times into strings of equal length. What is the length of each small piece of string?
B3 / G2 What is the smallest perfect square that is also a perfect cube?
B4 What is the probability a 55-sided die with sides labeled from 11 through 55 rolls an odd number?
B5 / G3 Hanfei spent 1414 dollars on chicken nuggets at McDonalds. 44 nuggets cost 33 dollars, 66 nuggets cost 44 dollars, and 1212 nuggets cost 99 dollars. How many chicken nuggets did Hanfei buy?
Set 2
B6 What is the probability a randomly chosen positive integer less than or equal to 1515 is prime?
B7 Andrew flips a fair coin with sides labeled 0 and 1 and also rolls a fair die with sides labeled 11 through 66. What is the probability that the sum is greater than 55?
B8 / G4 What is the radius of a circle with area 44?
B9 What is the maximum number of equilateral triangles on a piece of paper that can share the same corner?
B10 / G5 Bob likes to make pizzas. Bab also likes to make pizzas. Bob can make a pizza in 2020 minutes. Bab can make a pizza in 3030 minutes. If Bob and Bab want to make 5050 pizzas in total, how many hours would that take them?
Set 3
B11 / G6 Find the area of an equilateral rectangle with perimeter 2020.
B12 / G7 What is the minimum possible number of divisors that the sum of two prime numbers greater than 22 can have?
B13 / G8 Kwu and Kz play rock-paper-scissors-dynamite, a variant of the classic rock-paperscissors in which dynamite beats rock and paper but loses to scissors. The standard rock-paper-scissors rules apply, where rock beats scissors, paper beats rock, and scissors beats paper. If they throw out the same option, they keep playing until one of them wins. If Kz randomly throws out one of the four options with equal probability, while Kwu only throws out dynamite, what is the probability Kwu wins?
B14 / G9 Aven has 44 distinct baguettes in a bag. He picks three of the bagged baguettes at random and lays them on a table in random order. How many possible orderings of three baguettes are there on the table?
B15 / G10 Find the largest 77-digit palindrome that is divisible by 1111.
PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132170p28376644]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2023 MBMT Guts Round B16-30 G11-30 Montgomery Blair Math Tournament

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names
Set 4
B16 / G11 Let triangle ABCABC be an equilateral triangle with side length 66. If point DD is on ABAB and point EE is on BCBC, find the minimum possible value of AD+DE+CEAD + DE + CE.
B17 / G12 Find the smallest positive integer nn with at least seven divisors.
B18 / G13 Square AA is inscribed in a circle. The circle is inscribed in Square BB. If the circle has a radius of 1010, what is the ratio between a side length of Square AA and a side length of Square BB?
B19 / G14 Billy Bob has 55 distinguishable books that he wants to place on a shelf. How many ways can he order them if he does not want his two math books to be next to each other?
B20 / G15 Six people make statements as follows: Person 11 says “At least one of us is lying.” Person 22 says “At least two of us are lying.” Person 33 says “At least three of us are lying.” Person 44 says “At least four of us are lying.” Person 55 says “At least five of us are lying.” Person 66 says “At least six of us are lying.” How many are lying?
Set 5
B21 / G16 If xx and yy are between 00 and 11, find the ordered pair (x,y)(x, y) which maximizes (xy)2(x2−y2)(xy)^2(x^2 - y^2).
B22 / G17 If I take all my money and divide it into 1212 piles, I have 1010 dollars left. If I take all my money and divide it into 1313 piles, I have 1111 dollars left. If I take all my money and divide it into 1414 piles, I have 1212 dollars left. What’s the least amount of money I could have?
B23 / G18 A quadratic equation has two distinct prime number solutions and its coefficients are integers that sum to a prime number. Find the sum of the solutions to this equation.
B24 / G20 A regular 1212-sided polygon is inscribed in a circle. Gaz then chooses 33 vertices of the polygon at random and connects them to form a triangle. What is the probability that this triangle is right?
B25 / G22 A book has at most 77 chapters, and each chapter is either 33 pages long or has a length that is a power of 22 (including 11). What is the least positive integer nn for which the book cannot have nn pages?
Set 6
B26 / G26 What percent of the problems on the individual, team, and guts rounds for both divisions have integer answers?
B27 / G27 Estimate 12345112312345^{\frac{1}{123}}.
B28 / G28 Let OO be the center of a circle ω\omega with radius 33. Let A,B,CA, B, C be randomly selected on ω\omega. Let MM, NN be the midpoints of sides BCBC, CACA, and let AMAM, BNBN intersect at GG. What is the probability that OG≤1OG \le 1?
B29 / G29 Let r(a,b)r(a, b) be the remainder when aa is divided by bb. What is ∑i=1100r(2i,i)\sum^{100}_{i=1} r(2^i , i)?
B30 / G30 Bongo flips 20232023 coins. Call a run of heads a sequence of consecutive heads. Say a run is maximal if it isn’t contained in another run of heads. For example, if he gets HHHTTHTTHHHHTHHHHT T HT T HHHHT H, he’d have maximal runs of length 3,1,4,13, 1, 4, 1. Bongo squares the lengths of all his maximal runs and adds them to get a number MM. What is the expected value of MM?
- - - - - -
G19 Let ABCDABCD be a square of side length 22. Let MM be the midpoint of ABAB and NN be the midpoint of ADAD. Let the intersection of BNBN and CMCM be EE. Find the area of quadrilateral NECDNECD.
G21 Quadrilateral ABCDABCD has ∠A=∠D=60o\angle A = \angle D = 60^o. If AB=8AB = 8, CD=10CD = 10, and BC=3BC = 3, what is length ADAD?
G23 △ABC\vartriangle ABC is an equilateral triangle of side length xx. Three unit circles ωA\omega_A, ωB\omega_B, and ωC\omega_C lie in the plane such that ωA\omega_A passes through AA while ωB\omega_B and ωC\omega_C are centered at BB and CC, respectively. Given that ωA\omega_A is externally tangent to both ωB\omega_B and ωC\omega_C, and the center of ωA\omega_A is between point AA and line BC‾\overline{BC}, find xx.
G24 For some integers nn, the quadratic function f(x)=x2−(6n−6)x−(n2−12n+12)f(x) = x^2 - (6n - 6)x - (n^2 - 12n + 12) has two distinct positive integer roots, exactly one out of which is a prime and at least one of which is in the form 2k2^k for some nonnegative integer kk. What is the sum of all possible values of nn?
G25 In a triangle, let the altitudes concur at HH. Given that AH=30AH = 30, BH=14BH = 14, and the circumradius is 2525, calculate CHCH
PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132167p28376626]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2022
3

