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November 11th, 2012, 12:13 PM   #1
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2012 HMMT November - Guts Round

I just participated in this tournament, note: the questions are fairly hard, do not feel bad if you could not solve some of them, and actually, if you could solve more than half of those, you are already amazing!

The rules: work 3 at a time, after you hand in the 3 questions you were working on, you cannot get it back... ever. You start with questions 1-3, have 2.5 hours for the whole round, a total 36 questions, from easy to hard.
the number in the bracket means the point value of the question (all or nothing except the last 3)

Here are most of the questions (trying to find the rest of it):
1.[5] Find the number of prime numbers less than 30.
2.[5] Albert is very hungry, and goes to his favorite burger shop to buy a meal, which consists of a burger, a side, and a drink. Given that there are 5 different types of burgers offered, 3 different types of sides, and 12 different types of drink, find the number of meals Albert could get.
3.[5] Find the area of the region between two concentric circles that have radii 100 and 99.

4.[6] ABCD is a concave quadrilateral such that <CBA = 50 degrees, <BAD = 80 degrees, <ADC = 30 degrees, and CB = CD. Find <CBD.
5.[6] a and b are numbers such that 2a + 3b = 10 and 4a^2 + 9b^2 = 20. Find ab.
6.[6] Given the following formulas:
1+2+...+n = n(n+1)/2
1^2+2^2+...+n^2 = n(n+1)(2n+1)/6
1^3+2^3+...+n^3 = (n(n+1)/2)^2
find (1^3+3x1^2+3x1)+(2^3+3x2^2+3x2)+...+(99^3+3x99^2+3 x99).

7.[7] Consider the sequence given by a0 = 1, a1 = 1+3, a2 = 1+3+3^2, a3 = 1+3+3^2+3^3,.... Find the number of terms among a0,a1,a2,...a2012 that are divisible by 7.
8.[7] Three cards are drawn from the top of a shuffled standard 52-card deck. Find the probability that they are all of different suits. (A suit is either spades, clubs, hearts, or diamonds.)
9.[7] How many sets consist of distinct composite numbers that add up to 23?

10.[8] Consider the sequence defined by a0 = 0, a1 = 1, and (a sub n) = 2013 (a sub n-1) + 2012 a sub n-2 + n for n greater or equal to 2. Find the remainder when a2012 is divide by 2012.
11.[8] Find the smallest positive integer n such that the number of zeros that n! ends with is a positive multiple of 5.
12.[8] Three circles of radius 1 are drawn, whose centers form the vertices of an equilateral triangle of side length 1. Find the area of the region common to at least two of the circles.

13.[9] Triangle ABC has a perimeter of 18. <A = 60 degrees, and BC = 7. Find the area of the triangle.
14.[9] Mark would like to place the numbers 1 to 7 on a circle such that the sequences along both arcs going from 1 to 7 are increasing. For example, one arc could be 1,2,4,7 and the other could be 1,3,5,6,7. In how many distinct ways can Mark place the numbers? Two arrangements are distinct if and only if one cannot be rotated to match the other.
15.[9] Find the area of the region in the xy-plane consisting of all points (a,b) such that the quadratic ax^2+2(a+b-7)x-2b = 0 has fewer than two real solutions of x.

16.[10] Yuhang is making a bracelet for his one true love using beads. How many distinct bracelets can be made from 2 red beads, 2 green beads, and 2 blue brads, if two bracelets are distinct if and only if one cannot be made into the other through rotations and reflections?
17.[10] Given that x = ln30, you = ln360, and z = ln270, and that there are rational numbers p,q,r such that ln5400 = px+qy+rz, find the ordered triple (pqr).
18.[10] Let triangle ABC be a triangle with AC = 1 and <ABC obtuse. Let D and E be points on AC such that <DBC = <ABE = 90 degrees. If AD =DE = EC, find AB + AC.

19.[11] Find the number of triples of non negative integers (x,y,z) such that 15x+21y+35z = 525.
20.[11] An elementary school teacher is taking a class of 20 students on a field trip. To make sure her students don't get lost, she uses the buddy system - using the complete class roster, she pairs the students into 10 pairs and leaves if no person reports that someone from his or her pair is missing. In particular, the teacher will not notice if both students from a pair go missing. Suppose each student independently gets lost with probability of 1/10. Given that the teacher leaves, what is the probability that no student got lost?
21.[11] ABC is a triangle with AB = 7, BC = 10, and CA = 13. Point D lies on segment BC such that DC = 2BD. Point E lies on segment AD such that AE = 4ED. Point F lies on segment AC such that FC = 5AF. Find the area of triangle EFC.


25.[13] Triangle ABC satisfies AB = 8, BC = 9, and CA = 10. Evaluate ((sinB)^2+(sinC)^2-(sinA)^2)/(sinB x sinC).
26.[13] x1, x2, x3,... is a sequence of real numbers satisfying x1 = 1, x2 = 2, and x (sub n+1) = 2x (sub n) - x (sub n-1) + 2^n
27.[13] Find the number of ordered triples of positive integers (a,b,c) satisfying a^2-b^2+ac-bc = 2012.

28.[15] Find the set of all possible values that can be attained by the expression (ab+b^2)/(a^2+b^), where a and b are positive real numbers. Express your answer in interval notation.
29.[15] Find the sum of the real values of x satisfying (x+1)(2x+1)(3x+1)(4x+1) = 16x^4.
30.[15] A monkey forms a string of letters by repeatedly choosing one of the letters a, b, or c to type at random. Find the probability that he first types the string aaa before he first types the string abc.


34.[20] For a positive integer n, let r(n0 be the number of divisors of n. Determine ? 2012 r(n).
your score will be max {0,?20(1-(|S-k|)/S)^3?}, where k is your answer and S is the actual answer.
35.[20] A regular 2012-gon has a circumcircle with radius 1. Compute the area of the 2012-gon. Your score will be min(20,?(k^2)/5?), where k is the consecutive correct digits immediately following the decimal point of your answer.
36.[20[ Let ?1 and ?2 be permutations of the numbers from 1 through 7. Call ?1 superior to ?2 if the sum of all i such that ?1(i) < ?2(i) exceeds the sum of all i such that ?2(i) > ?1(i). Write down a permutation of the integers from 1 through 7. Let N be the total number of answers submitted for this problem, and let n be the number of submitted answers your answer is superior to. You score will be ?20 n/N?.

I will fill up the questions I left blank in near future since I could not find those problems for now.
Note: some notations I used words instead of symbols, so they may look strange.
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