Prove that for each real number , there are exactly two or three positive real numbers satisfying the equation .

Given positive integer Let and Show that

How many people actually get a flat 0/42 on jmo? I took it for the first time this year and I had never done oly math before so I really only had 2 weeks to figure it out since I didn’t think I would qual. I went in not expecting much but I didn’t think I wouldn’t be able to get ANYTHING. So I’m pretty sure I got 0/42 (unless i get pity points for writing incorrect solutions). Is that bad, am I sped, and should I be embarrassed? Or do other people actually also get 0?

Find all solutions to
in integers.

Proposed by Seongjin Shim

Explain how you got to that answer

Let a,b,c be real numbers with (a^2+b^2)(b^2+c^2)(c^2+a^2)>0. Prove that \left(6ab+6bc+6ca-a^2-b^2-c^2\right)\cdot\left(\frac{1}{a^2+b^2}+\frac{1}{b^2+c^2}+\frac{1}{c^2+a^2}\right)\le \frac{45}{2}

, , . .
Prove that : -is integer.

Let , .
polynomial with only one root.

If are real numbers such that prove inequality

Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an chessboard is . (Two bishops are said to be attacking if they are on a common diagonal).

My dream is to be on IMO team ik thats not going to happen b/c the kids that make it are like 6th mop quals :play_ball:. I somehow got a index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs. made mc nats

Is there some ways to get good at combinatorics? Im tired of this shi. Also I have some cobinotoics prbems that I have collected. Please leave a comment.
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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.

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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.

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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. A southern plantation has a length of 50 meters and a width of 60 meters. On the plantation, there is 1 kg of cotton per square meter waiting to be picked. The master of the plantation initially calls over 25 cotton pickers, each picking cotton at a rate of 5 kg per hour starting at 9 AM. However, he wants all of the cotton to be picked by 9 PM, and realizes that he needs to speed up the process. At 12 PM, the master then encourages his pickers to work faster by whipping them, in which they then all speed up to 6 kg per hour. At 1 PM, the master calls in 15 more pickers which pick at 5 kg per hour. Unfortunately, at 3 PM, the clouds drift away and the hot sun starts beating down, which slows every picker down by 2 kg per hour. At 4 PM, the clouds return, and all pickers return to picking at 5 kg per hour. At 5 PM, the master calls in 30 more pickers, which again pick at 5 kg per hour. At 6 PM, he calls in 30 more pickers. At 7 PM, he whips all the pickers again, speeding them up to 6 kg per hour. But at 8 PM, n pickers suddenly crash out and stop working due to fatigue, and the rest all slow back down to 5 kg per hour because they are tired. The master does not have any more pickers, 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 cotton can still be picked 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 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