MathDB

2017 BmMT

Part of BmMT problems

Subcontests

(3)
Ind. Tie
1

2017 BmMT Individual Tiebreaker Round - Berkley mini Math Tournament

p1. Consider a 4×44 \times 4 lattice on the coordinate plane. At (0,0)(0,0) is Mori’s house, and at (4,4)(4,4) is Mori’s workplace. Every morning, Mori goes to work by choosing a path going up and right along the roads on the lattice. Recently, the intersection at (2,2)(2, 2) was closed. How many ways are there now for Mori to go to work?
p2. Given two integers, define an operation ∗* such that if a and b are integers, then a ∗* b is an integer. The operation ∗* has the following properties: 1. a∗aa * a = 0 for all integers aa. 2. (ka+b)∗a=b∗a(ka + b) * a = b * a for integers a,b,ka, b, k. 3. 0≤b∗a<a0 \le b * a < a. 4. If 0≤b<a0 \le b < a, then b∗a=bb * a = b. Find 2017∗162017 * 16.
p3. Let ABCABC be a triangle with side lengths AB=13AB = 13, BC=14BC = 14, CA=15CA = 15. Let A′A', B′B', C′C', be the midpoints of BCBC, CACA, and ABAB, respectively. What is the ratio of the area of triangle ABCABC to the area of triangle A′B′C′A'B'C'?
p4. In a strange world, each orange has a label, a number from 00 to 1010 inclusive, and there are an infinite number of oranges of each label. Oranges with the same label are considered indistinguishable. Sally has 3 boxes, and randomly puts oranges in her boxes such that (a) If she puts an orange labelled a in a box (where a is any number from 0 to 10), she cannot put any other oranges labelled a in that box. (b) If any two boxes contain an orange that have the same labelling, the third box must also contain an orange with that labelling. (c) The three boxes collectively contain all types of oranges (oranges of any label). The number of possible ways Sally can put oranges in her 33 boxes is NN, which can be written as the product of primes: p1e1p2e2...pkekp_1^{e_1} p_2^{e_2}... p_k^{e_k} where p1≠p2≠p3...≠pkp_1 \ne p_2 \ne p_3 ... \ne p_k and pip_i are all primes and eie_i are all positive integers. What is the sum e1+e2+e3+...+eke_1 + e_2 + e_3 +...+ e_k?
p5. Suppose I want to stack 20172017 identical boxes. After placing the first box, every subsequent box must either be placed on top of another one or begin a new stack to the right of the rightmost pile. How many different ways can I stack the boxes, if the order I stack them doesn’t matter? Express your answer as p1e1p2e2...pnenp_1^{e_1} p_2^{e_2}... p_n^{e_n} where p1,p2,p3,...,pnp_1, p_2, p_3, ... , p_n are distinct primes and eie_i are all positive integers.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Ind. Round
1

