Olympiad problems

2003 IMC, 2

Tags: calculus , integration , limit , trigonometry , inequalities , real analysis , logarithm

Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)

2007 Nicolae Coculescu, 2

Tags: limit , real analysis , cesaro-stolz , sequence

Let be two sequences $ \left( a_n \right)_{n\ge 0} , \left( b_n \right)_{n\ge 0} $ satisfying the following system: $$ \left\{ \begin{matrix} a_0>0,& \quad a_{n+1} =a_ne^{-a_n} , &\quad\forall n\ge 0 \\ b_{0}\in (0,1) ,& \quad b_{n+1} =b_n\cos \sqrt{b_n} ,& \quad\forall n\ge 0 \end{matrix} \right. $$ Calculate $ \lim_{n\to\infty} \frac{a_n}{b_n} . $ [i]Florian Dumitrel[/i]

2009 Kyrgyzstan National Olympiad, 8

Tags: function , limit , algebra

Does there exist a function $ f: {\Bbb N} \to {\Bbb N}$ such that $ f(f(n \minus{} 1)) \equal{} f(n \plus{} 1) \minus{} f(n)$ for all $ n > 2$.

2011 Miklós Schweitzer, 9

Tags: differential equation , limit , real analysis

Let $x: [0, \infty) \to\Bbb R$ be a differentiable function. Prove that if for all t>1 $$x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)$$ then $\lim_{t\to\infty} x(t) = 0$

1989 Bulgaria National Olympiad, Problem 2

Tags: limit , series , floor function , algebra , sequence

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.

1999 Vietnam Team Selection Test, 1

Tags: inequalities , limit , calculus , integration , algebra

Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying: \[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\] Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.

2000 Moldova National Olympiad, Problem 2

Tags: limit , real analysis

For $n\in\mathbb N$, define $$a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}.$$ (a) Prove that the sequence $b_n=a_n^n$ is convergent and determine the limit. (b) Show that $\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}$.

1988 Greece National Olympiad, 4

Tags: limit , real analysis , sequence

Let $a_1=5$ and $a_{n+1}= a^2_{n}-2$ for any $n=1,2,...$. a) Find $\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}$ b) Find $\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)$

1973 Miklós Schweitzer, 5

Tags: limit , integration , logarithm , real analysis

Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\] [i]P. Medgyessy[/i]

2011 Bulgaria National Olympiad, 3

Tags: function , limit , geometry

Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.

1997 IMC, 1

Tags: limit , calculus , integration , logarithm , real analysis

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

1995 Putnam, 5

Tags: function , limit , vector , linear algebra , matrix

Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy \begin{align*} & \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ & \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ & \;\qquad \vdots \\ & \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\ \end{align*} For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?

2012 Serbia Team Selection Test, 2

Tags: floor function , limit , inequalities , number theory

Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.

2007 Nicolae Coculescu, 3

Tags: limit , real analysis , seuqence

Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by $$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$ Calculate $ \lim_{n\to\infty } a_n ^n . $ [i]Florian Dumitrel[/i]

2011 Vietnam National Olympiad, 2

Tags: limit , algebra

Let $\langle x_n\rangle$ be a sequence of real numbers defined as \[x_1=1; x_n=\dfrac{2n}{(n-1)^2}\sum_{i=1}^{n-1}x_i\] Show that the sequence $y_n=x_{n+1}-x_n$ has finite limits as $n\to \infty.$

2013 China Team Selection Test, 2

Tags: limit , modular arithmetic , quadratic , number theory

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

2020 LIMIT Category 1, 8

Tags: limit , number theory , algebra

Find the greatest integer which doesn't exceed $\frac{3^{100}+2^{100}}{3^{96}+2^{96}}$ (A)$81$ (B)$80$ (C)$79$ (D)$82$

1974 Poland - Second Round, 5

Tags: limit , algebra , sequence

The given numbers are real numbers $ q,t \in \langle \frac{1}{2}; 1) $, $ t \in (0; 1 \rangle $. Prove that there is an increasing sequence of natural numbers $ {n_k} $ ($ k = 1,2, \ldots $) such that $$ t = \lim_{N\to \infty} \sum_{j=1}^N q^{n_j}.$$

2013 Today's Calculation Of Integral, 871

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

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

1953 Putnam, A6

Tags: limit , sequence

Show that the sequence $$ \sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots$$ converges and evaluate the limit.

2003 India IMO Training Camp, 7

Tags: algebra , polynomial , limit , number theory

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

1994 Putnam, 1

Tags: inequalities , limit

Suppose that a sequence $\{a_n\}_{n\ge 1}$ satisfies $0 < a_n \le a_{2n} + a_{2n+1}$ for all $n\in \mathbb{N}$. Prove that the series$\sum_{n=1}^{\infty} a_n$ diverges.

1971 Spain Mathematical Olympiad, 5

Tags: complex number , limit , analysis

Prove that whatever the complex number $z$ is, it is true that $$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$ Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds $$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$

1983 Iran MO (2nd round), 5

Tags: limit

Find the value of $S_n= \arctan \frac 12 + \arctan \frac 18+ \arctan \frac {1}{18} + \cdots + \arctan \frac {1}{2n^2}.$ Also find $\lim_{n \to \infty} S_n.$

1954 Putnam, A5

Tags: function , limit

Let $f(x)$ be a real-valued function defined for $0<x<1.$ If $$ \lim_{x \to 0} f(x) =0 \;\; \text{and} \;\; f(x) - f \left( \frac{x}{2} \right) =o(x),$$ prove that $f(x) =o(x),$ where we use the O-notation.