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

[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.

2022 MBMT Guts Round Z15-25 Montgomery Blair Math Tournament

[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.

2022 MBMT Guts Round D16-30/ Z9-14,17,26-30 Montgomery Blair Math Tournament

[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.
2019
3

2019 MBMT Guts Round D16-30/ L10-15 Montgomery Blair Math Tournament

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
Set 4
D.16 / L.6 Alex has 100100 Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most 11 spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them?
D.17 I start with the number 11 in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket?
D.18 / L.12 Point PP inside square ABCDABCD is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area 2525, 2525, and 1515, what are all the possible values for the area of the fourth triangle?
D.19 Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at 100100 or more decibels, then they lose the game. Mr. Stein yells first, at 8888 decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win?
D.20 / L.15 A semicircle of radius 11 has line ℓ\ell along its base and is tangent to line mm. Let rr be the radius of the largest circle tangent to ℓ\ell, mm, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of rr is the interval [a,b][a, b]. Find b−ab - a.
Set 5
D.21 / L.14 Hungryman starts at the tile labeled “SS”. On each move, he moves 11 unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats.
https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png
D.22 / L.11 How many triples of nonnegative integers (x,y,z)(x, y, z) satisfy the equation 6x+10y+15z=3006x + 10y +15z = 300?
D.23 / L.16 Anson, Billiam, and Connor are looking at a 3D3D figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a 5×55 \times 5 square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure.
D.24 / L.13 Tse and Cho are playing a game. Cho chooses a number x∈[0,1]x \in [0, 1] uniformly at random, and Tse guesses the value of x(1−x)x(1 - x). Tse wins if his guess is at most 150\frac{1}{50} away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins?
D.25 / L.20 Find the largest solution to the equation 2019(x2019x2019−20192+2019)2019)=2019x2019+1.2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.
Set 6
This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.
D.26 / L.26 What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were 4747 volunteers; this is the fifth MBMT.
D.27 / L.27 William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than 10%10\% of the entire chocolate bar.
D.28 / L.28 Let xx be the positive solution to the equation xxxx=1.1x^{x^{x^x}}= 1.1. Estimate 1x−1\frac{1}{x-1}.
D.29 / L.29 Estimate the number of dots in the following box: https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png It may be useful to know that this image was produced by plotting (4x,y)(4\sqrt{x}, y) some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval [0,1][0, 1].
D.30 / L.30 For a positive integer nn, let f(n)f(n) be the smallest prime greater than or equal to nn. Estimate (f(1)−1)+(f(2)−2)+(f(3)−3)+...+(f(10000)−10000).(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).
For 26≤i≤3026 \le i \le 30, let EiE_i be your team’s answer to problem ii and let AiA_i be the actual answer to problem ii. Your score SiS_i for problem ii is given by S26=max⁡(0,12−∣E26−A26∣/5)S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5) S27=max⁡(0,12−100∣E27−A27∣)S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|) S28=max⁡(0,12−5∣E28−A28∣))S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|)) S29=12max⁡(0,1−3∣E29−A29∣A29)S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right) S30=max⁡(0,12−∣E30−A30∣/2000)S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)
PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2019 MBMT Guts Round D1-15/ L1-9 Montgomery Blair Math Tournament

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
Set 1
D.1 / L.1 Find the units digit of 313373^{1^{3^{3^7}}}.
D.2 Find the positive solution to the equation x3−x2=x−1x^3 - x^2 = x - 1.
D.3 Points AA and BB lie on a unit circle centered at O and are distance 11 apart. What is the degree measure of ∠AOB\angle AOB?
D.4 A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between 11 and 20192019, inclusive?
D.5 Ted has four children of ages 1010, 1212, 1515, and 1717. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now?
Set 2
D.6 Mr. Schwartz is on the show Wipeout, and is standing on the first of 55 balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is 1/21/2, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish.
D.7 / L. 5 Kevin has written 55 MBMT questions. The shortest question is 55 words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question?
D.8 / L. 3 Square ABCDABCD with side length 11 is rolled into a cylinder by attaching side ADAD to side BCBC. What is the volume of that cylinder?
D.9 / L.4 Haydn is selling pies to Grace. He has 44 pumpkin pies, 33 apple pies, and 11 blueberry pie. If Grace wants 33 pies, how many different pie orders can she have?
D.10 Daniel has enough dough to make 88 1212-inch pizzas and 1212 88-inch pizzas. However, he only wants to make 1010-inch pizzas. At most how many 1010-inch pizzas can he make?
Set 3
D.11 / L.2 A standard deck of cards contains 1313 cards of each suit (clubs, diamonds, hearts, and spades). After drawing 5151 cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs?
D.12 / L. 7 Let s(n)s(n) be the sum of the digits of nn. Let g(n)g(n) be the number of times s must be applied to n until it has only 11 digit. Find the smallest n greater than 20192019 such that g(n)≠g(n+1)g(n) \ne g(n + 1).
D.13 / L. 8 In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the 20192019 contestants, is secretly told that her score is SS. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of SS?
D.14 / L. 9 Let AA and BB be opposite vertices on a cube with side length 11, and let XX be a point on that cube. Given that the distance along the surface of the cube from AA to XX is 11, find the maximum possible distance along the surface of the cube from BB to XX.
D.15 A function ff with f(2)>0f(2) > 0 satisfies the identity f(ab)=f(a)+f(b)f(ab) = f(a) + f(b) for all a,b>0a, b > 0. Compute f(22019)f(23)\frac{f(2^{2019})}{f(23)}.

PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2019 MBMT Guts Round L10,16-30 Montgomery Blair Math Tournament

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
L.10 Given the following system of equations where x,y,zx, y, z are nonzero, find x2+y2+z2x^2 + y^2 + z^2. x+2y=xyx + 2y = xy 3y+z=yz3y + z = yz 3x+2z=xz3x + 2z = xz
Set 4
L.16 / D.23 Anson, Billiam, and Connor are looking at a 3D3D figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a 5×55 \times 5 square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure.
L.17 The repeating decimal 0.MBMT‾0.\overline{MBMT} is equal to pq\frac{p}{q}, where pp and qq are relatively prime positive integers, and M,B,TM, B, T are distinct digits. Find the minimum value of qq.
L.18 Annie, Bob, and Claire have a bag containing the numbers 1,2,3,...,91, 2, 3, . . . , 9. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so 123123, 213213, and 321321 all count as arithmetic sequences.)
L.19 Consider a set SS of positive integers. Define the operation f(S)f(S) to be the smallest integer n>1n > 1 such that the base 2k2^k representation of nn consists only of ones and zeros for all k∈Sk \in S. Find the size of the largest set SS such that f(S)<22019f(S) < 2^{2019}.
L.20 / D.25 Find the largest solution to the equation 2019(x2019x2019−20192+2019)2019=2019x2019+1.2019(x^{2019x^{2019}-2019^2+2019})^{2019} = 2019^{x^{2019}+1}.
Set 5
L.21 Steven is concerned about his artistic abilities. To make himself feel better, he creates a 100×100100 \times 100 square grid and randomly paints each square either white or black, each with probability 12\frac12. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png
L.22 Let x be chosen uniformly at random from [0,1][0, 1]. Let n be the smallest positive integer such that 3nx3^n x is at most 14\frac14 away from an integer. What is the expected value of nn?
L.23 Let AA and BB be two points in the plane with AB=1AB = 1. Let ℓ\ell be a variable line through AA. Let ℓ′\ell' be a line through BB perpendicular to ℓ\ell. Let X be on ℓ\ell and YY be on ℓ′\ell' with AX=BY=1AX = BY = 1. Find the length of the locus of the midpoint of XYXY .
L.24 Each of the numbers aia_i, where 1≤i≤n1 \le i \le n, is either −1-1 or 11. Also, a1a2a3a4+a2a3a4a5+...+an−3an−2an−1an+an−2an−1ana1+an−1ana1a2+ana1a2a3=0.a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0. Find the number of possible values for nn between 44 and 100100, inclusive.
L.25 Let SS be the set of positive integers less than 320193^{2019} that have only zeros and ones in their base 33 representation. Find the sum of the squares of the elements of SS. Express your answer in the form ab(cd−1)(ef−1)a^b(c^d - 1)(e^f - 1), where a,b,c,d,e,fa, b, c, d, e, f are positive integers and a,c,ea, c, e are not perfect powers.
PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2018
3

