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*I don't know if I can ask these type of questions here. Tell me and I will delete it right away if it's doesn't belong here.

I'm preparing for a math contest, but I'm done answering all the previous contest problems and trying to find similar problems to practice, sadly I either find much easier questions or much harder problems like the IMO ones.

Here are example problems:

(1). For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6) = 1$ since the only such triangle with a perimeter of 6 has side lengths 2, 2 and 2.

(a) Determine the values of $T(10)$, $T(11)$ and $T(12)$.

(b) If $m$ is a positive integer with $m \geq 3$, prove that $T(2m) = T(2m − 3)$.

(c) Determine the smallest positive integer $n$ such that $T(n) > 2010$.


(2). For each positive integer $N$, an Eden sequence from $\{1, 2, 3, \ldots, N\}$ is defined to be a sequence that satisfies the following conditions: (i) each of its terms is an element of the set of consecutive integers $\{1, 2, 3, \ldots, N\}$, (ii) the sequence is increasing, and (iii) the terms in odd numbered positions are odd and the terms in even numbered positions are even. For example, the four Eden sequences from $\{1, 2, 3\}$ are 1 3 1,2 1,2,3

(a) Determine the number of Eden sequences from $\{1, 2, 3, 4, 5\}$.

(b) For each positive integer $N$, define $e(N)$ to be the number of Eden sequences from $\{1, 2, 3, \ldots, N\}$.

If $e(17) = 4180$ and $e(20) = 17710$, determine $e(18)$ and $e(19)$.


(3). A multiplicative partition of a positive integer $n \geq 2$ is a way of writing $n$ as a product of one or more integers, each greater than 1. Note that we consider a positive integer to be a multiplicative partition of itself. Also, the order of the factors in a partition does not matter; for example, $2 \times 3 \times 5$ and $2 \times 5 \times 3$ are considered to be the same partition of 30. For each positive integer $n \geq 2$, define $P(n)$ to be the number of multiplicative partitions of $n$. We also define $P(1) = 1$. Note that $P(40) = 7$, since the multiplicative partitions of 40 are 40, $2 \times 20$, $4 \times 10$, $5 \times 8$, $2 \times 2 \times 10$, $2 \times 4 \times 5$, and $2 \times 2 \times 2 \times 5$.

(a) Determine the value of $P(64)$.

(b) Determine the value of $P(1000)$.

(c) Determine, with proof, a sequence of integers $a_0, a_1, a_2, a_3, \ldots$ with the property that $P(4 \times 5m) = a_0P(2m) + a_1P (2m − 1) + a_2P(2m − 2) + \ldots + a_{m − 1}P(21) + a_mP(20)$ for every positive integer $m$.

What branch of math are these types of problems from? Where can I find similar problems?

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  • $\begingroup$ math.stackexchange.com/questions/tagged/contest-math $\endgroup$ Mar 28, 2015 at 1:04
  • $\begingroup$ All the problems you listed are about combinatorics. $\endgroup$
    – Nathanson
    Mar 28, 2015 at 1:12
  • $\begingroup$ Which branch of combinatorics? I can't seem to find similar questions. $\endgroup$
    – Lenol
    Mar 28, 2015 at 1:14
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    $\begingroup$ Just enumerative combinatorics. In combinatorics, to get a sense of type it is better to look at the types of techniques involved in the solutions, rather than the type of ways the problem looks like. Very dissimilar looking problems can admit the same technique for their solutions. In these problems they seem to all be solvable using the basic combinatorial principles, generating functions, or recurrences probably can be used as well. $\endgroup$
    – Nathanson
    Mar 28, 2015 at 1:41

1 Answer 1

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Sometimes it can be hard to find resources to study from, especially for areas as diverse as contest math. A particularly good start is to look into problem books aimed at the level of the contest you are considering.

I have 4 references for problems that I really like, plus two periodicals. Let me say what they are and a bit about each.

Books

  1. Problem Solving through Problems

    This was one of the first problem books I started from when I became interested in competition math. It's aimed at competition style math, and is very well written. Something I found deceptively helpful was that many of the problem sources are also mentioned, so this can be an entry point into more sources of problems.

  2. Problem-Solving Strategies

    I worked through this book in conjunction with the book above. In general, this is a more diverse and slightly more challenging collection of problems, but it still has an instructional bent to it.

  3. Collections of Putnam Exams

    This could be a stand-in for the types of competition you are interested in. But almost any competition has a longstanding history, and therefore an extensive record. They can be daunting to understand at first, which is why I might recommend trying to get familiar with some of the general problem solving techniques out there first. Collections of IMO, Putnam, and other competitions are all available.

  4. The USSR Olympiad Book

    I like these questions in that they are particularly aimed at elementary questions with elementary answers, but which have an undeniable competition flavor to them. This might be colored by my own interests in competition mathematics.

Journals

  1. Crux Mathematicorum

    This is a periodical from the Canadian Mathematical Society that is uniquely aimed almost entirely at competition math across different levels. It happens to be in a strange transition point and is in deep want of more subscribers (I infer, but do not know). Their archives are available [but perhaps more people should subscribe]

  2. The American Mathematical Monthly

    This is a periodical from the Mathematical Association for America that comes out almost-monthly. Most of it is dedicated to expository overviews accessible to undergraduate or very-advanced high school math enthusiasts. They have a problem section each month that I find rather inspiring, and often has a competition feel. [But not always]. Similar are Mathematics Magazine and the College Mathematics Journal, which are also by the MAA, and which are pitched a bit lower. They each have their own problem sections too.

In fact, there are many good periodicals out there. The Mathematical Gazette from the UK is an exceptionally good periodical, for instance. But they do not offer electronic subscriptions, and past experience has taught me to be a bit dubious of their international shipping.

If you read a non-English language, there are famous Hungarian, Turkish, and Russian periodicals that are full of great mathematics and contest-style problems. I would suspect that each region has its own math journal with this type of content, but I cannot say other than these that I've mentioned here.

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