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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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How to solve this equation? $P(x)^2+P(\frac1x)^2= P(x^2)P(\frac1{x^2})$

How to solve this equation? Find all polynomials $P$ such that $P(x)^2+P(\frac1x)^2= P(x^2)P(\frac1{x^2})$ Please step by step
FMath
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Formula for smallest multiple of given number, whose every digit is 1

Introduction I've been solving a problem, which says which number is the smallest multiple of $x$ which only has digits with value 1. For example: $minOnes(3) = 3 -> 111$; $minOnes(7) = 6 -> 111111$$minOnes(11) = 2 -> 11$; $minOnes(2601) = 2448$. I…
Santiago Gil
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discount and percentage question, how to solve this

To attract more visitors, Zoo authority announces $20\%$ discount on every ticket which cost $\$25$. For this reason, sales of tickets increases by $28\%$. Find the $\%$ of increase in the number of visitors. A $40\%$ B $50\%$ C $60\%$ D no change…
kuberkashyap
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What is the average or above-average score range for the AMC 1o and AIME?

I'm not sure if this is the right place to ask, but what is the average/above-average score range for the AMC 10 and AIME? For example, what points would you say are below-average, average, above-average, and top-of-the-line? As a starter, I'd say…
crownusa
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How to find minimum number with max trailing zeros when multiplying with 4 or 7?

For example , 15 - 15*4=60 - minimum number with max trailing zeros when multiplying with 4 or 7 125 - 125*4*4=2000 400 - 400 will be the answer as its the minimum number with max trailing zeros. IF possible , I also want to know another different…
Shubham Marathia
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another olympiad problem from Arthur Engel related to invariant

The vertices of an n-gon are labeled by real numbers $x_1,...,x_n$ . Let $a, b, c, d$ be four successive labels. If $(a − d)(b − c) < 0$, then we may switch $b$ with $c$. Decide if this switching operation can be performed infinitely often. How to…
Bluey
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A function $y(x)$ satisfies the differential equation $y^{\prime}=4\sqrt{y-x^2}$ It is known that $y(1)=2$. Find $y(3)$.

A function $y(x)$ satisfies the differential equation $$y^{\prime}=4\sqrt{y-x^2}$$ It is known that $y(1)=2$. Find $y(3)$. My attempt: Clearly $y^{\prime}=4$ at $x=1$. That's all(LOL). Any hint to proceed will be much appreciated.
Idonknow
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let $q$ be the number of pairs of linearly independent vectors from $S$. What is the smallest and the largest possible value of $q$?

Let $S$ be a set of $n$ nonzero vectors in $\mathbb{R}^2$ such that $S$ spans the whole $\mathbb{R}^2$ and let $q$ be the number of pairs of linearly independent vectors from $S$. What is the smallest and the largest possible value of $q$? My…
Idonknow
  • 15,643
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Show that the equation has at least two solutions on the interval $0 \leq x \leq 1$

Let $0 < a < 1$. Show that the equation $$\int_0^x{\left( \sin \left(\frac{\pi \sin\frac{\pi t}{2}}{2} \right)+ \frac{2}{\pi} \sin^{-1} \left( \frac{2}{\pi} \sin^{-1}(t) \right) -2t \right)}dt = \frac{1}{2} \left( a-\frac{2}{\pi}\sin^{-1} \left(…
Idonknow
  • 15,643
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Let $H = \frac{A+A^T}{2}$. Assume that $H$ is positive definite. Prove that $\det(H) \geq \det(A)$.

Let $A$ be an $n \times n$ matrix with real entries and let $H = \frac{A+A^T}{2}$. Assume that $H$ is positive definite. Prove that $\det(H) \geq \det(A)$. This question is obtained from Moscow (I don't have the specific source). My attempt: Note…
Idonknow
  • 15,643
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show that $\sum_{n=1}^{2015}a_{n}\equiv 3\pmod 4$

Assmue that real sequence $\{a_{n}\}$ such $$a_{1}=1,|a_{n+1}|=2|a_{n}|$$ show that $$\sum_{n=1}^{2015}a_{n}\equiv 3\pmod 4$$ I have solve $$|a_{n}|=|a_{1}|\cdot 2^{n-1}=2^{n-1}\Longrightarrow a_{n}=\pm 2^{n-1}$$ Well and now I'm stuck and don't…
user225250
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Evaluate powers in fraction

This is abstracted from 2007 British Mathematical Olympiad Question 1.I wish to practice mathematics olympiad question for the upcoming Singapore Mathematics Olympiad Secondary 2 (Grade 8). Find the value of $$\frac {1^4+2007^4+2008^4}…
ministic2001
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Find a number that is evenly divisible by all numbers between 1 and 20

I'm solving this for a programming challenge, in fact I already solved it but I'd like to know if there's some kind of rule that could improve such thing? For example if I needed the numbers divisible by 2,4 and 8, they are all multiples of 2, so I…
juliano.net
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All means integer

$a$ and $b$ are distinct positive integers such that $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are integers. Find the smallest possible value of $|a-b|$. My work led me to find that one possible pair is $a=90, b=10$, but…
ether
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