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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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Sylvester problem.

A finite set $S$ of points in a plane has the property that a line passing through two of these points passes through a third point. Prove that all the points in $S$ are collinear. I saw a proof of this problem using extremal principle. Suppose that…
Aditya
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CIME Money problem

Question:At the local Blast Store, there are sufficiently many items with a price of $n.99 for each nonnegative integer n. A sales tax of 7.5% is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find…
Michael Li
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Set of Integers. How many does it contain? AMC 2003 Senior(Australia)

A set of positive integers has the properties that Every member in th set, apart form 1, is divisible by at least one of $2,3,$ or $5$. If the set contains $2n, 3n,$ or $5n$ for some integer $n$, then it contains all three and $n$ as well. The set…
Oziter
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Max continuous subarray sum + minimum continuous subarray sum = total sum in a circular array

I have an array $A$. This array is circular. By this, I mean you can arrange the array in a circle such that the end of the array is connected to the beginning of the array. I want to prove that $T$, the total sum of the array, is equal to the…
user5965026
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Putnam and Beyond Problem 21

I am having a hard time solving this rather interesting question below: Prove that any function defined on the entire real axis can be written as the sum of two functions whose graphs admit centers of symmetry. I know we have to set the two…
matcha_
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Confusion about A1, IMO 2002

The following is question A1 from the 2002 IMO: $S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h+k
user67803
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How to find a complex root of $x^{2021}=x^{2020}+1$ while this complex root also satisfies a quadratic equation with integer coefficients?

How to find a complex root of $x^{2021}=x^{2020}+1$ while this complex root also satisfies a quadratic equation with integer coefficients? I have no previous experience in solving complex equations so just have no clue on this kind of question... Is…
WWMASK
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2017 AMC 12A problem 6 (triangle inequality)

Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm. She places the rods with lengths 3, 7 and 15 on the table. She then wants to choose a fourth rod that she can put with these 3 such that they form a quadrilateral with…
SuperMage1
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Olympiad of May 2016

We say that a four-digit number $\overline{abcd} $, which starts at $ a $ and ends at $ d $, is interchangeable if there is an integer $ n> 1 $ such that $ n * \overline {abcd} $ is a four-digit number that starts in $ d $ and ends in $ a $. For…
aksjdnkajsnd
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Currency conversion problem confusion (shrams, shrims, shrums)

So me and a mock test creator were having this discussion about one of his problems. Here it is: In shrom currency, $\frac37 \text{ shrims}$, $4 \frac57\text{ shrums}$, and $14 \frac17\text{ shrams}$ have the same value. How many shrums are in…
asdf334
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If $\int_0^\pi{f(\theta)\cos\theta}=\int_0^\pi{f(\theta)\sin\theta}$, prove that $f$ has two zeroes in $(0,\pi)$

I read a Putnam question recently. I am having trouble remembering which year it was, but it was probably from before the 1950's Let $f\in[0,\pi]$ be a continuous function. If $$\int_0^\pi{f(\theta)\cos\theta \;{\rm…
user67803
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Expected values in coin flipping

David repeatedly flips a fair coin. Find the expected value of the total number of heads he will flip before flipping two consecutive tails. (A) $2$ (B) $5/2$ (C) $3$ (D) $4$ (E) $9/2$ I'm totally confused on this problem. Here's the two posts that…
asdf334
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Candle problem about burn rates

Two candles, one of which is two centimeters longer than the other, are lit. The longer and thinner one is lit at noon and the shorter but fatter one is lit 15 minutes later. Each candle burns at a steady rate, and by 4 PM both are the same length.…
asdf334
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How can I slice 'quarter ticket', 'half ticket', 'full ticket' price from 75000₺

I have 75000 ₺ prize. I want to give prize by prize type: 1 Full ticket = 9 result 2 Half ticket = 14 result 3 Quarter ticket = 201 result $total = 75000; $sql = $db->query("SELECT * FROM winners WHERE sayisi='5'"); while($list =…
Ismail Altunören
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Motivation for BMO1 2007-2008 Question 6

The function $f$ is defined on the set of positive integers by $$f(1) = 1,$$ $$f(2n) = 2f(n),$$ $$nf(2n + 1) = (2n + 1)(f(n) +n), n \geq 1$$ i) Prove that $f(n)$ is always an integer. ii) For how many positive integers less than 2007 is $f(n) = 2n$…
Hector Lombard
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