Olympiad problems

1990 Dutch Mathematical Olympiad, 2

Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$ $ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$. $ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit. $ (c)$ Determine $ \displaystyle\lim_{n\to\infty}a_1a_2a_3...a_n$.

1997 AIME Problems, 12

Tags: function , limit , quadratic , linear algebra

The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.

Today's calculation of integrals, 853

Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2005 Today's Calculation Of Integral, 76

Tags: calculus , integration , function , limit , inequalities , calculus computations

The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]

1974 Miklós Schweitzer, 6

Tags: induction , function , integration , limit , logarithm , real analysis

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

2007 Gheorghe Vranceanu, 3

Tags: limit , calculus

$ \lim_{n\to\infty } \frac{1}{2^n}\left( \left( \frac{a}{a+b}+\frac{b}{b+c} \right)^n +\left( \frac{b}{b+c}+\frac{c}{c+a} \right)^n +\left( \frac{c}{c+a}+\frac{a}{a+b} \right)^n \right) ,\quad a,b,c>0 $

1989 Spain Mathematical Olympiad, 3

Tags: inequalities , geometry , 3d geometry , induction , limit

Prove $ \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $

2010 Today's Calculation Of Integral, 598

Tags: calculus , integration , trigonometry , limit , calculus computations

For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam

1973 Poland - Second Round, 4

Tags: algebra , limit

Let $ x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] $ for $ n = 1, 2, 3, \ldots $. Prove that if $ p $, $ q $ are natural numbers satisfying the condition $ p - 1 < \sqrt{q} < p $, then $ \lim_{n\to \infty} x_n = 1 $. Attention. The symbol $ [a] $ denotes the largest integer not greater than $ a $.

1982 Putnam, B5

Tags: limit , function , real analysis , analysis , sequence

For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$, and for $n\ge0$, $u_{n+1}=\log_{u_n}x$. Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$.

1992 Dutch Mathematical Olympiad, 5

Tags: limit , algebra

We consider regular $ n$-gons with a fixed circumference $ 4$. Let $ r_n$ and $ a_n$ respectively be the distances from the center of such an $ n$-gon to a vertex and to an edge. $ (a)$ Determine $ a_4,r_4,a_8,r_8$. $ (b)$ Give an appropriate interpretation for $ a_2$ and $ r_2$ $ (c)$ Prove that $ a_{2n}\equal{}\frac{1}{2} (a_n\plus{}r_n)$ and $ r_{2n}\equal{}\sqrt{a_2n r_n}.$ $ (d)$ Define $ u_0\equal{}0, u_1\equal{}1$ and $ u_n\equal{}\frac{1}{2}(u_{n\minus{}2}\plus{}u_{n\minus{}1})$ for $ n$ even or $ u_n\equal{}\sqrt{u_{n\minus{}2} u_{n\minus{}1}}$ for $ n$ odd. Determine $ \displaystyle\lim_{n\to\infty}u_n$.

2007 District Olympiad, 3

Tags: calculus , integration , function , limit , real analysis , inequalities

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that: (a) $\lim_{x \to \infty}f(x)$ exists; (b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.

2006 VJIMC, Problem 2

Tags: limit , convergence

Suppose that $(a_n)$ is a sequence of real numbers such that the series $$\sum_{n=1}^\infty\frac{a_n}n$$is convergent. Show that the sequence $$b_n=\frac1n\sum^n_{j=1}a_j$$is convergent and find its limit.

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: limit , nested radical , real analysis , weierstrass , sequence

Show that if $ \left( s_n \right)_{n\ge 0} $ is a sequence that tends to $ 6, $ then, the sequence $$ \left( \sqrt[3]{s_n+\sqrt[3]{s_{n-1}+\sqrt[3]{s_{n-2}+\sqrt[3]{\cdots +\sqrt[3]{s_0}}}}} \right)_{n\ge 0} $$ tends to $ 2. $ [i]Mihai Bălună[/i]

