Practice AMC 12A

1. Find the sum of the infinite geometric series 1 + 7/18 + 49/324 + …
A - 36/11, B - 9/22, C - 18/11, D - 18/7, E - 9/14

2. What is the first digit after the decimal point in the square root of 420?
A - 1, B - 2, C - 3, D - 4, E - 5

3. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW/PX)*(PY/PZ).
A - 50/5863, B - 47/96, C - 1, D - 96/47, E - 5863/50

4. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4 + 20d, where a, b, and d are positive integers?
A - 29, B - 38, C - 43, D - 76, E - 82

5. What is the absolute difference of the probabilities of getting at least 6/10 on a 10-question true or false test and at least 3/5 on a 5-question true or false test?
A - 63/1024, B - 63/512, C - 63/256, D - 63/128, E - 0

6. How many arrangements of the letters in the word “sensor” are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original “sensor”)?
A - 72, B - 108, C - 144, D - 216, E - 432

7. Find the value of 0.9 * 0.97 + 0.5 * 0.1 * (0.5 * 0.97 + 0.5 * 0.2) rounded to the nearest tenth of a percent.
A - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%

8. Two painters are painting a room. Painter 1 takes 52:36 to paint the room, and painter 2 takes 26:18 to paint the room. With these two painters working together, how long should the job take?
A - 9:16, B - 10:52, C - 17:32, D - 35:02, E - 39:44

9. Statistics show that people who work out n days a week have a (1/10)(n+2) chance of getting a 6-pack, and the number of people who exercise n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person is selected. Find the probability that they have a 6-pack.
A - 13/30, B - 17/30, C - 19/30, D - 23/30, E - 29/30

10. A factory must produce 3,000 items today. The manager of the factory initially calls over 25 employees, each producing 5 items per hour starting at 9 AM. However, he needs all of the items to be produced by 9 PM, and realizes that he must speed up the process. At 12 PM, the manager then encourages his employees to work faster by increasing their pay, in which they then all speed up to 6 items per hour. At 1 PM, the manager calls in 15 more employees which make 5 items per hour each. Unfortunately, at 3 PM, the AC stops working and the hot sun starts taking its toll, which slows every employee down by 2 items per hour. At 4 PM, the technician fixes the AC, and all employees return to producing 5 items per hour. At 5 PM, the manager calls in 30 more employees, which again make 5 items per hour. At 6 PM, he calls in 30 more employees. At 7 PM, he rewards all the pickers again, speeding them up to 6 items per hour. But at 8 PM, n employees suddenly crash out and stop working due to fatigue, and the rest all slow back down to 5 items per hour because they are tired. The manager does not have any more employees, so if too many of them drop out, he is screwed and will have to go overtime. Find the maximum value of n such that all of the items can still be produced on time, done no later than 9 PM.
A - 51, B - 52, C - 53, D - 54, E - 55

11. Two congruent right rectangular prisms stand near each other. Both have the same orientation and altitude. A plane that cuts both prisms into two pieces passes through the vertical axes of symmetry of both prisms and does not cross the bottom or top faces of either prism. Let the point that the plane crosses the axis of symmetry of the first prism be A, and the point that the plane crosses the axis of symmetry of the second prism be B. A is 81% of the way from the bottom face to the top face of the first prism, and B is 69% of the way from the bottom face to the top face of the second prism. What percent of the total volume of both prisms combined is above the plane?
A - 19%, B - 25%, C - 50%, D - 75%, E - 81%

12. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?
A - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40

13. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?
A - 2, B - 3, C - 4, D - 5, E - 6

14. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.
A - 15, B - 18, C - 21, D - 24, E - 27

15. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn’t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes the shot?
A - 3/128, B - 3/64, C - 3/32, D - 3/16, E - 3/8

16. Amy purchases 6 fruits from a store. At the store, they have 5 of each of 5 different fruits. How many different combinations of fruits could Amy buy?
A - 210, B - 205, C - 195, D - 185, E - 180

