# Tagged Questions

Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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### Polynomial game problem: do we have winning strategy for this game?

I'm thinking about some game theory problem. Here it is, Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and ...
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### if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1$ and $A+B+C = 0$ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1$$ and $$A+B+C = 0$$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
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### No eight digit perfect fourth powers with distinct digits and not containing 3.

Problem: Prove that there are no perfect fourth powers that have eight distinct digits in their base 10 representation and also don't contain 3 as a digit. My attempt: Since the problem is about ...
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### Modular Arithmetic Related Question

I've been trying to solve this problem but I couldn't and I don't get the solutions either (I don't think I get how to use modular arithmetic to solve problems in general. And though I tried to ...
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### $15$ integers $m_1 \ldots m_{15}$such that $\sum _{k=1} ^{15} m_k \arctan {k} = \arctan 16$

Determine whether or not there exist $15$ integers $m_1 \ldots m_{15}$ such that $\sum _{k=1} ^{15} m_k \cdot \arctan (k) = \arctan (16)$. This is a question from IMC 2015 Day 1 Problem. Here ...
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### What is the probability that the two-seed makes the finals?

I can solve the following problem using brute-force combinatorics, but I'm looking for an elegant way to think about it, since there is a rather elegant answer. Suppose there is a tournament of ...
### Cauchy like inequality $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$
Problem: Prove that for real $x, y, \alpha, \beta$, $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$. I am looking for an elegant (non-bashy) ...