Mathematics Stack Exchange
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.

9758 questions
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How to solve $\ln x+x=1$

How can I solve this equation: $$\ln x+x=1$$ We had it on a local Olympiad math contest problem.
user144251
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How to find the 31st root of a large number without using calculator or logarithm in 30 seconds?

There is a recent contest in our school. Let $$ \begin{align} N=&\ 330450543498916787547705429904371272937979054655009\\&\ 6798365073248346973369906393714646262613023152668672 \end{align} $$ I am asked to find the $31$st root of $N$ without using…
Thirdy Yabata
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What does it take to get a perfect score on the Putnam?

There are many guides on how to prepare for the Putnam, and so many people participate in the Putnam. However scoring perfect scores or anything reasonably close to a perfect score on the Putnam is so rare. It seems to me that the present guides…
raindrop
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date and days of week problem

A competition in 2017 starts on Thursday, Mar 2nd. If the committee decides to set the start date for the next competition to be on the first Thursday of March in 2018, what is this date? A year ($365$ days) later it would be March 2nd 2018.…
space
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Put $2^{600}$, $3^{500}$, $4^{400}$, $5^{300}$, and $6^{200}$ in order from least to greatest

Put $2^{600}$, $3^{500}$, $4^{400}$, $5^{300}$, and $6^{200}$ in order. Problem I found while looking at old problems from math competitions. Clearly a simple solution would be to compare $600\ln2$, $500\ln3$, etc. But how would one go about…
deadgrips
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Math Olympiad Divisor Problem

The sum of the two smallest positive divisors of an integer $N$ is $6$, while the sum of the two largest positive divisors of $N$ is $1122$. Find $N$. I came across this question in a Math Olympiad Competition. I am able to find out that the…
snivysteel
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Given $p(x)$ is a polynomial with integer coefficients and that $p(a)=1$ for some integer $a$ prove that $p(x)$ has no more than two integral roots.

Given $p(x)$ is a polynomial with integer coefficients and that $p(a)=1$ for some integer $a$ prove that $p(x)$ has no more than two integral roots. I've attempted a proof by contradiction assuming $p(x)$ has three or more roots, but haven't gotten…
Rahul
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AMC 12 2010B Problem Help #18

Can someone explain this solution? A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting…
user87611
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Contest math problem algebra proof

Let $r, s$ be integers and let $$a = (2011)^2 + (2011)r + s$$ and $$b = (2012)^2 + (2012)r + s$$ Show that there exists an integer $c$ with $c^2 + rc + s = ab$. Can anyone help me with this?
user87611
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Bulgaria olympiad $2000$

There are $2000$ white balls in a box. There are also sufficiently many white, green and red balls outside the box. The following operations are allowed Replacement of two white balls with a green ball, Replacement of two red balls with a green…
Danlo
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Tips on this olympiad problem

My brother recently brought this problem to me, and while I found it quite interesting I cannot figure out how to solve it: For any positive integer $n$, let $f(n)$ be defined by $$ f(n) = \begin{cases} 3n+1 & \text{if } n \text{ is odd}…
Javier
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Method to solve missing numbers

The numbers from $1$ to $8$ are entered into the eight circles in this diagram, with the number $3$ placed as shown. In each triangle, the sum of the three numbers is the same. The sum of the four numbers which are at the corners of the central…
Daveo
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Olympiad Algebra Practice Question

I just needed some help for this problem they gave us at our mathematics class, its’s an Olympiad type question, I suspect the answer is 2019, I tried with smaller cases and tried to use a recursive relation, or some type of induction. I also tried…
Eurekaepicstyle
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An interesting inequality

if $n$ be give postive integers,and let $a=(a_{1},a_{2},\cdots,a_{n})$,and define $$S(a)=\sum_{i=1}^{n}3^{i-1}a_{i},~~~T(a)=\sum_{i=1}^{n}\dfrac{a_{i}}{3^{i-1}}$$ Assmue $m,k$ be postive integer such $m\ge 2k$,and define …
wightahtl
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Interesting Olympiad style problem.

Problem: Seven vertices of a cube are marked by $0$ and one by $1$. You may repeatedly select an edge and increase the numbers at its both ends by $1$. Your goal is to reach $8$ equal numbers. Solution: My solution does not match with the…
Student
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