2018 MBMT Guts Round C16-30, G10-15,25-30 Montgomery Blair Math Tournament

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
Set 4
C.16 / G.6 Let a,ba, b, and cc be real numbers. If a3+b3+c3=64a^3 + b^3 + c^3 = 64 and a+b=0a + b = 0, what is the value of cc?
C.17 / G.8 Bender always turns 6060 degrees clockwise. He walks 33 meters, turns, walks 22 meters, turns, walks 11 meter, turns, walks 44 meters, turns, walks 11 meter, and turns. How many meters does Bender have to walk to get back to his original position?
C.18 / G.13 Guang has 44 identical packs of gummies, and each pack has a red, a blue, and a green gummy. He eats all the gummies so that he finishes one pack before going on to the next pack, but he never eats two gummies of the same color in a row. How many different ways can Guang eat the gummies?
C.19 Find the sum of all digits qq such that there exists a perfect square that ends in qq.
C.20 / G.14 The numbers 55 and 77 are written on a whiteboard. Every minute Stev replaces the two numbers on the board with their sum and difference. After 20172017 minutes the product of the numbers on the board is mm. Find the number of factors of mm.
Set 5
C.21 / G.10 On the planet Alletas, 3233\frac{32}{33} of the people with silver hair have purple eyes and 811\frac{8}{11} of the people with purple eyes have silver hair. On Alletas, what is the ratio of the number of people with purple eyes to the number of people with silver hair?
C.22 / G.15 Let PP be a point on y=−1y = -1. Let the clockwise rotation of PP by 60o60^o about (0,0)(0, 0) be P′P'. Find the minimum possible distance between P′P' and (0,−1)(0, -1).
C.23 / G.18 How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.
C.24 Jeremy and Kevin are arguing about how cool a sweater is on a scale of 1−51-5. Jeremy says “33”, and Kevin says “44”. Jeremy angrily responds “3.53.5”, to which Kevin replies “3.753.75”. The two keep going at it, responding with the average of the previous two ratings. What rating will they converge to (and settle on as the coolness of the sweater)?
C.25 / G.20 An even positive integer nn has an odd factorization if the largest odd divisor of nn is also the smallest odd divisor of nn greater than 11. Compute the number of even integers nn less than 5050 with an odd factorization.
Set 6
C.26 / G.26 When 2018!=2018×2017×...×12018! = 2018 \times 2017 \times ... \times 1 is multiplied out and written as an integer, find the number of 44’s.
If the correct answer is AA and your answer is EE, you will receive 12min⁡  (A/E,E/A)312 \min\, \, (A/E, E/A)^3points.
C.27 / G.27 A circle of radius 1010 is cut into three pieces of equal area with two parallel cuts. Find the width of the center piece. https://cdn.artofproblemsolving.com/attachments/e/2/e0ab4a2d51052ee364dd14336677b053a40352.png If the correct answer is AA and your answer is EE, you will receive max⁡  (0,12−6∣A−E∣)\max \, \,(0, 12 - 6|A - E|)points.
C.28 / G.28 An equilateral triangle of side length 11 is randomly thrown onto an infinite set of lines, spaced 11 apart. On average, how many times will the boundary of the triangle intersect one of the lines? https://cdn.artofproblemsolving.com/attachments/0/1/773c3d3e0dfc1df54945824e822feaa9c07eb7.png For example, in the above diagram, the boundary of the triangle intersects the lines in 22 places.
If the correct answer is AA and your answer is EE, you will receive max⁡  (0,12−120∣A−E∣/A)\max\, \,(0, 12-120|A-E|/A) points.
C.29 / G.29 Call an ordered triple of integers (a,b,c)(a, b, c) nice if there exists an integer xx such that ax2+bx+c=0ax^2 + bx + c = 0. How many nice triples are there such that −100≤a,b,c≤100-100 \le a, b, c \le 100?
If the correct answer is AA and your answer is EE, you will receive 12min⁡  (A/E,E/A)12 \min\, \,(A/E, E/A) points.
C.30 / G.30 Let f(i)f(i) denote the number of MBMT volunteers to be born in the iith state to join the United States. Find the value of 1f(1)+2f(2)+3f(3)+...+50f(50)1f(1) + 2f(2) + 3f(3) + ... + 50f(50).
Note 1: Maryland was the 77th state to join the US. Note 2: Last year’s MBMT competition had 4242 volunteers.
If the correct answer is AA and your answer is EE, you will receive max⁡  (0,12−500(∣A−E∣/A)2)\max\, \,(0, 12 - 500(|A -E|/A)^2) points.
PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2018 MBMT Guts Round C1-15/ G1-10 Montgomery Blair Math Tournament

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
Set 1
C.1 / G.1 Daniel is exactly one year younger than his friend David. If David was born in the year 20082008, in what year was Daniel born?
C.2 / G.3 Mr. Pham flips three coins. What is the probability that no two coins show the same side?
C.3 / G.2 John has a sheet of white paper which is 33 cm in height and 44 cm in width. He wants to paint the sky blue and the ground green so the entire paper is painted. If the ground takes up a third of the page, how much space (in cm2^2) does the sky take up?
C.4 / G.5 Jihang and Eric are busy fidget spinning. While Jihang spins his fidget spinner at 1515 revolutions per second, Eric only manages 1010 revolutions per second. How many total revolutions will the two have made after 55 continuous seconds of spinning?
C.5 / G.4 Find the last digit of 1333337777⋅209347802⋅3940704⋅23094760911333337777 \cdot 209347802 \cdot 3940704 \cdot 2309476091.
Set 2
C.6 Evan, Chloe, Rachel, and Joe are splitting a cake. Evan takes 13\frac13 of the cake, Chloe takes 14\frac14, Rachel takes 15\frac15, and Joe takes 16\frac16. There is 1x\frac{1}{x} of the original cake left. What is xx?
C.7 Pacman is a 330o330^o sector of a circle of radius 44. Pacman has an eye of radius 11, located entirely inside Pacman. Find the area of Pacman, not including the eye.
C.8 The sum of two prime numbers aa and bb is also a prime number. If a<ba < b, find aa.
C.9 A bus has 5454 seats for passengers. On the first stop, 3636 people get onto an empty bus. Every subsequent stop, 11 person gets off and 33 people get on. After the last stop, the bus is full. How many stops are there?
C.10 In a game, jumps are worth 11 point, punches are worth 22 points, and kicks are worth 33 points. The player must perform a sequence of 11 jump, 11 punch, and 11 kick. To compute the player’s score, we multiply the 1st action’s point value by 11, the 22nd action’s point value by 22, the 3rd action’s point value by 33, and then take the sum. For example, if we performed a punch, kick, jump, in that order, our score would be 1×2+2×3+3×1=111 \times 2 + 2 \times 3 + 3 \times 1 = 11. What is the maximal score the player can get?
Set 3
C.11 66 students are sitting around a circle, and each one randomly picks either the number 11 or 22. What is the probability that there will be two people sitting next to each other who pick the same number?
C.12 / G. 8 You can buy a single piece of chocolate for 6060 cents. You can also buy a packet with two pieces of chocolate for $1.00\$1.00. Additionally, if you buy four single pieces of chocolate, the fifth one is free. What is the lowest amount of money you have to pay for 4444 pieces of chocolate? Express your answer in dollars and cents (ex. $3.70\$3.70).
C.13 / G.12 For how many integers kk is there an integer solution xx to the linear equation kx+2=14kx + 2 = 14?
C.14 / G.9 Ten teams face off in a swim meet. The boys teams and girls teams are ranked independently, each team receiving some number of positive integer points, and the final results are obtained by adding the points for the boys and the points for the girls. If Blair’s boys got 77th place while the girls got 55th place (no ties), what is the best possible total rank for Blair?
C.15 / G.11 Arlene has a square of side length 11, an equilateral triangle with side length 11, and two circles with radius 1/61/6. She wants to pack her four shapes in a rectangle without items piling on top of each other. What is the minimum possible area of the rectangle?
PS. You should use hide for answers. C16-30/G10-15, G25-30 have been posted [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2018 MBMT Guts Round G16-25 Montgomery Blair Math Tournament

