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2018 BmMT Individual Round - Berkley mini Math Tournament

Source:

November 5, 2023
bmmtalgebrageometrycombinatoricsnumber theory

Problem Statement

p1. If xx is a real number that satisfies 48x=16\frac{48}{x} = 16, find the value of xx.
p2. If ABCABC is a right triangle with hypotenuse BCBC such that ∠ABC=35o\angle ABC = 35^o, what is ∠BCA\angle BCA in degrees? https://cdn.artofproblemsolving.com/attachments/a/b/0f83dc34fb7934281e0e3f988ac34f653cc3f1.png
p3. If a△b=a+b−aba\vartriangle b = a + b - ab, find 4△94\vartriangle 9.
p4. Grizzly is 66 feet tall. He measures his shadow to be 44 feet long. At the same time, his friend Panda helps him measure the shadow of a nearby lamp post, and it is 66 feet long. How tall is the lamp post in feet?
p5. Jerry is currently twice as old as Tom was 77 years ago. Tom is 66 years younger than Jerry. How many years old is Tom?
p6. Out of the 10,00010, 000 possible four-digit passcodes on a phone, how many of them contain only prime digits?
p7. It started snowing, which means Moor needs to buy snow shoes for his 66 cows and 77 sky bison. A cow has 44 legs, and a sky bison has 66 legs. If Moor has 36 snow shoes already, how many more shoes does he need to buy? Assume cows and sky bison wear the same type of shoe and each leg gets one shoe.
p8. How many integers nn with 1≤n≤1001 \le n \le 100 have exactly 33 positive divisors?
p9. James has three 33 candies and 33 green candies. 33 people come in and each randomly take 22 candies. What is the probability that no one got 22 candies of the same color? Express your answer as a decimal or a fraction in lowest terms.
p10. When Box flips a strange coin, the coin can land heads, tails, or on the side. It has a 110\frac{1}{10}probability of landing on the side, and the probability of landing heads equals the probability of landing tails. If Box flips a strange coin 33 times, what is the probability that the number of heads flipped is equal to the number of tails flipped? Express your answer as a decimal or a fraction in lowest terms.
p11. James is travelling on a river. His canoe goes 44 miles per hour upstream and 66 miles per hour downstream. He travels 88 miles upstream and then 88 miles downstream (to where he started). What is his average speed, in miles per hour? Express your answer as a decimal or a fraction in lowest terms.
p12. Four boxes of cookies and one bag of chips cost exactly 10001000 jelly beans. Five bags of chips and one box of cookies cost less than 10001000 jelly beans. If both chips and cookies cost a whole number of jelly beans, what is the maximum possible cost of a bag of chips?
p13. June is making a pumpkin pie, which takes the shape of a truncated cone, as shown below. The pie tin is 1818 inches wide at the top, 1616 inches wide at the bottom, and 11 inch high. How many cubic inches of pumpkin filling are needed to fill the pie? https://cdn.artofproblemsolving.com/attachments/7/0/22c38dd6bc42d15ad9352817b25143f0e4729b.png
p14. For two real numbers aa and bb, let a#b=ab−2a−2b+6a\# b = ab - 2a - 2b + 6. Find a positive real number xx such that (x#7)#x=82(x\#7) \#x = 82.
p15. Find the sum of all positive integers nn such that n2+20n+51n2+4n+3\frac{n^2 + 20n + 51}{n^2 + 4n + 3} is an integer.
p16. Let ABCABC be a right triangle with hypotenuse ABAB such that AC=36AC = 36 and BC=15BC = 15. A semicircle is inscribed in ABCABC as shown, such that the diameter XCXC of the semicircle lies on side ACAC and that the semicircle is tangent to ABAB. What is the radius of the semicircle? https://cdn.artofproblemsolving.com/attachments/4/2/714f7dfd09f6da1d61a8f910b5052e60dcd2fb.png
p17. Let aa and bb be relatively prime positive integers such that the product abab is equal to the least common multiple of 1650016500 and 990990. If 16500a\frac{16500}{a} and 990b\frac{990}{b} are both integers, what is the minimum value of a+ba + b?
p18. Let xx be a positive real number so that x−1x=1x - \frac{1}{x} = 1. Compute x8−1x8x^8 - \frac{1}{x^8} .
p19. Six people sit around a round table. Each person rolls a standard 66-sided die. If no two people sitting next to each other rolled the same number, we will say that the roll is valid. How many di erent rolls are valid?
p20. Given that 131=0.a1a2a3a4a5...an‾\frac{1}{31} = 0.\overline{a_1a_2a_3a_4a_5... a_n} (that is, 131\frac{1}{31} can be written as the repeating decimal expansion 0.a1a2...ana1a2...ana1a2...0.a_1a_2... a_na_1a_2... a_na_1a_2... ), what is the minimum value of nn?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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