MathDB

2015 BmMT

Part of BmMT problems

Subcontests

(2)
Ind. Round
1

2015 BmMT Individual Round - Berkley mini Math Tournament

p1. What is the units digit of 1+9+92+...+920151 + 9 + 9^2 +... + 9^{2015} ?
p2. In Fourtown, every person must have a car and therefore a license plate. Every license plate must be a 44-digit number where each digit is a value between 00 and 99 inclusive. However 00000000 is not a valid license plate. What is the minimum population of Fourtown to guarantee that at least two people who have the same license plate?
p3. Two sides of an isosceles triangle △ABC\vartriangle ABC have lengths 99 and 44. What is the area of △ABC\vartriangle ABC?
p4. Let xx be a real number such that 101x=x10^{\frac{1}{x}} = x. Find (x3)2x(x^3)^{2x}.
p5. A Berkeley student and a Stanford student are going to visit each others campus and go back to their own campuses immediately after they arrive by riding bikes. Each of them rides at a constant speed. They first meet at a place 17.517.5 miles away from Berkeley, and secondly 1010 miles away from Stanford. How far is Berkeley away from Stanford in miles?
p6. Let ABCDEFABCDEF be a regular hexagon. Find the number of subsets SS of {A,B,C,D,E,F}\{A,B,C,D,E, F\} such that every edge of the hexagon has at least one of its endpoints in SS.
p7. A three digit number is a multiple of 3535 and the sum of its digits is 1515. Find this number.
p8. Thomas, Olga, Ken, and Edward are playing the card game SAND. Each draws a card from a 5252 card deck. What is the probability that each player gets a di erent rank and a different suit from the others?
p9. An isosceles triangle has two vertices at (1,4)(1, 4) and (3,6)(3, 6). Find the xx-coordinate of the third vertex assuming it lies on the xx-axis.
p10. Find the number of functions from the set {1,2,...,8}\{1, 2,..., 8\} to itself such that f(f(x))=xf(f(x)) = x for all 1≤x≤81 \le x \le 8.
p11. The circle has the property that, no matter how it's rotated, the distance between the highest and the lowest point is constant. However, surprisingly, the circle is not the only shape with that property. A Reuleaux Triangle, which also has this constant diameter property, is constructed as follows. First, start with an equilateral triangle. Then, between every pair of vertices of the triangle, draw a circular arc whose center is the 33rd vertex of the triangle. Find the ratio between the areas of a Reuleaux Triangle and of a circle whose diameters are equal.
p12. Let aa, bb, cc be positive integers such that gcd (a,b)=2(a, b) = 2, gcd (b,c)=3(b, c) = 3, lcm (a,c)=42(a, c) = 42, and lcm (a,b)=30(a, b) = 30. Find abcabc.
p13. A point PP is inside the square ABCDABCD. If PA=5PA = 5, PB=1PB = 1, PD=7PD = 7, then what is PCPC?
p14. Find all positive integers nn such that, for every positive integer xx relatively prime to nn, we have that nn divides x2−1x^2 - 1. You may assume that if n=2kmn = 2^km, where mm is odd, then nn has this property if and only if both 2k2^k and mm do.
p15. Given integers a,b,ca, b, c satisfying abc+a+c=12abc + a + c = 12 bc+ac=8bc + ac = 8 b−ac=−2,b - ac = -2, what is the value of aa?
p16. Two sides of a triangle have lengths 2020 and 3030. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?
p17. Find the number of non-negative integer solutions (x,y,z)(x, y, z) of the equation xyz+xy+yz+zx+x+y+z=2014.xyz + xy + yz + zx + x + y + z = 2014.
p18. Assume that AA, BB, CC, DD, EE, FF are equally spaced on a circle of radius 11, as in the figure below. Find the area of the kite bounded by the lines EAEA, ACAC, FCFC, BEBE. https://cdn.artofproblemsolving.com/attachments/7/7/57e6e1c4ef17f84a7a66a65e2aa2ab9c7fd05d.png
p19. A positive integer is called cyclic if it is not divisible by the square of any prime, and whenever p<qp < q are primes that divide it, qq does not leave a remainder of 11 when divided by pp. Compute the number of cyclic numbers less than or equal to 100100.
p20. On an 8×88\times 8 chess board, a queen can move horizontally, vertically, and diagonally in any direction for as many squares as she wishes. Find the average (over all 6464 possible positions of the queen) of the number of squares the queen can reach from a particular square (do not count the square she stands on).
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Team Round
1