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
Set 4
G.16 A number kk is the product of exactly three distinct primes (in other words, it is of the form pqrpqr, where p,q,rp, q, r are distinct primes). If the average of its factors is 6666, find kk.
G.17 Find the number of lattice points contained on or within the graph of x23+y22=12\frac{x^2}{3} +\frac{y^2}{2}= 12. Lattice points are coordinate points (x,y)(x, y) where xx and yy are integers.
G.18 / C.23 How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.
G.19 Cindy has a cone with height 1515 inches and diameter 1616 inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint?
G.20 / C.25 An even positive integer nn has an odd factorization if the largest odd divisor of nn is also the smallest odd divisor of n greater than 1. Compute the number of even integers nn less than 5050 with an odd factorization.
Set 5
G.21 In the magical tree of numbers, nn is directly connected to 2n2n and 2n+12n + 1 for all nonnegative integers n. A frog on the magical tree of numbers can move from a number nn to a number connected to it in 11 hop. What is the least number of hops that the frog can take to move from 10001000 to 20182018?
G.22 Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for 1010 days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy 10241024 dollars, but then he learns the catch - after 1010 days, the amount of money he has must be a multiple of 1111 or he loses all his money. What is the largest amount of money Stan can have after the 1010 days are up?
G.23 Let Γ1\Gamma_1 be a circle with diameter 22 and center O1O_1 and let Γ2\Gamma_2 be a congruent circle centered at a point O2∈Γ1O_2 \in \Gamma_1. Suppose Γ1\Gamma_1 and Γ2\Gamma_2 intersect at AA and BB. Let Ω\Omega be a circle centered at AA passing through BB. Let PP be the intersection of Ω\Omega and Γ1\Gamma_1 other than BB and let QQ be the intersection of Ω\Omega and ray AO1→\overrightarrow{AO_1}. Define RR to be the intersection of PQPQ with Γ1\Gamma_1. Compute the length of O2RO_2R.
G.24 88 people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible?
G.25 Let SS be the set of points (x,y)(x, y) such that y=x3−5xy = x^3 - 5x and x=y3−5yx = y^3 - 5y. There exist four points in SS that are the vertices of a rectangle. Find the area of this rectangle.

PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here
2017
3

2017 MBMT Guts Round R1-15/ P1-5 Montgomery Blair Math Tournament

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names
Set 1
R1.1 / P1.1 Find 291+503−91+492−103−392291 + 503 - 91 + 492 - 103 - 392.
R1.2 Let the operation aa & bb be defined to be a−ba+b\frac{a-b}{a+b}. What is 33 & −2-2?
R1.3. Joe can trade 55 apples for 33 oranges, and trade 66 oranges for 55 bananas. If he has 2020 apples, what is the largest number of bananas he can trade for?
R1.4 A cone has a base with radius 33 and a height of 55. What is its volume? Express your answer in terms of π\pi.
R1.5 Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded 34\frac34 of his dumplings for Arman’s samosas, then he gave 33 dumplings to Anish, and lastly he gave David 12\frac12 of the dumplings he had left. How many dumplings did Guang bring to school?
Set 2
R2.6 / P1.3 In the recording studio, Kanye has 1010 different beats, 99 different manuscripts, and 8 different samples. If he must choose 11 beat, 11 manuscript, and 11 sample for his new song, how many selections can he make?
R2.7 How many lines of symmetry does a regular dodecagon (a polygon with 1212 sides) have?
R2.8 Let there be numbers a,b,ca, b, c such that ab=3ab = 3 and abc=9abc = 9. What is the value of cc?
R2.9 How many odd composite numbers are there between 11 and 2020?
R2.10 Consider the line given by the equation 3x−5y=23x - 5y = 2. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point?
Set 3
R3.11 Let ABCDABCD be a rectangle such that AB=4AB = 4 and BC=3BC = 3. What is the length of BD?
R3.12 Daniel is walking at a constant rate on a 100100-meter long moving walkway. The walkway moves at 33 m/s. If it takes Daniel 2020 seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s.
R3.13 / P1.3 Pratik has a 66 sided die with the numbers 1,2,3,4,61, 2, 3, 4, 6, and 1212 on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to 1212?
R3.14 / P1.5 Find the two-digit number such that the sum of its digits is twice the product of its digits.
R3.15 If a2+2a=120a^2 + 2a = 120, what is the value of 2a2+4a+12a^2 + 4a + 1?

PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here, and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2017 MBMT Guts Round R16-30/ P6-10/P26-30 Montgomery Blair Math Tournament

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names
Set 4
R4.16 / P1.4 Adam and Becky are building a house. Becky works twice as fast as Adam does, and they both work at constant speeds for the same amount of time each day. They plan to finish building in 66 days. However, after 22 days, their friend Charlie also helps with building the house. Because of this, they finish building in just 55 days. What fraction of the house did Adam build?
R4.17 A bag with 1010 items contains both pencils and pens. Kanye randomly chooses two items from the bag, with replacement. Suppose the probability that he chooses 11 pen and 11 pencil is 2150\frac{21}{50} . What are all possible values for the number of pens in the bag?
R4.18 / P2.8 In cyclic quadrilateral ABCDABCD, ∠ABD=40o\angle ABD = 40^o, and ∠DAC=40o\angle DAC = 40^o. Compute the measure of ∠ADC\angle ADC in degrees. (In cyclic quadrilaterals, opposite angles sum up to 180o180^o.)
R4.19 / P2.6 There is a strange random number generator which always returns a positive integer between 11 and 75007500, inclusive. Half of the time, it returns a uniformly random positive integer multiple of 2525, and the other half of the time, it returns a uniformly random positive integer that isn’t a multiple of 2525. What is the probability that a number returned from the generator is a multiple of 3030?
R4.20 / P2.7 Julia is shopping for clothes. She finds TT different tops and SS different skirts that she likes, where T≥S>0T \ge S > 0. Julia can either get one top and one skirt, just one top, or just one skirt. If there are 5050 ways in which she can make her choice, what is T−ST - S?
Set 5
R5.21 A 5×5×55 \times 5 \times 5 cube’s surface is completely painted blue. The cube is then completely split into 1×1×1 1 \times 1 \times 1 cubes. What is the average number of blue faces on each 1×1×1 1 \times 1 \times 1 cube?
R5.22 / P2.10 Find the number of values of nn such that a regular nn-gon has interior angles with integer degree measures.
R5.23 44 positive integers form an geometric sequence. The sum of the 44 numbers is 255255, and the average of the second and the fourth number is 102102. What is the smallest number in the sequence?
R5.24 Let SS be the set of all positive integers which have three digits when written in base 20162016 and two digits when written in base 20172017. Find the size of SS.
R5.25 / P3.12 In square ABCDABCD with side length 1313, point EE lies on segment CDCD. Segment AEAE divides ABCDABCD into triangle ADEADE and quadrilateral ABCEABCE. If the ratio of the area of ADEADE to the area of ABCEABCE is 4:114 : 11, what is the ratio of the perimeter of ADEADE to the perimeter of ABCEABCE?
Set 6
R6.26 / P6.25 Submit a decimal n to the nearest thousandth between 00 and 200200. Your score will be min⁡(12,S)\min (12, S), where SS is the non-negative difference between nn and the largest number less than or equal to nn chosen by another team (if you choose the smallest number, S=nS = n). For example, 1.414 is an acceptable answer, while 2\sqrt2 and 1.41421.4142 are not.
R6.27 / P6.27 Guang is going hard on his YNA project. From 1:001:00 AM Saturday to 1:001:00 AM Sunday, the probability that he is not finished with his project xx hours after 1:001:00 AM on Saturday is 1x+1\frac{1}{x+1} . If Guang does not finish by 1:00 AM on Sunday, he will stop procrastinating and finish the project immediately. Find the expected number of minutes AA it will take for him to finish his project. An estimate of EE will earn 12⋅2−∣E−A∣/6012 \cdot 2^{-|E-A|/60} points.
R6.28 / P6.28 All the diagonals of a regular 100100-gon (a regular polygon with 100100 sides) are drawn. Let AA be the number of distinct intersection points between all the diagonals. Find AA. An estimate of EE will earn 12⋅(16log⁡10(max⁡(EA,AE))+1)−1212 \cdot \left(16 \log_{10}\left(\max \left(\frac{E}{A},\frac{A}{E}\right)\right)+ 1\right)^{-\frac12} or 00 points if this expression is undefined.
R6.29 / P6.29 Find the smallest positive integer AA such that the following is true: if every integer 1,2,...,A1, 2, ..., A is colored either red or blue, then no matter how they are colored, there are always 6 integers among them forming an increasing arithmetic progression that are all colored the same color. An estimate of EE will earn 12min(EA,AE)12 min \left(\frac{E}{A},\frac{A}{E}\right) points or 00 points if this expression is undefined.
R6.30 / P6.30 For all integers n≥2n \ge 2, let f(n)f(n) denote the smallest prime factor of nn. Find A=∑n=2106f(n)A =\sum^{10^6}_{n=2}f(n). In other words, take the smallest prime factor of every integer from 22 to 10610^6 and sum them all up to get AA. You may find the following values helpful: there are 7849878498 primes below 10610^6, 95929592 primes below 10510^5, 12291229 primes below 10410^4, and 168168 primes below 10310^3. An estimate of EE will earn max⁡(0,12−4log⁡10(max(EA,AE))\max \left(0, 12-4 \log_{10}(max \left(\frac{E}{A},\frac{A}{E}\right)\right) or 00 points if this expression is undefined.
PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here, and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2017 MBMT Guts Round P11-25 Montgomery Blair Math Tournament

