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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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Traveling Salesman Variant - Greedy Algorithm based on Least Cost better than that based on Highest

This is a rather intuitive statement about the traveling salesman problem that is surprisingly tricky to prove. I haven't seen this anywhere else, so I thought I would share it. The source is an old Russian olympiad (1970s?). There are two Russian…
cats
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Logic problem involving sum of digits

Good one guys! I'm studying to the national maths olympiad (brazil) by myself, and I ran up to the following question: Let $S(n)$ be the sum of the digits of n. For example $S(77) = 14$ and $S(2003) = 5 $ . Tell if there is a n that is…
John Silva
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Problem from Iran Olympiad?

Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer. I gussed the answer is no. Let $\overline{a_n ,...,a_1 ,a_0}$ be in the form $2^k$ and $\sigma\in S_n$ be sch…
Ramand
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Factoring $x^5 + x^4 + x^3 + x^2 + x + 1$ without using $\frac{x^n - 1}{x-1}$?

I was at a math team meet today and one of the problems was to factor $x^5 + x^4 + x^3 + x^2 + x + 1$. It also gave the hint that it decomposes into two trinomials and a binomial. The solution they gave was based on the fact that $\frac{x^6 -…
user99185
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The square of a number's last few digits remain the same.

The number $9376$ has a property that the last four digits of $9376^2$ remain the same. How many $4$ digit numbers have this property? Are there values of $n>4$ such that a $n$-digit number has $n$ digits unchanged at the end after squaring? Thank…
Yang Lu
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Prove $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ for $a,b,c>0$ Any hints/solutions?
user85798
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AMC Problem Help 12B 2010

A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$ , and $a_3=\tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$? The AMC website has a solution to this, and I wish I could post that here. I would like to know…
user87611
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Exercise sum equal to 1 using only the digits 1,2, 3,...,9

Give a method to write the number one as the sum of three fractions, where each fraction the numerator is a one-digit number the denominator is a two-digit number and numbers that can be used are from 1 to 9 without repeating numbers.
Roiner Segura Cubero
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Frobenius number for three set of numbers

I came across the below problem In the olden days, there were only $4$ paise, $7$ paisa, and $11$ paisa coins. What is the maximum amount you can't exactly pay using just the three sets of coins? I recognized that this is somewhat related to the…
Ganit
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Area of trapezoid given all sides: Why is this an Olympiad problem?

I was going through some Olympiad Maths and found this question: Given a trapezoid with its upper base $5$ cm long, lower base $10$ cm long, and its legs are $3$ and $4$ cm long. What is the area of this trapezoid? Yeah, I know. There are…
Tyrcnex
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Number of coins to pickup in the beginning to ensure a definite win : Coins Game

$A$ and $B$ were playing a game which involved picking up coins from a table. Each player in his turn was to pick a minimum of $2$ and a maximum of $6$ coins, except when there is only $1$ coin left in which case, he has to pick up that coin. The…
Ganit
  • 1,689
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Contest Math: construct equations with$ \sum_{i=1}^k a_i = \prod_{j=1}^ka_j$ where $a_k$ are all positive rational number.

The question is to find all possible integer n such that there exists at least 2 positive rational numbers $a_k$ such that $n = \sum_{i=1}^k a_i = \prod_{j=1}^ka_j.$ What I think is obvious is, for any composite numbers >=6, they all qualify…
WWMASK
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How many coefficients of the polynomial $P_n(x_1,...,x_n) = \prod_{1\leq i < j \leq n}(x_i+x_j)$ are odd?

How many coefficients of the polynomial $P_n(x_1,...,x_n) = \prod_{1\leq i < j \leq n}(x_i+x_j)$ are odd? I computed a couple small cases first. for $n = 1$ the problem doens't make sense. For $n=2$ there are 2 odd coefficients, and for $n = 3$…
Math_Day
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Generating Function question, need help

Let $$h_{n} = \sum_{k=0}^{n} \dbinom{n}{k} \frac{2^{k+1}}{k+1}.$$ If $$S= \sum_{n=0}^{\infty} \frac{h_{n}}{n!},$$ find $\lfloor S \rfloor $. I calculated the first few values of n. I got, when n=0, we get 2 when n=1, we get 4 when n=2, we get 26/3 i…
user829751
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How many positive integer values of n exist such that $\frac{4^n+2^n+1}{n^2+n+1}$ also a positive integer?

My question is as follows: find the number of values of the positive integers $n$ such that the fraction below is also a positive integer: $$\frac{4^n+2^n+1}{n^2+n+1}$$ I have only been able to establish the bases cases for $n=2$ and $n=4$ which…
sphere jerry
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