Find all prime number pairs such that is a perfect square.

Proposed by chengbilly

Let be positive numbers such that . Find the maximum value of

Given a triangle with circumcircle , and two arbitrary points on . Let , , be points on lines , , , respectively, such that , , and concur at a point . Let be a point on line such that , , , are concyclic. Similarly, let be a point on line such that , , , are concyclic, and let be a point on line such that , , , are concyclic. Prove that , , concur at a single point.

Proposed by chengbilly

Problem. Given be non-negative real numbers such that Prove that

Let Prove that

Points lie on a circle and point lies outside the circle. The given points are such that (i) lines and are tangent to , (ii) are collinear, and (iii) . Prove that bisects .

is a triangle with incenter . The foot of the perpendicular from to is , and the foot of the perpendicular from to is . Prove that .

Alex Zhu.

Prove that
a+b+c\ge 2\left(\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}\right)

holds \forall a,b,c\ge 0: a^3+b^3+c^3+2abc=5.

Let be an acute triangle with . Denote by and points on the segment such that . is a point on segment . intersects and at and , respectively. The angle bisectors of and intersect at . If and , prove that is cyclic.

Let be an acute triangle. Let be the altitude on , and let be any interior point on . Lines , when extended, intersect at respectively. Prove that .

Given with , let be the bisector of with on the side . Let be a circle passing through and which is tangent to at . Suppose cuts the side again at . The tangent to the circumcircle of at intersects again at . Let be the intersection point of the segments and . Prove that is perpendicular to .

The Lexington High School Math Team is proud to announce LMT Spring 2025 and our inaugural Girls’ LMT 2025! LMT is a competition for middle school students interested in math. Students can participate individually, or on teams of 4-6 members. This announcement contains information for BOTH competitions.

LMT Spring 2025 will take place from 8:30 AM-5:00 PM on Saturday, May 3rd at Lexington High School, 251 Waltham St., Lexington, MA 02421.

The competition will include two individual rounds, a Team Round, and a Guts Round, with a break for lunch and mini-events. A detailed schedule is available at https://lhsmath.org/LMT/Schedule.

There is a $15 fee per participant, paid on the day of the competition. Pizza will be provided for lunch, at no additional cost.

Register for LMT at https://lhsmath.org/LMT/Registration/Home.

Girls’ LMT 2025 will be held ONLINE on MathDash from 11:00 AM-4:15 PM EST on Saturday, April 19th, 2025. Participation is open to middle school students who identify as female or non-binary. The competition will include an individual round and a team round with a break for lunch and mini-events. It is free to participate.

Register for GLMT at https://www.lhsmath.org/LMT/Girls_LMT.

More information is available on our website: https://lhsmath.org/LMT/Home. Email lmt.lhsmath@gmail.com with any questions.

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 60 meters and a width of 50 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

Hi everyone,
I just have a question. I live in PA and I sadly didn't make it to nationals this year. Is PA a competitive state? I'm new into mathcounts and not sure