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names
Set 3
P3.11 Find all possible values of cc in the following system of equations: a2+ab+c2=31a^2 + ab + c^2 = 31 b2+ab−c2=18b^2 + ab - c^2 = 18 a2−b2=7a^2 - b^2 = 7
P3.12 / R5.25 In square ABCDABCD with side length 1313, point EE lies on segment CDCD. Segment AEAE divides ABCDABCD into triangle ADEADE and quadrilateral ABCEABCE. If the ratio of the area of ADEADE to the area of ABCEABCE is 4:114 : 11, what is the ratio of the perimeter of ADEADE to the perimeter ofABCE ABCE?
P3.13 Thomas has two distinct chocolate bars. One of them is 11 by 55 and the other one is 11 by 33. If he can only eat a single 11 by 11 piece off of either the leftmost side or the rightmost side of either bar at a time, how many different ways can he eat the two bars?
P3.14 In triangle ABCABC, AB=13AB = 13, BC=14BC = 14, and CA=15CA = 15. The entire triangle is revolved about side BCBC. What is the volume of the swept out region?
P3.15 Find the number of ordered pairs of positive integers (a,b)(a, b) that satisfy the equation a(a−1)+2ab+b(b−1)=600a(a -1) + 2ab + b(b - 1) = 600.
Set 4
P4.16 Compute the sum of the digits of (102017−1)2(10^{2017} - 1)^2 .
P4.17 A right triangle with area 210210 is inscribed within a semicircle, with its hypotenuse coinciding with the diameter of the semicircle. 22 semicircles are constructed (facing outwards) with the legs of the triangle as their diameters. What is the area inside the 22 semicircles but outside the first semicircle?
P4.18 Find the smallest positive integer nn such that exactly 110\frac{1}{10} of its positive divisors are perfect squares.
P4.19 One day, Sambuddha and Jamie decide to have a tower building competition using oranges of radius 11 inch. Each player begins with 1414 oranges. Jamie builds his tower by making a 33 by 33 base, placing a 22 by 22 square on top, and placing the last orange at the very top. However, Sambuddha is very hungry and eats 44 of his oranges. With his remaining 1010 oranges, he builds a similar tower, forming an equilateral triangle with 33 oranges on each side, placing another equilateral triangle with 22 oranges on each side on top, and placing the last orange at the very top. What is the positive difference between the heights of these two towers?
P4.20 Let r,sr, s, and tt be the roots of the polynomial x3−9x+42x^3 - 9x + 42. Compute the value of (rs)3+(st)3+(tr)3(rs)^3 + (st)^3 + (tr)^3.
Set 5
P5.21 For all integers k>1k > 1, ∑n=0∞k−n=kk−1\sum_{n=0}^{\infty}k^{-n} =\frac{k}{k -1}. There exists a sequence of integers j0,j1,...j_0, j_1, ... such that ∑n=0∞jnk−n=(kk−1)3\sum_{n=0}^{\infty}j_n k^{-n} =\left(\frac{k}{k -1}\right)^3 for all integers k>1k > 1. Find j10j_{10}.
P5.22 Nimi is a triangle with vertices located at (−1,6)(-1, 6), (6,3)(6, 3), and (7,9)(7, 9). His center of mass is tied to his owner, who is asleep at (0,0)(0, 0), using a rod. Nimi is capable of spinning around his center of mass and revolving about his owner. What is the maximum area that Nimi can sweep through?
P5.23 The polynomial x19−x−2x^{19} - x - 2 has 1919 distinct roots. Let these roots be a1,a2,...,a19a_1, a_2, ..., a_{19}. Find a137+a237+...+a1937a^{37}_1 + a^{37}_2+...+a^{37}_{19}.
P5.24 I start with a positive integer nn. Every turn, if nn is even, I replace nn with n2\frac{n}{2}, otherwise I replace nn with n−1n-1. Let kk be the most turns required for a number n<500n < 500 to be reduced to 11. How many values of n<500n < 500 require k turns to be reduced to 11?
P5.25 In triangle ABCABC, AB=13AB = 13, BC=14BC = 14, and AC=15AC = 15. Let II and OO be the incircle and circumcircle of ABCABC, respectively. The altitude from AA intersects II at points PP and QQ, and OO at point RR, such that QQ lies between PP and RR. Find PRPR.
PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here, and R16-30 /P6-10/ P26-30 [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2016
2

2016 MBMT Guts Round - p1- p15 - Montgomery Blair Math Tournament

Set 1
p1. Arnold is currently stationed at (0,0)(0, 0). He wants to buy some milk at (3,0)(3, 0), and also some cookies at (0,4)(0, 4), and then return back home at (0,0)(0, 0). If Arnold is very lazy and wants to minimize his walking, what is the length of the shortest path he can take?
p2. Dilhan selects 11 shirt out of 33 choices, 11 pair of pants out of 44 choices, and 22 socks out of 66 differently-colored socks. How many outfits can Dilhan select? All socks can be worn on both feet, and outfits where the only difference is that the left sock and right sock are switched are considered the same.
p3. What is the sum of the first 100100 odd positive integers?
p4. Find the sum of all the distinct prime factors of 15911591.
p5. Let set S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. From SS, four numbers are selected, with replacement. These numbers are assembled to create a 44-digit number. How many such 44-digit numbers are multiples of 33?
Set 2
p6. What is the area of a triangle with vertices at (0,0)(0, 0), (7,2)(7, 2), and (4,4)(4, 4)?
p7. Call a number nn “warm” if n−1n - 1, nn, and n+1n + 1 are all composite. Call a number mm “fuzzy” if mm may be expressed as the sum of 33 consecutive positive integers. How many numbers less than or equal to 3030 are warm and fuzzy?
p8. Consider a square and hexagon of equal area. What is the square of the ratio of the side length of the hexagon to the side length of the square?
p9. If x2+y2=361x^2 + y^2 = 361, xy=−40xy = -40, and x−yx - y is positive, what is x−yx - y?
p10. Each face of a cube is to be painted red, orange, yellow, green, blue, or violet, and each color must be used exactly once. Assuming rotations are indistinguishable, how many ways are there to paint the cube?
Set 3
p11. Let DD be the midpoint of side BCBC of triangle ABCABC. Let PP be any point on segment ADAD. If MM is the maximum possible value of [PAB][PAC]\frac{[PAB]}{[PAC]} and mm is the minimum possible value, what is M−mM - m?
Note: [PQR][PQR] denotes the area of triangle PQRPQR.
p12. If the product of the positive divisors of the positive integer nn is n6n^6, find the sum of the 33 smallest possible values of nn.
p13. Find the product of the magnitudes of the complex roots of the equation (x−4)4+(x−2)4+14=0(x - 4)^4 +(x - 2)^4 + 14 = 0.
p14. If xy−20x−16y=2016xy - 20x - 16y = 2016 and xx and yy are both positive integers, what is the least possible value of max⁡(x,y)\max (x, y)?
p15. A peasant is trying to escape from Chyornarus, ruled by the Tsar and his mystical faith healer. The peasant starts at (0,0)(0, 0) on a 6×66 \times 6 unit grid, the Tsar’s palace is at (3,3)(3, 3), the healer is at (2,1)(2, 1), and the escape is at (6,6)(6, 6). If the peasant crosses the Tsar’s palace or the mystical faith healer, he is executed and fails to escape. The peasant’s path can only consist of moves upward and rightward along the gridlines. How many valid paths allow the peasant to escape?
PS. You should use hide for answers. Rest sets have been posted [url=https://artofproblemsolving.com/community/c3h2784259p24464954]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2016 MBMT Guts Round - p16 - p30 - Montgomery Blair Math Tournament