17. Find the area of a cyclic quadrilateral with side lengths 6, 9, 4, and 2, rounded to the nearest integer.
A - 16, B - 19, C - 22, D - 25, E - 28

18. Find the slope of the line tangent to the graph of y = x^2 + x + 1 at the point (2, 7).
A - 2, B - 3, C - 4, D - 5, E - 6

19. Let f(n) = 4096n/(2^n). Find f(1) + f(2) + … + f(12).
A - 8142, B - 8155, C - 8162, D - 8169, E - 8178

20. Find the sum of all positive integers n greater than 1 and less than 16 such that (n-1)! + 1 is divisible by n.
A - 41, B - 44, C - 47, D - 50, E - 53

21. In a list of integers where every integer in the list ranges from 1 to 200, inclusive, and the chance of randomly drawing an integer n from the list is proportional to n if n <= 100 and to 201 - n if n >= 101, what is the sum of the numerator and denominator of the probability that a random integer drawn from the list is greater than 30, when expressed as a common fraction in lowest terms?
A - 1927, B - 2020, C - 2025, D - 3947, E - 3952

22. In a small town, there were initially 9 people who did not have a certain bacteria and 3 people who did. Denote this group to be the first generation. Then those 12 people would randomly get into 6 pairs and reproduce, making the second generation, consisting of 6 people. Then the process repeats for the second generation, where they get into 3 pairs. Of the 3 people in the third generation, what is the probability that exactly one of them does not have the bacteria? Assume that if at least one parent has the bacteria, then the child is guaranteed to get it.
A - 8/27, B - 1/3, C - 52/135, D - 11/27, E - 58/135

23. Amy, Steven, and Melissa each start at the point (0, 0). Assume the coordinate axes are in miles. At t = 0, Amy starts walking along the x-axis in the positive x direction at 0.6 miles per hour, Steven starts walking along the y-axis in the positive y direction at 0.8 miles per hour, and Melissa starts walking along the x-axis in the negative x direction at 0.4 miles per hour. However, a club that does not like them patrols the circumference of the circle x^2 + y^2 = 1. Three officers of the club, equally spaced apart on the circumference of the circle, walk counterclockwise along its circumference and make one revolution every hour. At t = 0, one of the officers of the club is at (1, 0). Any of Amy, Steven, and Melissa will be caught by the club if they walk within 50 meters of one of their 3 officers. How many of the three will be caught by the club?
A - 0, B - 1, C - 2, D - 3, E - Not enough info to determine

24.
A list of 9 positive integers consists of 100, 112, 122, 142, 152, and 160, as well as a, b, and c, with a <= b <= c. The range of the list is 70, both the mean and median are multiples of 10, and the list has a unique mode. How many ordered triples (a, b, c) are possible?
A - 1, B - 2, C - 3, D - 4, E - 5

25. What is the integer closest to the value of tan(83)? (The 83 is in degrees)
A - 2, B - 3, C - 4, D - 6, E - 8

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Practice AMC 10A

1. Find the sum of the infinite geometric series 1 + 7/18 + 49/324 + …
A - 36/11, B - 9/22, C - 18/11, D - 18/7, E - 9/14

2. What is the first digit after the decimal point in the square root of 420?
A - 1, B - 2, C - 3, D - 4, E - 5

3. Caden’s calculator is broken and two of the digits are swapped for some reason. When he entered in 9 + 10, he got 21. What is the sum of the two digits that got swapped?
A - 2, B - 3, C - 4, D - 5, E - 6

4. Two circles with radiuses 47 and 96 intersect at two points A and B. Let P be the point 82% of the way from A to B. A line is drawn through P that intersects both circles twice. Let the four intersection points, from left to right be W, X, Y, and Z. Find (PW/PX)*(PY/PZ).
A - 50/5863, B - 47/96, C - 1, D - 96/47, E - 5863/50

5. Two dice are rolled, and the two numbers shown are a and b. How many possible values of ab are there?
A - 17, B - 18, C - 19, D - 20, E - 21

6. What is the largest positive integer that cannot be expressed in the form 6a + 9b + 4 + 20d, where a, b, and d are positive integers?
A - 29, B - 38, C - 43, D - 76, E - 82

7. What is the absolute difference of the probabilities of getting at least 6/10 on a 10-question true or false test and at least 3/5 on a 5-question true or false test?
A - 63/1024, B - 63/512, C - 63/256, D - 63/128, E - 0

8. How many arrangements of the letters in the word “sensor” are there such that the two vowels have an even number of letters (remember 0 is even) between them (including the original “sensor”)?
A - 72, B - 108, C - 144, D - 216, E - 432

9. Find the value of 0.9 * 0.97 + 0.5 * 0.1 * (0.5 * 0.97 + 0.5 * 0.2) rounded to the nearest tenth of a percent.
A - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%

10. Two painters are painting a room. Painter 1 takes 52:36 to paint the room, and painter 2 takes 26:18 to paint the room. With these two painters working together, how long should the job take?
A - 9:16, B - 10:52, C - 17:32, D - 35:02, E - 39:44

11. Suppose that on the coordinate grid, the x-axis represents climate, and the y-axis represents landscape, where -1 <= x, y <= 1 and a higher number for either coordinate represents better conditions along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent cities, plains, desert, snowy lands, and mountains, respectively. An area is classified as whichever point it is closest to. Suppose a theoretical new area is selected by picking a random point within the square bounded by plains, desert, snowy lands, and mountains as its vertices. What is the probability that it is a plains?
A - 1 - (1/4)pi, B - 1/5, C - (1/16)pi, D - 1/4, E - 1/8

12. Statistics show that people who work out n days a week have a (1/10)(n+2) chance of getting a 6-pack, and the number of people who exercise n days a week is directly proportional to 8 - n (Note that n can only be an integer from 0 to 7, inclusive). A random person is selected. Find the probability that they have a 6-pack.
A - 13/30, B - 17/30, C - 19/30, D - 23/30, E - 29/30

13. A factory must produce 3,000 items today. The manager of the factory initially calls over 25 employees, each producing 5 items per hour starting at 9 AM. However, he needs all of the items to be produced by 9 PM, and realizes that he must speed up the process. At 12 PM, the manager then encourages his employees to work faster by increasing their pay, in which they then all speed up to 6 items per hour. At 1 PM, the manager calls in 15 more employees which make 5 items per hour each. Unfortunately, at 3 PM, the AC stops working and the hot sun starts taking its toll, which slows every employee down by 2 items per hour. At 4 PM, the technician fixes the AC, and all employees return to producing 5 items per hour. At 5 PM, the manager calls in 30 more employees, which again make 5 items per hour. At 6 PM, he calls in 30 more employees. At 7 PM, he rewards all the pickers again, speeding them up to 6 items per hour. But at 8 PM, n employees suddenly crash out and stop working due to fatigue, and the rest all slow back down to 5 items per hour because they are tired. The manager does not have any more employees, so if too many of them drop out, he is screwed and will have to go overtime. Find the maximum value of n such that all of the items can still be produced on time, done no later than 9 PM.
A - 51, B - 52, C - 53, D - 54, E - 55

14. Find the number of positive integers n less than 69 such that the average of all the squares from 1^2 to n^2, inclusive, is an integer.
A - 11, B - 12, C - 23, D - 24, E - 48

15. Find the number of ordered pairs (a, b) of integers such that (a - b)^2 = 625 - 2ab.
A - 6, B - 10, C - 12, D - 16, E - 20

