p1. What is the sum of the first 12 positive integers?
p2. How many positive integers less than or equal to 100 are multiples of both 2 and 5?
p3. Alex has a bag with 4 white marbles and 4 black marbles. She takes 2 marbles from the bag without replacement. What is the probability that both marbles she took are black? Express your answer as a decimal or a fraction in lowest terms.
p4. How many 5-digit numbers are there where each digit is either 1 or 2?
p5. An integer a with 1≤a≤10 is randomly selected. What is the probability that a100 is an integer? Express your answer as decimal or a fraction in lowest terms.
p6. Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let P be the number of points of intersection between any two circles. How many possible values of P are there?
p7. Let x,y,z be nonzero real numbers such that x+y+z=xyz. Compute yz1+yz+xz1+xz+xy1+xy.
p8. How many positive integers less than 106 are simultaneously perfect squares, cubes, and fourth powers?
p9. Let C1 and C2 be two circles centered at point O of radii 1 and 2, respectively. Let A be a point on C2. We draw the two lines tangent to C1 that pass through A, and label their other intersections with C2 as B and C. Let x be the length of minor arc BC, as shown. Compute x.
https://cdn.artofproblemsolving.com/attachments/7/5/915216d4b7eba0650d63b26715113e79daa176.png
p10. A circle of area π is inscribed in an equilateral triangle. Find the area of the triangle.
p11. Julie runs a 2 mile route every morning. She notices that if she jogs the route 2 miles per hour faster than normal, then she will finish the route 5 minutes faster. How fast (in miles per hour) does she normally jog?
p12. Let ABCD be a square of side length 10. Let EFGH be a square of side length 15 such that E is the center of ABCD, EF intersects BC at X, and EH intersects CD at Y (shown below). If BX=7, what is the area of quadrilateral EXCY ?
https://cdn.artofproblemsolving.com/attachments/d/b/2b2d6de789310036bc42d1e8bcf3931316c922.png
p13. How many solutions are there to the system of equations
a2+b2=c2
(a+1)2+(b+1)2=(c+1)2 if a,b, and c are positive integers?
p14. A square of side length s is inscribed in a semicircle of radius r as shown. Compute rs.
https://cdn.artofproblemsolving.com/attachments/5/f/22d7516efa240d00d6a9743a4dc204d23d190d.png
p15. S is a collection of integers n with 1≤n≤50 so that each integer in S is composite and relatively prime to every other integer in S. What is the largest possible number of integers in S?
p16. Let ABCD be a regular tetrahedron and let W,X,Y,Z denote the centers of faces ABC, BCD, CDA, and DAB, respectively. What is the ratio of the volumes of tetrahedrons WXYZ and WAYZ? Express your answer as a decimal or a fraction in lowest terms.
p17. Consider a random permutation {s1,s2,...,s8} of {1,1,1,1,−1,−1,−1,−1}. Let S be the largest of the numbers s1, s1+s2, s1+s2+s3, ... , s1+s2+...+s8. What is the probability that S is exactly 3? Express your answer as a decimal or a fraction in lowest terms.
p18. A positive integer is called almost-kinda-semi-prime if it has a prime number of positive integer divisors. Given that there are168 primes less than 1000, how many almost-kinda-semi-prime numbers are there less than 1000?
p19. Let ABCD be a unit square and let X,Y,Z be points on sides AB, BC, CD, respectively, such that AX=BY=CZ. If the area of triangle XYZ is 31 , what is the maximum value of the ratio XB/AX?
https://cdn.artofproblemsolving.com/attachments/5/6/cf77e40f8e9bb03dea8e7e728b21e7fb899d3e.png
p20. Positive integers a≤b≤c have the property that each of a+b, b+c, and c+a are prime. If a+b+c has exactly 4 positive divisors, find the fourth smallest possible value of the product c(c+b)(c+b+a).
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. algebrageometrycombinatoricsnumber theorybmmt