Set 4
p16. Albert, Beatrice, Corey, and Dora are playing a card game with two decks of cards numbered 1−501-50 each. Albert, Beatrice, and Corey draw cards from the same deck without replacement, but Dora draws from the other deck. What is the probability that the value of Corey’s card is the highest value or is tied for the highest value of all 44 drawn cards?
p17. Suppose that ss is the sum of all positive values of xx that satisfy 2016{x}=x+[x]2016\{x\} = x+[x]. Find {s}\{s\}. (Note: [x][x] denotes the greatest integer less than or equal to xx and {x}\{x\} denotes x−[x]x - [x].)
p18. Let ABCABC be a triangle such that AB=41AB = 41, BC=52BC = 52, and CA=15CA = 15. Let H be the intersection of the BB altitude and CC altitude. Furthermore let PP be a point on AHAH. Both PP and HH are reflected over BCBC to form P′P' and H′H' . If the area of triangle P′H′CP'H'C is 6060, compute PHPH.
p19. A random integer nn is chosen between 11 and 3030, inclusive. Then, a random positive divisor of n,kn, k, is chosen. What is the probability that k2>nk^2 > n?
p20. What are the last two digits of the value 33613^{361}?
Set 5
p21. Let f(n)f(n) denote the number of ways a 3×n3 \times n board can be completely tiled with 1×31 \times 3 and 1×41 \times 4 tiles, without overlap or any tiles hanging over the edge. The tiles may be rotated. Find ∑i=09f(i)=f(0)+f(1)+...+f(8)+f(9)\sum^9_{i=0} f(i) = f(0) + f(1) + ... + f(8) + f(9). By convention, f(0)=1f(0) = 1.
p22. Find the sum of all 55-digit perfect squares whose digits are all distinct and come from the set {0,2,3,5,7,8}\{0, 2, 3, 5, 7, 8\}.
p23. Mary is flipping a fair coin. On average, how many flips would it take for Mary to get 44 heads and 22 tails?
p24. A cylinder is formed by taking the unit circle on the xyxy-plane and extruding it to positive infinity. A plane with equation z=1−xz = 1 - x truncates the cylinder. As a result, there are three surfaces: a surface along the lateral side of the cylinder, an ellipse formed by the intersection of the plane and the cylinder, and the unit circle. What is the total surface area of the ellipse formed and the lateral surface? (The area of an ellipse with semi-major axis aa and semi-minor axis bb is πab\pi ab.)
p25. Let the Blair numbers be defined as follows: B0=5B_0 = 5, B1=1B_1 = 1, and Bn=Bn−1+Bn−2B_n = B_{n-1} + B_{n-2} for all n≥2n \ge 2. Evaluate ∑i=0∞Bi51i=B0+B151+B2512+B3513+...\sum_{i=0}^{\infty} \frac{B_i}{51^i}= B_0 +\frac{B_1}{51} +\frac{B_2}{51^2} +\frac{B_3}{51^3} +...
Estimation
p26. Choose an integer between 11 and 1010, inclusive. Your score will be the number you choose divided by the number of teams that chose your number.
p27. 20162016 blind people each bring a hat to a party and leave their hat in a pile at the front door. As each partier leaves, they take a random hat from the ones remaining in a pile. Estimate the probability that at least 11 person gets their own hat back.
p28. Estimate how many lattice points lie within the graph of ∣x3∣+∣y3∣<2016|x^3| + |y^3| < 2016.
p29. Consider all ordered pairs of integers (x,y)(x, y) with 1≤x,y≤20161 \le x, y \le 2016. Estimate how many such ordered pairs are relatively prime.
p30. Estimate how many times the letter “e” appears among all Guts Round questions.
PS. You should use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c3h2779594p24402189]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2015.30
1
2015.29
1
2015.28
1
2015.27
1
2015.26
1
2015.25
1
2015.24
1
2015.23
1
2015.22
1
2015.21
1
2015.20
1
2015.19
1
2015.18
1
2015.17
1
2015.16
1
2015.15
1
2015.14
1
2015.13
1
2015.12
1
2015.11
1
2015.10
1
2015.9
1
2015.8
1
2015.7
1
2015.6
1
2015.5
1
2015.4
1
2015.3
1
2015.2
1
2015.1
1