2017 BmMT Individual Round - Berkley mini Math Tournament

p1. It’s currently 6:006:00 on a 1212 hour clock. What time will be shown on the clock 100100 hours from now? Express your answer in the form hh : mm.
p2. A tub originally contains 1010 gallons of water. Alex adds some water, increasing the amount of water by 20%. Barbara, unhappy with Alex’s decision, decides to remove 20%20\% of the water currently in the tub. How much water, in gallons, is left in the tub? Express your answer as an exact decimal.
p3. There are 20002000 math students and 40004000 CS students at Berkeley. If 55805580 students are either math students or CS students, then how many of them are studying both math and CS?
p4. Determine the smallest integer xx greater than 11 such that x2x^2 is one more than a multiple of 77.
p5. Find two positive integers x,yx, y greater than 11 whose product equals the following sum: 9+11+13+15+17+19+21+23+25+27+29.9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29. Express your answer as an ordered pair (x,y)(x, y) with x≤yx \le y.
p6. The average walking speed of a cow is 55 meters per hour. If it takes the cow an entire day to walk around the edges of a perfect square, then determine the area (in square meters) of this square.
p7. Consider the cube below. If the length of the diagonal ABAB is 333\sqrt3, determine the volume of the cube. https://cdn.artofproblemsolving.com/attachments/4/d/3a6fdf587c12f2e4637a029f38444914e161ac.png
p8. I have 1818 socks in my drawer, 66 colored red, 88 colored blue and 44 colored green. If I close my eyes and grab a bunch of socks, how many socks must I grab to guarantee there will be two pairs of matching socks?
p9. Define the operation a@ba @ b to be 3+ab+a+2b3 + ab + a + 2b. There exists a number xx such that x@b=1x @ b = 1 for all bb. Find xx.
p10. Compute the units digit of 2017(20172)2017^{(2017^2)}.
p11. The distinct rational numbers −−x-\sqrt{-x}, xx, and −x-x form an arithmetic sequence in that order. Determine the value of xx.
p12. Let y=x2+bx+cy = x^2 + bx + c be a quadratic function that has only one root. If bb is positive, find b+2c+1\frac{b+2}{\sqrt{c}+1}.
p13. Alice, Bob, and four other people sit themselves around a circular table. What is the probability that Alice does not sit to the left or right of Bob?
p14. Let f(x)=∣x−8∣f(x) = |x - 8|. Let pp be the sum of all the values of xx such that f(f(f(x)))=2f(f(f(x))) = 2 and qq be the minimum solution to f(f(f(x)))=2f(f(f(x))) = 2. Compute p⋅qp \cdot q.
p15. Determine the total number of rectangles (1×11 \times 1, 1×21 \times 2, 2×22 \times 2, etc.) formed by the lines in the figure below: \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular}
p16. Take a square ABCDABCD of side length 11, and let PP be the midpoint of ABAB. Fold the square so that point DD touches PP, and let the intersection of the bottom edge DCDC with the right edge be QQ. What is BQBQ? https://cdn.artofproblemsolving.com/attachments/1/1/aeed2c501e34a40a8a786f6bb60922b614a36d.png
p17. Let AA, BB, and kk be integers, where kk is positive and the greatest common divisor of AA, BB, and kk is 11. Define x#yx\# y by the formula x#y=Ax+Bykxyx\# y = \frac{Ax+By}{kxy} . If 8#4=128\# 4 = \frac12 and 3#1=1363\# 1 = \frac{13}{6} , determine the sum A+B+kA + B + k.
p18. There are 2020 indistinguishable balls to be placed into bins AA, BB, CC, DD, and EE. Each bin must have at least 22 balls inside of it. How many ways can the balls be placed into the bins, if each ball must be placed in a bin?
p19. Let TiT_i be a sequence of equilateral triangles such that (a) T1T_1 is an equilateral triangle with side length 1. (b) Ti+1T_{i+1} is inscribed in the circle inscribed in triangle TiT_i for i≥1i \ge 1. Find ∑i=1∞Area(Ti).\sum^{\infty}_{i=1} Area (T_i).
p20. A gorgeous sequence is a sequence of 11’s and 00’s such that there are no consecutive 11’s. For instance, the set of all gorgeous sequences of length 33 is {[1,0,0]\{[1, 0, 0],[1,0,1] [1, 0, 1], [0,1,0][0, 1, 0], [0,0,1][0, 0, 1], [0,0,0]}[0, 0, 0]\}. Determine the number of gorgeous sequences of length 77.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Team Round
1

