such that 
.
Where 
with legs 
and 
is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?
be non-negative real numbers such that 
.![\[
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\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
be a permutation of 
such that there are exactly 
ordered pairs of integers 
satisfying 
and 
.
students, 
students play cricket, 
students play hockey and 
students play basketball. 
students play both basketball and hockey, 
students play both cricket and basketball, 
students play both cricket and hockey, and 
students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.
be reals such that 
. Prove that
Let 
. Prove that
,
,
is not greater than 
.
Write only the answers to the questions given.
to be more exact:
in years 2002-08 time was 90' for part A and 120' for part B
white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...
unit. The area of the hexagon is ...
and 
.
is ...
so that 
and 
is ...
,
,
,
has an average of 
and a median of 
.
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
.
and 
are squares of two consecutive natural numbers, then the smallest value of 
is ...
,
and 
.
and 
lies on the line segment 
.
and 
.
is ...
meet 
for each natural number 
.
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ...
and 
.
.
is a point on the side extension 
so that 
is perpendicular to 
and 
.
is ...
consists of 
with 
positive reals is ....
with 
is ....
square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of 
coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of 
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I am going to rant about how the AMC is very stupid. I welcome your comments and let me know if you agree with me.
I love doing math. I love thinking about olympiad type problems over several days; I often look forward to weekends when I have time to work on IMO Shortlists, Russian and Romanian problems from the Contests page, and USAMTS problems. These Olympiad type problems are sometimes stupid/contrived/there's really no use in doing them except to practice for high school math competitions; especially when it comes to inequalities and advanced euclidean geometry. I don't waste my time with these, because I don't do Olympiads to practice for math competitons. I do Olympiad problems to improve my problem solving skills, for fun, and because they are challenging yet the mathematics involved is elementary. I tend to work on combinatorics and number theory problems which require problem solving only, and not some ridiculous "trick" or pointless bashes. And I don't care about speed either - I take my time, I like to ponder.
And I can say I have greatly improved my problem solving skills in the past year by doing these kinds of problems. And so I look forward to taking the USAMO, if I qualify for it, as I believe I do stand a chance. The problem for me, is qualifying. We all know that to qualify for the USAMO, you have to have a superb combined performance on the AIME and the AMC. Now, I have never really enjoyed computational math - its boring, its easy to make mistakes, and it often tests concepts which have been recycled from one contest to another, many times over - you might call these concepts "tricks". And so every year I dread the AMC, and the AIME - I don't do well with the time constraints, I always make mistakes, and I end up not making the USA(J)MO.
So I would like to rant about the stupidity of the AMC, both as a test of skill in mathematics, and its relation to the culture of math competitions in the U.S. In many other countries with national math competitions, there is no computational exam - only Olympiad. Computational contests make little sense in the first place, as they do not produce "thinkers", "mathematicians", or even "problem-solvers". From my experience, at least with the AMC, they instead produce "m(athletes)", "expert test-takers", and "top-scorers". This is NOT what real math is about, and so it is my belief that these tests do not produce the kind of thinkers that other countries, especially those in Eastern Europe, do so well. The majority of scores on the USAMO are close to 0 because the AMC selects speedy, "tricksters", trained to solve the same kinds of problems very quickly, and not ponderers, creative thinkers, and problem-solvers. One can look at where the Eastern European olympiad veterans end up - as emmenint mathematicians, many with Fields medals; and where AMC top scorers end up - on Wall Street.
The AMC is also just a ridiculous method of selecting people for the USAMO. For me it is quite difficult to make the USAMO through the AMC 12. I don't believe I can get more than 18 questions right on the AMC 12, due to time constraints, and I'm sick of practicing for this boring competition. I am afraid I will be dissapointed again this year, after I fail to make USAMO, as it will have been another oppurtunity lost to practice my Olympiad problem solving skills. I don't want to leave high school never having been able to show my skills in a contest setting.
Please comment.
I love doing math. I love thinking about olympiad type problems over several days; I often look forward to weekends when I have time to work on IMO Shortlists, Russian and Romanian problems from the Contests page, and USAMTS problems. These Olympiad type problems are sometimes stupid/contrived/there's really no use in doing them except to practice for high school math competitions; especially when it comes to inequalities and advanced euclidean geometry. I don't waste my time with these, because I don't do Olympiads to practice for math competitons. I do Olympiad problems to improve my problem solving skills, for fun, and because they are challenging yet the mathematics involved is elementary. I tend to work on combinatorics and number theory problems which require problem solving only, and not some ridiculous "trick" or pointless bashes. And I don't care about speed either - I take my time, I like to ponder.
And I can say I have greatly improved my problem solving skills in the past year by doing these kinds of problems. And so I look forward to taking the USAMO, if I qualify for it, as I believe I do stand a chance. The problem for me, is qualifying. We all know that to qualify for the USAMO, you have to have a superb combined performance on the AIME and the AMC. Now, I have never really enjoyed computational math - its boring, its easy to make mistakes, and it often tests concepts which have been recycled from one contest to another, many times over - you might call these concepts "tricks". And so every year I dread the AMC, and the AIME - I don't do well with the time constraints, I always make mistakes, and I end up not making the USA(J)MO.
So I would like to rant about the stupidity of the AMC, both as a test of skill in mathematics, and its relation to the culture of math competitions in the U.S. In many other countries with national math competitions, there is no computational exam - only Olympiad. Computational contests make little sense in the first place, as they do not produce "thinkers", "mathematicians", or even "problem-solvers". From my experience, at least with the AMC, they instead produce "m(athletes)", "expert test-takers", and "top-scorers". This is NOT what real math is about, and so it is my belief that these tests do not produce the kind of thinkers that other countries, especially those in Eastern Europe, do so well. The majority of scores on the USAMO are close to 0 because the AMC selects speedy, "tricksters", trained to solve the same kinds of problems very quickly, and not ponderers, creative thinkers, and problem-solvers. One can look at where the Eastern European olympiad veterans end up - as emmenint mathematicians, many with Fields medals; and where AMC top scorers end up - on Wall Street.
The AMC is also just a ridiculous method of selecting people for the USAMO. For me it is quite difficult to make the USAMO through the AMC 12. I don't believe I can get more than 18 questions right on the AMC 12, due to time constraints, and I'm sick of practicing for this boring competition. I am afraid I will be dissapointed again this year, after I fail to make USAMO, as it will have been another oppurtunity lost to practice my Olympiad problem solving skills. I don't want to leave high school never having been able to show my skills in a contest setting.
Please comment.
,
and some other equations that you add together and then you complete the square to find there is only one solution.
to derive the inequality.
.
.
. The intent of my question was to take the test officially and have it graded, so that they are in contention for JMO. I am seeing posts where folks say that they guessed right on the last 10 (string of D's or C's or whatever) and they make AIME. Then they end up struggling on the AIME. Allowing every AMC10/12 test taker to the AIME, the stress due to the AMC10/12 is much lower...
