This is a challenge to the whole AoPS community I bet you can't prove this basic inequality: by different methods. The same inequality can be used twice but in different ways, dm me if you have any question about the rules.
Feelin good for this one.

I've heard using colored pencils is really useful for geometry problems. Is this only for very hard problems, or can it be used in MATHCOUNTS/AMC 8/10? An example problem would be much appreciated.

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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!

Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas.
Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas.

Find all functions such that for all , we have .

Find all functions such that, for all integers that satisfy , the following equality holds:

(Here denotes the set of integers.)

Proposed by Liam Baker, South Africa

Let be a board. Some of its cells are colored black in such a way that every board of has at most black cells. Find the maximum amount of black cells that the board may have.

Show that there exists an integer ? such that
[tex3]\frac{a^{29} - 1}{a - 1}[/tex3]
have at least 2007 distinct prime factors.

Let a ≥b ≥c ≥0 be real numbers such that a^2 +b^2 +c^2 +abc = 4. Prove that
a+b+c+(√a−√c)^2 ≥3.

Find all functions such that is unbounded and
is a perfect square for all integer

Let a,b,c be positive real numbers such that x +y+z = 3. Prove that
yx^3 +zy^3+xz^3+9xyz≤ 12.

In triangle , let be the -excenter. Points and are placed on line such that is between and , and is between and . Moreover, and are the contact points of with the -excircle of triangles and , respectively. Let be the -excenter of triangle , and let be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.

Proposed by Ali Nazarboland

Practice AMC 12A

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. 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 “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

7. 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%

8. 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

9. 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

10 (Main). PM me for problem (I copied over this problem from the 10A but just found out a “sheriff” removed it for some reason so I don’t want to take any risks)
A - 51, B - 52, C - 53, D - 54, E - 55

10 (Alternate). 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

11. 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%

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 a green FN?
A - 3/128, B - 3/64, C - 3/32, D - 3/16, E - 3/8

16. Martin decides to rob 6 packages of Kool-Aid from a store. At the store, they have 5 packages each of 5 different flavors of Kool-Aid. How many different combinations of Kool-Aid could Martin rob?
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. Suppose that the strength of a protest is measured in “effectiveness points”. Malcolm gathers 2048 people for a protest. During the first hour of the protest, all 2048 people protest with an effectiveness of 1 point per person. At the start of each hour of the protest after the first, half of the protestors will leave, but the ones remaining will gain one effectiveness point per person. For example, that means that during the second hour, there will be 1024 people protesting at 2 effectiveness points each, during the third hour, there will be 512 people protesting at 3 effectiveness points each, and so on. The protest will conclude at the end of the twelfth hour. After the protest is over, how many effectiveness points did it earn in total?
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. Scientific research suggests that Stokely Carmichael had an IQ of 30. Given that IQ ranges from 1 to 200, inclusive, goes in integer increments, and the chance of having an IQ of n 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 person is smarter than Stokely Carmichael, when expressed as a common fraction in lowest terms?
A - 1927, B - 2020, C - 2025, D - 3947, E - 3952

22. In Alabama, Jim Crow laws apply to anyone who has any positive amount of Jim Crow ancestry, no matter how small the fraction, as long as it is greater than zero. In a small town in Alabama, there were initially 9 Non-Jim Crows and 3 Jim Crows. 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 is Non-Jim Crow?
A - 8/27, B - 1/3, C - 52/135, D - 11/27, E - 58/135

23. Goodman, Chaney, and Schwerner each start at the point (0, 0). Assume the coordinate axes are in miles. At t = 0, Goodman starts walking along the x-axis in the positive x direction at 0.6 miles per hour, Chaney starts walking along the y-axis in the positive y direction at 0.8 miles per hour, and Schwerner starts walking along the x-axis in the negative x direction at 0.4 miles per hour. However, a clan that does not like them patrols the circumference of the circle x^2 + y^2 = 1. Three knights of the clan, 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 knights of the clan is at (1, 0). Any of Goodman, Chaney, and Schwerner will be caught by the clan if they walk within 50 meters of one of their 3 knights. How many of the three will be caught by the clan?
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

For any positive integer , let denote the number of its positive divisors (including 1 and itself). Determine all positive integers for which there exists a positive integer such that .