2017 BmMT Team Round - Berkley mini Math Tournament Fall

p1. Suppose a1⋅2=a2⋅3=a3a_1 \cdot 2 = a_2 \cdot 3 = a_3 and a1+a2+a3=66a_1 + a_2 + a_3 = 66. What is a3a_3?
p2. Ankit buys a see-through plastic cylindrical water bottle. However, in coming home, he accidentally hits the bottle against a wall and dents the top portion of the bottle (above the 77 cm mark). Ankit now wants to determine the volume of the bottle. The area of the base of the bottle is 2020 cm2^2 . He fills the bottle with water up to the 55 cm mark. After flipping the bottle upside down, he notices that the height of the empty space is at the 77 cm mark. Find the total volume (in cm3^3) of this bottle. https://cdn.artofproblemsolving.com/attachments/1/9/f5735c77b056aaf31b337ea1b777a591807819.png
p3. If PP is a quadratic polynomial with leading coefficient 1 1 such that P(1)=1P(1) = 1, P(2)=2P(2) = 2, what is P(10)P(10)?
p4. Let ABC be a triangle with AB=1AB = 1, AC=3AC = 3, and BC=3BC = 3. Let DD be a point on BCBC such that BD=13BD =\frac13 . What is the ratio of the area of BADBAD to the area of CADCAD?
p5. A coin is flipped 12 12 times. What is the probability that the total number of heads equals the total number of tails? Express your answer as a common fraction in lowest terms.
p6. Moor pours 33 ounces of ginger ale and 1 1 ounce of lime juice in cup AA, 33 ounces of lime juice and 1 1 ounce of ginger ale in cup BB, and mixes each cup well. Then he pours 1 1 ounce of cup AA into cup BB, mixes it well, and pours 1 1 ounce of cup BB into cup AA. What proportion of cup AA is now ginger ale? Express your answer as a common fraction in lowest terms.
p7. Determine the maximum possible area of a right triangle with hypotenuse 77. Express your answer as a common fraction in lowest terms.
p8. Debbie has six Pusheens: 22 pink ones, 22 gray ones, and 22 blue ones, where Pusheens of the same color are indistinguishable. She sells two Pusheens each to Alice, Bob, and Eve. How many ways are there for her to do so?
p9. How many nonnegative integer pairs (a,b)(a, b) are there that satisfy ab=90−a−bab = 90 - a - b?
p10. What is the smallest positive integer a1...ana_1...a_n (where a1,...,ana_1, ... , a_n are its digits) such that 9⋅a1...an=an...a19 \cdot a_1 ... a_n = a_n ... a_1, where a1a_1, an≠0a_n \ne 0?
p11. Justin is growing three types of Japanese vegetables: wasabi root, daikon and matsutake mushrooms. Wasabi root needs 22 square meters of land and 44 gallons of spring water to grow, matsutake mushrooms need 33 square meters of land and 33 gallons of spring water, and daikon need 1 1 square meter of land and 1 1 gallon of spring water to grow. Wasabi sell for 6060 per root, matsutake mushrooms sell for 6060 per mushroom, and daikon sell for 22 per root. If Justin has 500500 gallons of spring water and 400400 square meters of land, what is the maximum amount of money, in dollars, he can make?
p12. A prim number is a number that is prime if its last digit is removed. A rime number is a number that is prime if its first digit is removed. Determine how many numbers between 100100 and 999999 inclusive are both prim and rime numbers.
p13. Consider a cube. Each corner is the intersection of three edges; slice off each of these corners through the midpoints of the edges, obtaining the shape below. If we start with a 2×2×22\times 2\times 2 cube, what is the volume of the resulting solid? https://cdn.artofproblemsolving.com/attachments/4/8/856814bf99e6f28844514158344477f6435a3a.png
p14. If a parallelogram with perimeter 1414 and area 12 12 is inscribed in a circle, what is the radius of the circle?
p15. Take a square ABCDABCD of side length 11, and draw AC‾\overline{AC}. Point EE lies on BC‾\overline{BC} such that AE‾\overline{AE} bisects ∠BAC\angle BAC. What is the length of BEBE?
p16. How many integer solutions does f(x)=(x2+1)(x2+2)+(x2+3)(x+4)=2017f(x) = (x^2 + 1)(x^2 + 2) + (x^2 + 3)(x + 4) = 2017 have?
p17. Alice, Bob, Carol, and Dave stand in a circle. Simultaneously, each player selects another player at random and points at that person, who must then sit down. What is the probability that Alice is the only person who remains standing?
p18. Let xx be a positive integer with a remainder of 22 when divided by 33, 33 when divided by 44, 44 when divided by 55, and 55 when divided by 66. What is the smallest possible such xx?
p19. A circle is inscribed in an isosceles trapezoid such that all four sides of the trapezoid are tangent to the circle. If the radius of the circle is 1 1, and the upper base of the trapezoid is 1 1, what is the area of the trapezoid?
p20. Ray is blindfolded and standing 1 1 step away from an ice cream stand. Every second, he has a 1/41/4 probability of walking 1 1 step towards the ice cream stand, and a 3/43/4 probability of walking 1 1 step away from the ice cream stand. When he is 00 steps away from the ice cream stand, he wins. What is the probability that Ray eventually wins?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.