MathDB

2017

Part of MBMT Guts Rounds

Problems(3)

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

Source:

2/22/2022
[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.
algebrageometrycombinatoricsnumber theoryMBMT
2017 MBMT Guts Round R16-30/ P6-10/P26-30 Montgomery Blair Math Tournament

Source:

2/22/2022
[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.
algebrageometrycombinatoricsnumber theoryMBMT
2017 MBMT Guts Round P11-25 Montgomery Blair Math Tournament

Source:

2/22/2022
[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.
algebrageometrycombinatoricsnumber theoryMBMT