2015 BmMT Team Round - Berkley mini Math Tournament Fall

p1. Let ff be a function such that f(x+y)=f(x)+f(y)f(x + y) = f(x) + f(y) for all xx and yy. Assume f(5)=9f(5) = 9. Compute f(2015)f(2015).
p2. There are six cards, with the numbers 2,2,4,4,6,62, 2, 4, 4, 6, 6 on them. If you pick three cards at random, what is the probability that you can make a triangles whose side lengths are the chosen numbers?
p3. A train travels from Berkeley to San Francisco under a tunnel of length 1010 kilometers, and then returns to Berkeley using a bridge of length 77 kilometers. If the train travels at 3030 km/hr underwater and 60 km/hr above water, what is the train’s average speed in km/hr on the round trip?
p4. Given a string consisting of the characters A, C, G, U, its reverse complement is the string obtained by first reversing the string and then replacing A’s with U’s, C’s with G’s, G’s with C’s, and U’s with A’s. For example, the reverse complement of UAGCAC is GUGCUA. A string is a palindrome if it’s the same as its reverse. A string is called self-conjugate if it’s the same as its reverse complement. For example, UAGGAU is a palindrome and UAGCUA is self-conjugate. How many six letter strings with just the characters A, C, G (no U’s) are either palindromes or self-conjugate?
p5. A scooter has 22 wheels, a chair has 66 wheels, and a spaceship has 1111 wheels. If there are 1010 of these objects, with a total of 5050 wheels, how many chairs are there?
p6. How many proper subsets of {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\} are there such that the sum of the elements in the subset equal twice a number in the subset?
p7. A circle and square share the same center and area. The circle has radius 11 and intersects the square on one side at points AA and BB. What is the length of AB‾\overline{AB} ?
p8. Inside a circle, chords ABAB and CDCD intersect at PP in right angles. Given that AP=6AP = 6, BP=12BP = 12 and CD=15CD = 15, find the radius of the circle.
p9. Steven makes nonstandard checkerboards that have 2929 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
p10. John is organizing a race around a circular track and wants to put 33 water stations at 99 possible spots around the track. He doesn’t want any 22 water stations to be next to each other because that would be inefficient. How many ways are possible?
p11. In square ABCDABCD, point EE is chosen such that CDECDE is an equilateral triangle. Extend CECE and DEDE to FF and GG on ABAB. Find the ratio of the area of △EFG\vartriangle EFG to the area of △CDE\vartriangle CDE.
p12. Let SS be the number of integers from 22 to 84628462 (inclusive) which does not contain the digit 1,3,5,7,91,3,5,7,9. What is SS?
p13. Let x, y be non zero solutions to x2+xy+y2=0x^2 + xy + y^2 = 0. Find x2016+(xy)1008+y2016(x+y)2016\frac{x^{2016} + (xy)^{1008} + y^{2016}}{(x + y)^{2016}} .
p14. A chess contest is held among 1010 players in a single round (each of two players will have a match). The winner of each game earns 22 points while loser earns none, and each of the two players will get 11 point for a draw. After the contest, none of the 1010 players gets the same score, and the player of the second place gets a score that equals to 4/54/5 of the sum of the last 55 players. What is the score of the second-place player?
p15. Consider the sequence of positive integers generated by the following formula a1=3a_1 = 3, an+1=an+an2a_{n+1} = a_n + a^2_n for n=2,3,...n = 2, 3, ... What is the tens digit of a1007a_{1007}?
p16. Let (x,y,z)(x, y, z) be integer solutions to the following system of equations x2z+y2z+4xy=48x^2z + y^2z + 4xy = 48 x2+y2+xyz=24x^2 + y^2 + xyz = 24 Find ∑x+y+z\sum x + y + z where the sum runs over all possible (x,y,z)(x, y, z).
p17. Given that x+y=ax + y = a and xy=bxy = b and 1≤a,b≤501 \le a, b \le 50, what is the sum of all a such that x4+y4−2x2y2x^4 + y^4 - 2x^2y^2 is a prime squared?
p18. In △ABC\vartriangle ABC, MM is the midpoint of AB‾\overline{AB}, point NN is on side BC‾\overline{BC}. Line segments AN‾\overline{AN} and CM‾\overline{CM} intersect at OO. If AO=12AO = 12, CO=6CO = 6, and ON=4ON = 4, what is the length of OMOM?
p19. Consider the following linear system of equations. 1+a+b+c+d=11 + a + b + c + d = 1 16+8a+4b+2c+d=216 + 8a + 4b + 2c + d = 2 81+27a+9b+3c+d=381 + 27a + 9b + 3c + d = 3 256+64a+16b+4c+d=4256 + 64a + 16b + 4c + d = 4 Find a−b+c−da - b + c - d.
p20. Consider flipping a fair coin 8 8 times. How many sequences of coin flips are there such that the string HHH never occurs?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.