16. What is the 420th digit after the decimal point in the decimal expansion of 1/13?
A - 4, B - 5, C - 6, D - 7, E - 8

17. Two congruent right rectangular prisms stand near each other. Both have the same orientation and altitude. A plane that cuts both prisms into two pieces passes through the vertical axes of symmetry of both prisms and does not cross the bottom or top faces of either prism. Let the point that the plane crosses the axis of symmetry of the first prism be A, and the point that the plane crosses the axis of symmetry of the second prism be B. A is 81% of the way from the bottom face to the top face of the first prism, and B is 69% of the way from the bottom face to the top face of the second prism. What percent of the total volume of both prisms combined is above the plane?
A - 19%, B - 25%, C - 50%, D - 75%, E - 81%

18. What is the greatest number of positive integer factors an integer from 1 to 100 can have?
A - 10, B - 12, C - 14, D - 15, E - 16

19. On an analog clock, the minute hand makes one full revolution every hour, and the hour hand makes one full revolution every 12 hours. Both hands move at a constant rate. During which of the following time periods does the minute hand pass the hour hand?
A - 7:35 - 7:36, B - 7:36 - 7:37, C - 7:37 - 7:38, D - 7:38 - 7:39, E - 7:39 - 7:40

20. Find the smallest positive integer that is a leg in three different Pythagorean triples.
A - 12, B - 14, C - 15, D - 20, E - 21

21. How many axes of symmetry does the graph of (x^2)(y^2) = 69 have?
A - 2, B - 3, C - 4, D - 5, E - 6

22. Real numbers a, b, and c are chosen uniformly and at random from 0 to 3. Find the probability that a + b + c is less than 2.
A - 4/81, B - 8/81, C - 4/27, D - 8/27, E - 2/3

23. Let f(n) be the sum of the positive integer divisors of n. Find the sum of the digits of the smallest odd positive integer n such that f(n) is greater than 2n.
A - 15, B - 18, C - 21, D - 24, E - 27

24. Find the last three digits of 24^10.
A - 376, B - 576, C - 626, D - 876, E - 926

25. A basketball has a diameter of 9 inches, and the hoop has a diameter of 18 inches. Peter decides to pick up the basketball and make a throw. Given that Peter has a 1/4 chance of accidentally hitting the backboard and missing the shot, but if he doesn’t, he is guaranteed that the frontmost point of the basketball will be within 18 inches of the center of the hoop at the moment when a great circle of the basketball crosses the plane containing the rim. No part of the ball will extend behind the backboard at any point during the throw, and the rim is attached directly to the backboard. What is the probability that Peter makes the shot?
A - 3/128, B - 3/64, C - 3/32, D - 3/16, E - 3/8

do they send the Interesting Test to everyone who applied or do they read the friendly letter first and only send to the kids they like?

Let and Prove that

A positive integer is given, find the largest real number such that there exists a geometric sequence with common ratio satisfying
for all positive integers . Here, denotes the distance from the real number to the nearest integer.

2 Let there be an infinite geometric sequence , where the common ratio . Given that


find the largest possible range of .

Suppose is your favourite function, is your mother's favourite function, and is your father's favourite function. It was discovered that for any reals , Prove that you are boring.

(Hint: you might need to use the quotable result that if someone's favourite function is a linear polynomial, they are boring)

If the sum of all positive divisors (including itself) of a positive integer is , then is called a perfect number. For example, the sum of the positive divisors of 6 is , hence 6 is a perfect number.
Prove: There does not exist a perfect number of the form , where are positive integers, and are odd primes.

Round 2

1 Define the isolation index of real number to be , where is the largest integer no greater than , and . For each positive integer , find the maximum possible real number such as there exists an infinite geometric sequence with common ratio and isolation index of each term being at least .

Let and prove

In the plane, there are two line segments and , with . Prove that there exists and only exists one point such that .( corresponds to , corresponds to )
Click to reveal hidden text

In triangle , let be the -excenter. Points and are placed on line such that is between and , and is between and . Moreover, and are the contact points of with the -excircle of triangles and , respectively. Let be the -excenter of triangle , and let be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.

Proposed by Ali Nazarboland

Let a,b,c be positive real numbers such that x +y+z = 3. Prove that
yx^3 +zy^3+xz^3+9xyz≤ 12.