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Mathematics Stack Exchange

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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A questions regarding specific math competition exercises

I have a somewhat unconventional question, and I apologize if it does not adhere to the criteria for inquiries on this website. However, as I have received this task on relatively short notice, I urgently need help from someone. I have to prepare a…
MathGeek
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Colorado Mathematics Olympiad problem review

Could you please check my solution for this problem from the Colorado Mathematics Olympiad. I'm new to mathematics problems and I'd like to know if it has any flaw or if it could be improved. "Twenty-three people of positive integral weight decide…
Csiki Szabi
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Looking for related (and hopefully easier) problems about subgrids of bigger grids.

I have been working on the following problem without much success: Counters are placed on 25 of the squares of a 6 × 7 chessboard. Prove that there exists a 2 × 2 subboard with at least three counters on its four squares. This problem comes from…
Will Octagon Gibson
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If we start with a set of non-zero numbers $S$ and keep applying the operation, show that we can never again obtain the set $S$.

Given some numbers, we may choose two of them, say $a$ and $b$, and replace them with the numbers $$a + \frac{b}{2} \ \text{ and } \ b-\frac{a}{2}.$$ If we start with a set of non-zero numbers $S$ and keep applying the operation, show that we can…
Louie
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All natural numbers $n$ such that $\frac{n^3+2021}{n+21}$ is a natural number.

The problem is as such: Find all natural numbers $n$ such that $$\dfrac{n^3+2021}{n+21}$$ is a natural number. So far, I've only found two solutions. My approach:…
ryan.zcd
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Problem C1 from IMO 2017 SL

Problem A reсtangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either…
Denis Krylov
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The value of $(a^2+b^2)(c^2+1)$.

The question is: Given real numbers $a,b,c$ that satisfy $$ab(c^2-1)+c(a^2-b^2)=12$$ $$(a+b)c+(a-b)=7$$ Find the value of $(a^2+b^2)(c^2+1)$ From what I've done, I got $7(3ac+3a+3bc-b)-2ab(c+1)(c-1)=(a^2+b^2)(c^2+1)$. I think I'm inching further…
ryan.zcd
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Is it possible to place $1995$ natural numbers on along a circle such that for two of these numbers the ratio of the greatest to the least is a prime?

Is it possible to place $1995$ different natural numbers on along a circle such that for any two of these numbers, the ratio of the greatest to the least is a prime? I'm confused about what does it mean for a natural number to be "along" a circle?…
Walker
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Math Olympiad problem

A year is peculiar if the sum of the first two digits and the last two digits is equal to the middle two digits. For example, 1978. When was the last peculiar year and is there an algorithm to find any peculiar year? How would I go about solving…
MathApprentice
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Number of Triples and a Limit

I came with a curious question, that I tried solving in $\mathbb{Z^3}$, the equation $2x^2 + 3y^2 + 5z^2 = R$, for $R > 0 $, where $n(R)$ is the quant. of triples that satisfy that equation. What I tried was to calculate the following: $$\lim_{R \to…
João Víctor Melo
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Geometric interpretation for the binary expansion using gradient

The following question is modified from the the 2021 Mathsbombe competition (now closed): Beatrice lines up an infinitely large piece of squared graph paper with red lines going horizontally at 1cm intervals and blue lines going vertically at 1cm…
Bysshed
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Existence of integer coefficient polynomial such that $f(\sqrt{2}+\sqrt{3})=\sqrt{2}$

The original problem asked to "Show that there exists a polynomial $f(x)$ with rational coefficients such that $f(\sqrt{2}+\sqrt{3})=\sqrt{2}$." I was wondering if there exists an integer coefficient polynomial. Intuitively, I think the answer…
1-___-
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Simplify the following expression!

Simplify the following expression $$\sqrt[3]{a+\frac{a+8}{3}\sqrt{\frac{a-1}{3}}}+\sqrt[3]{a-\frac{a+8}{3}\sqrt{\frac{a-1}{3}}}$$ I tried using the form $\displaystyle\frac{a^3+b^3}{a^2-ab+b^2}$, and i also tried to assume the requested expression…
user884324
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Find the value of $p_{2021}$.

Given a sequence $\{i\}_{i\in\mathbb{N}}$. A new sequence $\{p_n\}_{n\in\mathbb{N}}$ is obtained from $\{i\}_{i\in\mathbb{N}}$ by omitting all multiples of $3$ or $4$, but not $5$. Find $p_{2021}$. I try as follows. Let $S=\{1, 2, 3,…
Ongky Denny Wijaya
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What methods could I use to solve this question?

How would I go about solving this question without ‘brute forcing’ it? (By this, I mean is there a trick to doing so? I figured that it had something to do with halves and so this made it easier for me to lower the number of odds satisfying this…
user959782
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