[/quote]
,
having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer 
of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation 
.
that satisfies:![\[
\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k
\]](http://latex.artofproblemsolving.com/b/b/e/bbe7fe81e366fe390be5d17193d1581cc40dbece.png)
be a positive integer. Prove that for all integers 
,![\[
\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}
\]](http://latex.artofproblemsolving.com/0/4/4/04464c1a350e3f9a929ab51534018498680068fc.png)
be real positive numbers such that: 
.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of 
for some positive integer 
.
be integer numbers 
satisfy that
.
is a perfect square number
of a positive integer number
such that:
such that for all 
,![\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]](http://latex.artofproblemsolving.com/a/a/1/aa1bce053788a3efa59819407b41cfe472f544d0.png)
such that 
for every real 
.
,
.
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Hi everyone! I was talking to djmathman earlier today, and we both noticed an increase in threads this contest season along the lines of “I suck at math because I didn’t do well on the AMC 10/12 test”. This unfortunate thought pattern seems to be growing a lot as people associate self-worth with contest math performance. However, while it’s true that people who often do great on math contests go on to do amazing things in mathematics, doing poorly on math contests does not make a person any less of a mathematician. Contest mathematics isn’t the “be all end all” of mathematics performance: it’s merely a gateway into getting people to think about more interesting problems.
One huge contributing factor to success in contest mathematics is having seen a lot of problems. Many contest math problems are very similar to problems on previous year’s contests, and therefore, understanding a lot of problem solving techniques is critical to success. (For instance, consider 2016 AMC 12A Problem #22 . Having seen 1987 AIME I Problem #7, a student could solve this problem almost immediately. On the other hand, a student who has never seen a problem like this before would be at a huge disadvantage, because, while they could come up with a solution on the fly, they don’t have a lot of time to do so.) Time pressure is a huge element of the AMC tests; these contests don’t always allow students to fully think about problems. When I do math problems, one of my favorite things to do is sit down with an idea and work with it for a while until I really fully grasp that concept, and the MAA does a fantastic job of starting conversations about tons of interesting things in mathematics. However, during the actual contest, competitors don’t have enough time to do so. If you can’t figure out how to solve a problem on the test, while it’s natural to feel bad initially (I’ve kicked myself many times over the “could’ve would’ve” problems), remember the main goal of doing mathematics: to understand and enjoy the problems. Read the different solutions on the forums, research a topic more which you may have been unfamiliar with, read a book. Then, next time around, not only will you nail the problem on the test, but you will also understand the underlying idea and intuition behind it.
I’m currently a senior in high school, and am almost officially finished with high school math competitions. I’ve participated in the AIME for the past 5 years along with MATHCOUNTS Nationals in 8th grade. I rarely share my scores with others on these tests for two reasons: (1) they’re usually below the first quartile of scores posted on AoPS and, most importantly, (2) I don’t compete in contest math for the sake of having a good score. I do it because I enjoy the problems, and the underlying mathematical ideas which accompany them. I’m an avid lover of Number Theory problems (shameless self promotion) and problems like 2016 AMC 12B Problem #22 excite me a lot, because this problem combined ideas about repeating decimals, order of a number, and divisibility. I didn’t solve this problem until after the test was over; however, when I did, I excitedly shared it with everyone in my school’s math club while teaching them some new things in number theory. Sharing this problem with my friends and teachers is the essence of the beauty of mathematics for me, because it lends itself well to collaboration in problem solving. When I find interesting problems like these, I often have them queued up to show to various people I encounter because I love inspiring others to have this level of inquisitiveness about a mathematical idea.
I’ve also been on the writing end of several math contests, including many Mock AMC exams on the AoPS forum, most notably the 2015 Mock AIME I. I also help write problems for the NIMO contest. My favorite thing about writing these problems is allowing competitors to think about mathematical concepts in new ways. For instance, this polynomial transformation problem taught a very important idea in algebra, which is building a polynomial out of the roots (an idea which was also featured in a similar USAMO problem before). Contributing to these discussions and having people solve my problems in many different ways is incredibly humbling for me, and is part of the beauty of contest mathematics. For more information on this, I highly recommend reading djmathman’s post here .
One of the great things about contest math is it starts these discussions. And, while tons of team contests like ARML and MATHCOUNTS try to inspire this level of collaboration and communication, it seems like it is often the missing link for many students who may be kicking themselves over a low score. Instead of thinking of a 96 on the AMC 10 as a complete failure and a wasted 2 years preparing for the exam, don’t let this score define you. Instead, learn new ideas from the problems and share them with those around you. Teaching may be a passion which is just mine, but I hope that you all can learn to truly enjoy the problems. Maybe you couldn’t solve 2016 AMC 12A Problem 23 . Read the solutions online, try to understand what’s going on in the 3d graph for this problem. Study equations like these more, and understand their graphs (this is especially important when discussing the space of matrices later down the road). If you want to go way above and beyond, maybe try to start understanding double integrals, as in va2010’s post in that thread. The main point is, this problem alone can generate tons of interesting discussions, and missing out on these are a shame. Getting a bad score isn’t a bad thing, but not learning from it surely is.
