Hi AOPSers,

I have finished writing a mock geometry test for fun and practice for the real college competitions like HMMT/PUMaC/CMIMC... There would be 10 questions and you should finish the test in 60 minutes, the test would be close to the actual test (hopefully). You could sign up under this thread, PM me your answers!. The submission would close on March 31st at 11:59PM PST.

I would create a private discussion forum so everyone could discuss after finishing the test. This is the first mock I've written, please sign up and enjoy geometry!!

~Bluesoul

Discussion forum: Discussion forum

Leaderboard

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

Hello everyone! We are excited to announce a new competition, the Tennessee Math Tournament, created by the Tennessee Math Coalition! Anyone can participate in the virtual competition for free.

The testing window is from March 22nd to April 12th, 2025. Virtual competitors may participate in the competition at any time during that window.

The virtual competition consists of three rounds: Individual, Bullet, and Team. The Individual Round is 60 minutes long and consists of 30 questions (AMC 10 level). The Bullet Round is 20 minutes long and consists of 80 questions (Mathcounts Chapter level). The Team Round is 30 minutes long and consists of 16 questions (AMC 12 level). Virtual competitors may compete in teams of four, or choose to not participate in the team round.

To register and see more information, click here!

If you have any questions, please email connect@tnmathcoalition.org or reply to this thread!

Thank you to our lead sponsor, Jane Street!

IMAGE

Congratulations to the USA team for placing 1st at the 65th IMO that took place in Bath, United Kingdom.

The team members were:

Jordan Lefkowitz
Krishna Pothapragada
Jessica Wan
Alexander Wang
Qiao Zhang
Linus Tang

Let be an arbitrary interior point of , and , , intersect , , at , , , respectively. Suppose that be the midpoint of , and intersect at , intersects at , and intersects at . Show that is tangent to .

A positive integer with three digits is written on the board. Each second the number on the board gets replaced by , where is the largest prime divisor of .
Prove that either after 999 seconds or 1000 second the number on the board will be a power of two.

a, b, c are real numbers
|a| + |b| + |c| − |a + b| − |b + c| − |c + a| + |a + b + c| ≥ 0
hey everyone, so I came across this inequality, and I did make some progress:
Let (a+b), (b+c), (c+a) be three sums T1, T2 and T3. As there are 3 sums, but they can be of only 2 signs, by pigeon hole principle, atleast 2 of the 3 sums must be of the same sign.
But I'm getting stuck on the case work. Can anyone help?
Thnx a lot

Let be triangle with incenter and circumcircle . Let and , the line parallel to through cuts , in and . Prove that the circumradius of and are equal.

Let be an acute scalene triangle inscribed in a circle . The angle bisector of intersects at and at . The point is the midpoint of . Let be the altitude in , and the circumcircle of intersects again at . Let be the midpoint of , and let be the reflection of with respect to . Prove that the triangles and are similar.

In a school with double schooling, in the morning the language teacher divided the students into groups for an activity. In the afternoon, the math teacher divided the same students into groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.

Note: Each group has at least one student.

Positive real numbers with sum are such that the equation has a positive root smaller than .
Prove that there exists a positive integer such that the inequality holds.

Let triangle be inscribed in the circumcircle and circumscribed about the incircle , with . The incircle touches the sides , , and at , , and , respectively. A line through , perpendicular to , intersects , , and at , , and , respectively. The line meets at (distinct from ). The circumcircle of triangle intersects at (distinct from ). Let be the midpoint of the arc of . The line cuts segments and at and , respectively, and the tangents to the circle at and intersect at . Prove that .

Let be an acute scalene triangle. A point varies on its side . The points and are the midpoints of the arcs and (not containing ) of the circumcircles of triangles and , respectively. Prove that the circumcircle of triangle passes through a fixed point, independent of the choice of on .