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Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions. Provide enough information about the source to confirm the question doesn't come from a live contest.

This tag is intended for

  1. Questions from mathematics competitions.
  2. Inquiries about alternative proofs for problems that are from math contests.
  3. Questions that have been inspired by a contest problem, including practice problems.
  4. Questions requesting advice on competing in contests.

See this list of mathematics competitions to get an idea of the types of questions this tag is for.

Mathematics StackExchange has a policy on questions from current competitions. Questions from ongoing competitions will be locked and temporarily deleted until the end of the contest. It is a good idea to include information about a contest, such as a link to the contest webpage.

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The path must come close to itself after a long time.

IMO 1982, Problem 6. Let $S$ be a square with sides of length 100. Let $L$ be a path within $S$ that does not meet itself and that is composed of linear segments $A_0A_1, A_1A_2, \ldots, A_{n-1}A_n$ with $A_0\neq A_n$. Suppose that for every point…
caffeinemachine
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Decomposing a large block $7\times 4 \times 5$ into small ones $1\times 1 \times 1,\ 2\times 2 \times 2,\ 3\times 3 \times 3$

Consider a block of size $7\times 4 \times 5$. We want to make this with cubes of size $1\times 1 \times 1,\ 2\times 2 \times 2,\ 3\times 3 \times 3$. If $x,\ y,\ z$ cubes are needed respectively, then what is the smallest number $x$ ? Proof : Here…
HK Lee
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Find the value of $\frac{x^5+y^5+z^5-1}{xyz}.$

Let $x,y,z \in \mathbb{R}$ such that $xy+yz+zx=0$ and $xyz \neq 0.$ Find the value of $\frac{x^5+y^5+z^5-1}{xyz}.$ =$...\frac{(x+y+z)^5+5xyz(x+y+z)^2−1}{xyz}$ so for x+y+z=1 we have 5. Is it possible to find another value?
piteer
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2021 Mathcounts practice competition target #4 question #2

What is the greatest number of positive consecutive numbers that sum to 400? My approach: We have terms from : $(x-n)..... x.....(x+n)$ This would mean the sum of these terms is $(2n+1) * 2x * 1/2 = x(2n+1) = 400$ $2n+1$ MUST be an odd number, so…
Smartsav10
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How many positive multiples of 2013, and have exactly 2013 divisors?

The full question: How many positive integers are multiples of 2013 and have exactly 2013 divisors (including 1 and the number itself)? I thought of 2013!, but I think it wouldn't work out well, since it would have more than 2013 divisors. I don't…
Pete Justin Dagaraga
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Mathcounts National 2014 Sprint #15

This problem is from the 2014 Mathcounts National Sprint test. The correct answer is 13/1024, but I keep getting 17/2048s. My attempt at this problem was to use casework. Obviously, the denominator of this fraction will come out to 161616 = 2^12 =…
Smartsav10
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Multiple division of a cube

A cube with an edge length of 10 is divided into two cuboids with integer edge lengths by a flat cut. Afterwards, one of those cuboids is again being divided into two smaller cuboids with integer edge lengths by a second flat cut. What is the…
Martin Müller
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System with cubed and squared terms

Find the ordered pair (x,y) such that $x + y ^3 = 100$ $x^2 - y^2 = 100$ I decided to substitute $x = 100 - y^3$ into the second equation and got $10000 - 200y^3 + y^6 - y^2 = 100$ Is there any trick or method I can use to solve the 6th degree…
yvngyup
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if $a^{2}+b^{2}=1$ Prove that $-\sqrt{2} \le a+b \le \sqrt{2}$

$a,b \in \mathbb{R}$ My attempt: $$a^{2}+b^{2}=1 \iff a^{2}+2ab+b^{2}=1+2ab$$ $$(a+b)^2=1+2ab \iff \mid a+b \mid =\sqrt{1+2ab} \iff a+b = \pm\sqrt{1+2ab}$$ But notice that $\sqrt{1+2ab} \in \mathbb{R} \iff 1+2ab \geq 0 \iff 2ab \geq -1$ . let's take…
PNT
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Prove that if $(a + b + c)(ab + ac + bc) = abc$ then sum of some two numbers equals $0$

Prove that if $(a + b + c)(ab + ac + bc) = abc$ then sum of some two numbers equals $0$. Without loss of generality let's suppose that $a=0$ then $(b + c)bc = 0 \Rightarrow b+c = 0$ or $bc = 0 \Rightarrow b=0 \lor c = 0$ and $a+b=0 \lor a+c=0$…
ioleg19029700
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Question on the 2020 RMO.

In the “expression” each of the twelve @ symbols is replaced with either $\times$ or $÷$ such that the value of the resulting expression is an integer. 1@2@3@4@5@6@7@8@9@10@11@12@13 Find the greatest common factor of all such integer values.
Pinak Paliwal
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$n_1, n_2,...,n_s$ such $(n_1+k)(n_2+k)\cdots(n_s+k)$ integral multiple of $n_1n_2\cdots n_s$. Prove $|n_i| = 1$ for some $i$

Let $n_1, n_2,...,n_s$ be distinct integers such that \begin{equation} (n_1+k)(n_2+k)\cdots(n_s+k) \end{equation} is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$. Give a proof or a counterexample of the following statement:…
Math_Day
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Teams in a Volleyball Tournament ;Prove that in the entire tournament, there is a team that has lost to no more than $4$ of the remaining $109$ teams.

$110$ teams participate in a volleyball tournament. Every team has played every other team exactly once (there are no ties in volleyball). Turns out that in any set of $55$ teams, there is one which has lost to no more than $4$ of the remaining $54$…
Raheel
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Chess Game Winning Strategy

Two players take turns placing kings on a $9 \times 9$ chessboard so that no king can capture another one. The player who cannot win this loses. Is there a certain way to think about the problem?
Angelo Mark
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String of 0 and 1's question

There are sixteen different ways of writing four-digit strings using 1s and 0s. Three of these strings are 1010, 0100 and 1001. These three can be found as substrings of 101001. There is a string of nineteen 1s and 0s which contains all sixteen…
Selwyn Liu
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