For another post in a similar vein, I highly recommend reading this post by hyperbolictangent. Although it approaches the manner from the perspective of students who are trying to prove themselves by commenting on how they underperformed on a contest, the ideas present in that post are incredibly relevant to this topic as well. (I highly recommend reading the whole thread too, as it has many different perspectives from many successful students).
Good luck on your future endeavors, and don’t forget to sit back and enjoy the problems!
One huge contributing factor to success in contest mathematics is having seen a lot of problems. Many contest math problems are very similar to problems on previous year’s contests, and therefore, understanding a lot of problem solving techniques is critical to success. (For instance, consider 2016 AMC 12A Problem #22 . Having seen 1987 AIME I Problem #7, a student could solve this problem almost immediately. On the other hand, a student who has never seen a problem like this before would be at a huge disadvantage, because, while they could come up with a solution on the fly, they don’t have a lot of time to do so.) Time pressure is a huge element of the AMC tests; these contests don’t always allow students to fully think about problems. When I do math problems, one of my favorite things to do is sit down with an idea and work with it for a while until I really fully grasp that concept, and the MAA does a fantastic job of starting conversations about tons of interesting things in mathematics. However, during the actual contest, competitors don’t have enough time to do so. If you can’t figure out how to solve a problem on the test, while it’s natural to feel bad initially (I’ve kicked myself many times over the “could’ve would’ve” problems), remember the main goal of doing mathematics: to understand and enjoy the problems. Read the different solutions on the forums, research a topic more which you may have been unfamiliar with, read a book. Then, next time around, not only will you nail the problem on the test, but you will also understand the underlying idea and intuition behind it.
I’m currently a senior in high school, and am almost officially finished with high school math competitions. I’ve participated in the AIME for the past 5 years along with MATHCOUNTS Nationals in 8th grade. I rarely share my scores with others on these tests for two reasons: (1) they’re usually below the first quartile of scores posted on AoPS and, most importantly, (2) I don’t compete in contest math for the sake of having a good score. I do it because I enjoy the problems, and the underlying mathematical ideas which accompany them. I’m an avid lover of Number Theory problems (shameless self promotion) and problems like 2016 AMC 12B Problem #22 excite me a lot, because this problem combined ideas about repeating decimals, order of a number, and divisibility. I didn’t solve this problem until after the test was over; however, when I did, I excitedly shared it with everyone in my school’s math club while teaching them some new things in number theory. Sharing this problem with my friends and teachers is the essence of the beauty of mathematics for me, because it lends itself well to collaboration in problem solving. When I find interesting problems like these, I often have them queued up to show to various people I encounter because I love inspiring others to have this level of inquisitiveness about a mathematical idea.
I’ve also been on the writing end of several math contests, including many Mock AMC exams on the AoPS forum, most notably the 2015 Mock AIME I. I also help write problems for the NIMO contest. My favorite thing about writing these problems is allowing competitors to think about mathematical concepts in new ways. For instance, this polynomial transformation problem taught a very important idea in algebra, which is building a polynomial out of the roots (an idea which was also featured in a similar USAMO problem before). Contributing to these discussions and having people solve my problems in many different ways is incredibly humbling for me, and is part of the beauty of contest mathematics. For more information on this, I highly recommend reading djmathman’s post here .
One of the great things about contest math is it starts these discussions. And, while tons of team contests like ARML and MATHCOUNTS try to inspire this level of collaboration and communication, it seems like it is often the missing link for many students who may be kicking themselves over a low score. Instead of thinking of a 96 on the AMC 10 as a complete failure and a wasted 2 years preparing for the exam, don’t let this score define you. Instead, learn new ideas from the problems and share them with those around you. Teaching may be a passion which is just mine, but I hope that you all can learn to truly enjoy the problems. Maybe you couldn’t solve 2016 AMC 12A Problem 23 . Read the solutions online, try to understand what’s going on in the 3d graph for this problem. Study equations like these more, and understand their graphs (this is especially important when discussing the space of matrices later down the road). If you want to go way above and beyond, maybe try to start understanding double integrals, as in va2010’s post in that thread. The main point is, this problem alone can generate tons of interesting discussions, and missing out on these are a shame. Getting a bad score isn’t a bad thing, but not learning from it surely is.
For another post in a similar vein, I highly recommend reading this post by hyperbolictangent. Although it approaches the manner from the perspective of students who are trying to prove themselves by commenting on how they underperformed on a contest, the ideas present in that post are incredibly relevant to this topic as well. (I highly recommend reading the whole thread too, as it has many different perspectives from many successful students).
Good luck on your future endeavors, and don’t forget to sit back and enjoy the problems!
This post has been edited 4 times. Last edited by Binomial-theorem, Mar 16, 2018, 10:30 AM
Reason: good -> well
Reason: good -> well
.