1978 Miklós Schweitzer, 10

Tags: parameterization , limit , probability and stats

Let $ Y_n$ be a binomial random variable with parameters $ n$ and $ p$. Assume that a certain set $ H$ of positive integers has a density and that this density is equal to $ d$. Prove the following statements: (a) $ \lim _{n \rightarrow \infty}P(Y_n\in H)\equal{}d$ if $ H$ is an arithmetic progression. (b) The previous limit relation is not valid for arbitrary $ H$. (c) If $ H$ is such that $ P(Y_n \in H)$ is convergent, then the limit must be equal to $ d$. [i]L. Posa[/i]

1980 VTRMC, 3

Tags: limit , sequence

Let $$a_n = \frac{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot2n}.$$ (a) Prove that $\lim_{n\to \infty}a_n$ exists. (b) Show that $$a_n = \frac{\left(1-\frac1{2^2}\right)\left(1-\frac1{4^2}\right)\left(1-\frac1{6^2}\right)\cdots\left(1-\frac{1}{(2n)^2}\right)}{(2n+1)a_n}.$$ (c) Find $\lim_{n\to\infty}a_n$ and justify your answer

2019 District Olympiad, 4

Tags: improper integral , integral , limit , calculus

Let $a$ be a real number, $a>1.$ Find the real numbers $b \ge 1$ such that $$\lim_{x \to \infty} \int\limits_0^x (1+t^a)^{-b} \mathrm{d}t=1.$$

2006 Moldova National Olympiad, 11.1

Tags: limit , trigonometry , algebra

Let $n\in\mathbb{N}^*$. Prove that \[ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}. \]

2002 Iran MO (3rd Round), 2

Tags: function , limit , algebra

$f: \mathbb R\longrightarrow\mathbb R^{+}$ is a non-decreasing function. Prove that there is a point $a\in\mathbb R$ that \[f(a+\frac1{f(a)})<2f(a)\]

2017 Mathematical Talent Reward Programme, MCQ: P 10

Tags: limit , calculus , function , differentiable function

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $\lim \limits_{x\to \infty}f'(x)=1$, then [list=1] [*] $f$ is increasing [*] $f$ is unbounded [*] $f'$ is bounded [*] All of these [/list]

1997 South africa National Olympiad, 1

Tags: limit , geometry

From an initial triangle $\Delta A_0B_0C_0$, a sequence of triangles $\Delta A_1B_1C_1$, $A_2B_2C_2$, ... is formed such that, at each stage, $A_{k + 1}$, $B_{k + 1}$ and $C_{k + 1}$ are the points where the incircle of $\Delta A_kB_kC_k$ touches the sides $B_kC_k$, $C_kA_k$ and $A_kB_k$ respectively. (a) Express $\angle A_{k + 1}B_{k + 1}C_{k + 1}$ in terms of $\angle A_kB_kC_k$. (b) Deduce that, as $k$ increases, $\angle A_kB_kC_k$ tends to $60^{\circ}$.

2005 Today's Calculation Of Integral, 85

Tags: calculus , integration , limit , trigonometry , calculus computations

Evaluate \[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\] where $ [x] $ is the integer equal to $ x $ or less than $ x $.

2011 Today's Calculation Of Integral, 713

Tags: calculus , integration , limit , calculus computations

If a positive sequence $\{a_n\}_{n\geq 1}$ satisfies $\int_0^{a_n} x^{n}\ dx=2$, then find $\lim_{n\to\infty} a_n.$

2005 ISI B.Stat Entrance Exam, 6

Tags: function , limit , calculus , derivative , algebra

Let $f$ be a function defined on $(0, \infty )$ as follows: \[f(x)=x+\frac1x\] Let $h$ be a function defined for all $x \in (0,1)$ as \[h(x)=\frac{x^4}{(1-x)^6}\] Suppose that $g(x)=f(h(x))$ for all $x \in (0,1)$. (a) Show that $h$ is a strictly increasing function. (b) Show that there exists a real number $x_0 \in (0,1)$ such that $g$ is strictly decreasing in the interval $(0,x_0]$ and strictly increasing in the interval $[x_0,1)$.

2009 Today's Calculation Of Integral, 502

Tags: calculus , integration , limit , calculus computations

(1) For $ 0 < x < 1$, prove that $ (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}.$ (2) Find $ \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}$.