Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions.

This tag is intended for

  1. Actual problems from mathematics competitions
  2. Inquiries about alternative proofs for a particular problem that is from a math contest
  3. Questions that have been explicitly inspired by a contest problem, including practice problems
  4. Soft questions requesting advice on competing in contests or those inquiring into the general utility of such things in terms of encouraging the further pursuit of mathematics

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

Please note that 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. Therefore it is usually a good idea to include information about a contest, such as a link to the contest webpage. This is particularly true if the contest is a recent one.

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A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: $$\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x…
Vladimir Reshetnikov
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Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the advisability of participating, the career…
Potato
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Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem…
HN_NH
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111
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Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square

Question: For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ satisfies the given conditions, since: …
math110
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What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?

Problem 6 of the 1988 International Mathematical Olympiad notoriously asked: Let $a$ and $b$ be positive integers and $k=\frac{a^2+b^2}{1+ab}$. Show that if $k$ is an integer then $k$ is a perfect square. The usual way to show this involves a…
kdog
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Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
leava_sinus
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A goat tied to a corner of a rectangle

A goat is tied to an external corner of a rectangular shed measuring 4 m by 6 m. If the goat’s rope is 8 m long, what is the total area, in square meters, in which the goat can graze? Well, it seems like the goat can turn a full circle of radius…
space
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Graph theoretic proof: For six irrational numbers, there are three among them such that the sum of any two of them is irrational.

Problem. Let there be six irrational numbers. Prove that there exists three irrational numbers among them such that the sum of any two of those irrational numbers is also irrational. I have tried to prove it in the following way, but I am not…
Arpon Basu
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6 answers

Contest problem: Show $\sum_{n = 1}^\infty \frac{n^2a_n}{(a_1+\cdots+a_n)^2}<\infty$ s.t., $a_i>0$, $\sum_{n = 1}^\infty \frac{1}{a_n}<\infty$

The following is probably a math contest problem. I have been unable to locate the original source. Suppose that $\{a_i\}$ is a sequence of positive real numbers and the series $\displaystyle\sum_{n = 1}^\infty \frac{1}{a_n}$ converges. Show that…
Potato
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Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the real exam. After each mock exam the teacher tells…
Sergio
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Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by cubing both sides and using $x = \sqrt[3]{2}$ for…
Paramanand Singh
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75
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5 answers

Drunk man with a set of keys.

I found this problem in a contest of years ago, but I'm not very good at probability, so I prefer to see how you do it: A man gets drunk half of the days of a month. To open his house, he has a set of keys with $5$ keys that are all very similar,…
iam_agf
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Let, $A\subset\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$.

A challenge problem from Sally's Fundamentals of Mathematical Analysis. Problem reads: Suppose $A$ is a subset of $\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \setminus \{p\}$ with the usual…
David Bowman
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63
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7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish. Try: Suppose $k$th fisher caught $r_k$ fishes and that we…
nonuser
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63
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13 answers

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and the $x$-axis between $x=0$ and $x=1$ is equal to…
Daniel W. Farlow
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