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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.

9758 questions
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sums of squares of integers

We have to prove that there exists infinitely many integers $a,b,c$ such that $a^2 + b^2 = c^2 + 3$ . This looked like a very straight-forward question . I did some algebraic manipulations but couldn't reach the conclusion . Help me here
abkds
  • 2,210
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Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$

Let $a$, $b$ and $c$ be positive real numbers. $(\mathrm{i})$ Prove that $4(a^3 + b^3) \ge (a + b)^3$. $(\mathrm{ii})$Prove that $9(a^3 + b^3 + c^3) \ge (a + b + c)^3.$ For the first one I tried expanding to get $a^3 + b^3 \ge a^2b+ab^2$ but…
user113994
  • 21
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Diophantine Approximation and Liouville Theorem

I'm reading the alternative proof of the MO problem: http://koopakoo.wordpress.com/2008/09/03/cgmo-2007-problem-7-and-liouvilles-theorem/ . However, I have a problem, namely that in the alternative proof (just right below the official proof), I…
Chung Hei Wu
  • 21
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Calculating number of pages from sum of page numbers

A novel has 6 chapters. As usual, starting from the first page of the first chapter, the pages of the novel are numbered $1, 2, 3, 4, \ldots$. Also, each chapter begins on a new page. The last chapter is the longest and the page numbers of its…
user87611
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2 answers

The biggest number $N$ that satisfies certain requirements.

The problem is as such: Say a natural number $n$ as special if it does not have the digit $0$, has $2021$ as the sum of its digits, and the sum of the digits of $2n$ does not exceed 1202. Let $N$ be the largest number that is special. How many…
ryan.zcd
  • 403
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APMO $2022$ Problem $4$

Let $n$ and $k$ be positive integers. Cathy is playing the following game. There are $n$ marbles and $k$ boxes, with the marbles labelled $1$ to $n$. Initially, all marbles are placed inside one box. Each turn Cathy chooses a box and then moves the…
Aadi
  • 804
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Find the maximum value of an algebraic expression.

The question is: Determine the maximum value of $a, b, c$ from the following system of quadratic equations. $$\frac{4a^2}{1+4a^2}=b$$ $$\frac{4b^2}{1+4b^2}=c$$ $$\frac{4c^2}{1+4c^2}=a$$ I haven't an idea on how to do this, and my attempts led to a…
ryan.zcd
  • 403
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2 answers

Invariant Principle

On a paper there exist three distinct natural numbers. A "step" means that we choose two of them and replace one of these with their arithmetic mean. Prove that there exists a finite sequence of "steps" after which we will have two equal numbers. I…
Thhh2023
  • 21
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A Game to be played Blindfolded

This is a game that was re-created in a recreational math session. You are given four glass cylinders closed on top and bottom , and the curved edge being made of glass , with an arrow painted inside. The cylinders are placed on the four corners of…
Anuj jha
  • 157
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Take-away game problem - find the winning strategy

There are 200 coins in a pile. During a players turn they must take at least one coin and cannot take more than half of the coins. The person who takes the last coin wins. Devise a strategy so you always win. What strategy/method should I use to…
LS.88.99
  • 43
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The number of solutions of $a+b+c+d=n,\ a\geq b,\ d,\ {\rm and}\ c\geq b+1,\ d$

Define $P(n)$ to be the number of solutions : $a,\ b,\ c,\ d$ are nonnegative integers s.t. $$a+b+c+d=n $$ and $$ a\geq b,\ d,\ {\rm and}\ c\geq b+1,\ d$$ Define $Q(n)$ to be the number of solutions : $x,\ y,\ z,\ w $ are nonnegative integers…
HK Lee
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3 answers

Consecutive Number Divisibility

While I was solving a practice problem, I became interested in coming to the conclusion about the following: Is it possible for both $\frac{x+1}y$ AND $\frac x{y+1}$ to be integers, and if so, how would I find them. Looking at this, I was pretty…
Smartsav10
  • 177
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3 answers

What are the possible real values of $\frac{1}{x} + \frac{1}{y}$ given $x^3 +y^3 +3x^2y^2 = x^3y^3$?

Let $x^3 +y^3 +3x^2y^2 = x^3y^3$ for $x$ and $y$ real numbers different from $0$. Then determine all possible values of $\frac{1}{x} + \frac{1}{y}$ I tried to factor this polynomial but there's no a clear factors
user912835
  • 93
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1 answer

31st IMO 1990 shortlist p1

Question : Is there a positive integer which can be written as the sum of 1990 consecutive positive integers and which can be written as a sum of two or more consecutive positive integers in just 1990 ways? (IMO 1990 shortlisted) I know that…
user
  • 205
2
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2 answers

Cambridge entrance exam 1984

A particle of unit mass moves under the action of $n$ forces directed towards $n$ fixed points $A_1,A_2,...,A_n$. The force towards $A_i$ is of magnitude $k_i$ times the distance of the particle from $A_i$. When the particle is at B, it's…
Mdren
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