Product of two integers is less than three times of their sum. Find those integers.

Let be an acute-angled triangle with and incenter . Let be the mixtilinear circle at vertex , i.e. the circle internally tangent to the circumcircle of and also tangent to lines and . A circle passes through points and and is tangent to at point , with and being inside . Prove that:

Let , , , be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, . Prove that for all .

Proposed by Morteza Saghafian, Iran

With Find all functions satisfied

im in 7th grade and took the AMC 10 A/B with absouletely nauseating score, which i will not reveal. I wasnt even close to AIME frankly. My goals are the following:
7th grade: AMC 8 - DHR
8th grade:AIME qual, AMC 8 Perfect
9th grade: AMC 10 DHR maybe?, AIME 7+
10th grade: USAJMO, AIME 12+, AMC 10 DHR
11th grade: USAMO, AIME 12+, AMC 12 DHR
12th grade: USAMO, AIME great score, AMC 12 perfect or close?
These are the goals that i want to achieve. I will do literally anything to achieve them. Can someone please give me a good tip so i can follow it for the next 5 years of my life to become a 3 time USAMO qual and a 5 time AIME qual, and have an perfect AMC 8 under my belt?

I can’t find it anywhere

problem

How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

Let be positive real numbers.
Prove that :

links for details on each comp

i think thats all the major ones, comment if you have any more that i forgot to include
also it would really make my day if you

EDIT: i forgot ARML and JBMO, comment if you like those

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

Let triangle be an acute triangle with and let and be its orthocenter and circumcenter, respectively. Let be the circle . The line and the circle of radius centered at cross at and , respectively. Prove that , the circle on diameter and circle are concurrent.
Proposed by Romania, Radu-Andrew Lecoiu

Prove that for every odd prime p there are infinitely many positive integers k such that the exponents
of 2 and k in the prime factorization of k! are even.

Let . Prove that: