a) If I have three sheep, how can I arrange them so that they are all an equal distance away from each other?
b)If I have four sheep, how can I do the same thing as I wanted to do in part a)?

Make your argument here on why your AoPS user ID is the coolest!!!!! :pilot: :pilot: :pilot: :pilot: :pilot:

For instance, my ID, , can be written as , both which are prime.

is a cuban prime, prime index prime

is a Sophie Germain prime

I think my ID is pretty cool, but theres probably better IDs out there.

stuff

I came across this question from MathCounts Nationals 2022 Sprint #24, but I could not figure out how to do it.
Can someone help me on this question?

If S is a nonempty set, P(S) is defined to be the product of the elements in S. What is the sum of the values of P(S) over all the nonempty subsets S of {1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10}? Express your answer as a common fraction.

yay 1000 posts!

here's a problem:

The mean, median, and mode of the data values are all equal to . What is the value of ?



(AMC 10A 2016 #7

If f : R^+ --> R^+ satisfying f(f(x)/y ) = yf ( y ) + (f(x)). Then the value of α such that Sigma K = 1 to n [ 1 / f(K) ] = 420

Find all function satisfying:
.

Find all functions that satisfy:

Let be reals . Prove that

[center]

Me (PatTheKing806) and EaZ_Shadow have created a AMC 10/AIME Study Forum! Hopefully, this forum wont die quickly. To signup, do /join or \join.

Click here to join! (or do some pushups) :P

People should join this forum if they are wanting to do well on the AMC 10 next year, trying get into AIME, or loves math!

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Gunn Math Competition will take place at Gunn High School in Palo Alto, California on THIS Sunday, March 30th. Gather a team of up to four and compete for over $7,500 in prizes! The deadline to sign up is March 27th. We welcome participants of all skill levels, with separate Beginner and Advanced (AIME) divisions for all students, from advanced 4th graders to 12th graders.

For more information, check our website, [url][/url]compete.gunnmathcircle.org, where registration is free and now open. The deadline to sign up is this Friday, March 28th. If you are unable to make a team, register as an individual and we will be able to create teams for you.

Special Guest Speaker: Po-Shen LohIMAGE
We are honored to welcome Po-Shen Loh, a world-renowned mathematician, Carnegie Mellon professor, and former coach of the USA International Math Olympiad team. He will deliver a several 30-minute talks to both students and parents, offering deep insights into mathematical thinking and problem-solving in the age of AI!

For any questions, reach out at ghsmathcircle@gmail.com or ask in our Discord server, which you can join through the website.

Find information on our AoPS page too! https://artofproblemsolving.com/wiki/index.php/Gunn_Math_Competition_(GMC)
Thank you to our sponsors for making this possible!
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Check out our flyer! IMAGE

Hey everyone!

I’m back with another practice test. Sorry this one took a while to pump out since I have been busy lately.

Post your score/distribution, favorite problems, and thoughts on the difficulty of the test down below. Hope you enjoy!


Practice AMC 10A

1. Find the sum of the infinite geometric series 1/2 + 7/36 + 49/648 + …
A - 18/11, B - 9/22, C - 9/11, 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 + 4c + 20d, where a, b, c, 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 - 0, B - 1/504, C - 1/252, D - 1/126, E - 1/63

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

9. After opening his final exam, Jason does not know how to solve a single question. So he decides to pull out his phone and search up the answers. Doing this, Jason has a success rate of anywhere from 94-100% for any given question he uses his phone on. However, if the teacher sees his phone at any point during the test, then Jason gets a 0.5 multiplier on his final test score, as well as he must finish the rest of the test questions without his phone. (Assume Jason uses his phone on every question he does until he finishes the test or gets caught.) Every question is a 5-choice multiple choice question. Jason has a 90% chance of not being caught with his phone. What is the expected value of Jason’s test score, rounded to the nearest tenth of a percent?
A - 89.9%, B - 90.0%, C - 90.1%, D - 90.2%, E - 90.3%

10. A criminal is caught by a police officer. Due to a lack of cooperation, the officer calls in a second officer so they can start the arrest smoothly. Officer 1 takes 26:18 to arrest a criminal, and officer 2 takes 13:09 to arrest a criminal. With these two police officers working together, how long should the arrest take?
A - 4:23, B - 5:26, C - 8:46, D - 17:32, E - 19:44

11. Suppose that on the coordinate grid, the x-axis represents economic freedom, and the y-axis represents social freedom, where -1 <= x, y <= 1 and a higher number for either coordinate represents more freedom along that particular axis. Accordingly, the points (0, 0), (1, 1), (-1, 1), (-1, -1), and (1, -1) represent democracy, anarchy, socialism, communism, and fascism, respectively. A country is classified as whichever point it is closest to. Suppose a theoretical new country is selected by picking a random point within the square bounded by anarchy, socialism, communism, and fascism as its vertices. What is the probability that it is fascist?
A - 1 - (1/4)pi, B - 1/5, C - (1/16)pi, D - 1/4, E - 1/8

12. Statistics show that people in Memphis who eat at KFC n days a week have a (1/10)(n+2) chance of liking kool-aid, and the number of people who eat at KFC 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 in Memphis is selected. Find the probability that they like kool-aid.
A - 13/30, B - 17/30, C - 19/30, D - 23/30, E - 29/30

13. PM me for problem (I don’t want to post it on here again because apparently a “sheriff” got rid of it)
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 towers stand near each other. Both take the shape of a right rectangular prism. A plane that cuts both towers into two pieces passes through the vertical axes of symmetry of both towers and does not cross the floor or roof of either tower. Let the point that the plane crosses the axis of symmetry of the first tower be A, and the point that the plane crosses the axis of symmetry of the second tower be B. A is 81% of the way from the floor to the roof of the first tower, and B is 69% of the way from the floor to the roof of the second tower. What percent of the total mass of both towers 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 a green FN?
A - 3/128, B - 3/64, C - 3/32, D - 3/16, E - 3/8

There are bowls arranged in a row, numbered through , where and are given positive integers. Initially, each of the first bowls contains an apple, and each of the last bowls contains a pear.
A legal move consists of moving an apple from bowl to bowl and a pear from bowl to bowl , provided that the difference is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first bowls each containing a pear and the last bowls each containing an apple. Show that this is possible if and only if the product is even.

Everybody, with EGMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict

For any positive numbers and natural numbers prove the inequality

Let be reals . Prove that

Find all and

Show that J>0. For this to be true, I guess we also need the condition for c and d to be positive, isn't that right?

Prove where and

Show that the determinant: is non-negative, if its elements etc., are real.

If and are analytic functions, show that is also analytic.

Hi everyone!
I noticed that every time i try to answer a question and get it right/wrong the screen freezes and i can't scroll anymore...
100%
Chrome
PC
do u guys have the same issue?
I don't think it is like this for Alcumus but for normal class problems it does
*the challenge problems

EDIT: sry for the terrible title :blush:

What does this mean?

Is this a glitch

Several posts have been made about this years before, but the office hours banner hasn't appeared yet even though I refreshed the page multiple times. It has been 50 minutes since when office hours should have started and no mods have responded to a question I posted 3 hours ago (relative to last edit). Can others confirm?

This is very wrong