Olympiad problems

1986 Traian Lălescu, 1.1

Let $ a $ be a positive real number. Calculate $ \lim_{n\to\infty} \frac{a^n}{(1+a)(1+a^2)\cdots (1+a^n)} . $

Prove that if $ E \subset \mathbb{R}$ is a bounded set of positive Lebesgue measure, then for every $ u < 1/2$, a point $ x\equal{}x(u)$ can be found so that \[ |(x\minus{}h,x\plus{}h) \cap E| \geq uh\] and \[ |(x\minus{}h,x\plus{}h) \cap (\mathbb{R} \setminus E)| \geq uh\] for all sufficiently small positive values of $ h$. [i]K. I. Koljada[/i]

ICMC 7, 4

Tags: convergence , series , real analysis

Let $(t_n)_{n\geqslant 1}$ be the sequence defined by $t_1=1, t_{2k}=-t_k$ and $t_{2k+1}=t_{k+1}$ for all $k\geqslant 1.$ Consider the series \[\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.\]Prove that this series converges to a positive real number. [i]Proposed by Dylan Toh[/i]

2013 Taiwan TST Round 1, 2

Tags: polynomial , number theory , rational roots , real analysis , algebra

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

1980 Miklós Schweitzer, 8

Tags: function , trigonometry , real analysis

Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

2007 IMS, 3

Tags: search , real analysis

Prove that $\mathbb R^{2}$ has a dense subset such that has no three collinear points.

1982 Putnam, A6

Tags: limit , real analysis

Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties: $(\text i)$ $|x_n|$ is a strictly decreasing function of $n$; $(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$; $(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$. Prove or disprove that these conditions imply that $$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$

2000 District Olympiad (Hunedoara), 3

Tags: limit , calculus , real analysis , sequence

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

2013 BMT Spring, 3

Tags: real analysis , limit

Evaluate $$\lim_{x\to0}\frac{\sin2x}{e^{3x}-e^{-3x}}$$