# Tagged Questions

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

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### A fair die is rolled four times. What is the probability that each of the final three rolls is at least as large as the roll preceding it?

A fair die is rolled four times. What is the probability that each of the final three rolls is at least as large as the roll preceding it? This question is from the AIME 2001. I am looking for a ...
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### $f(x) \ge f(x + \sin x)$, nonconstant functions, infinite number of solutions to $f'(x) = 0$.

Let $\mathcal{F}$ be the set of all the differentiable functions $f: \mathbb{R} \to \mathbb{R}$, which have the property $f(x) \ge f(x + \sin x)$, for all $x \in \mathbb{R}$. Prove that ...
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### A polynomial specialized at (1, 1, …, 1)

Consider the polynomial \begin{eqnarray} P_{l, p}(\lambda_1, \cdots, \lambda_{2n+1})=\sum_{j=0}^{2n+1}(-1)^j\sum_{\stackrel{0\leq k_r\leq ...
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### Complex Numbers and their relationship with higher Mathematics

Let $z_1, z_2, \cdots, z_n$ be complex numbers satisfying $$|z_1|+|z_2|+\cdots +|z_n|=1.$$ Prove that there is a non-empty subset of $\{z_1,z_2,\cdots,z_n\}$ the sum of whose elements has modulus at ...
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### Pairs of cards from 1,2,…,n arranged according to rules on pairwise separation - is it possible for various n?

A deck of $2n+1$ cards consists of a joker and, for each number between $1$ and $n$ inclusive, two cards marked with that number. The $2n + 1$ cards are placed in a row, with the joker in the ...