Olympiad problems

2017 Romania National Olympiad, 1

Tags: limit , function , real analysis , definite integral , calculus , integration , sequence

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

2002 India Regional Mathematical Olympiad, 6

Tags: inequalities , limit

Prove that for any natural number $n > 1$, \[ \frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}. \]

2013 Brazil National Olympiad, 3

Tags: function , induction , limit , polynomial , functional equation , algebra

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.

1999 Harvard-MIT Mathematics Tournament, 1

Tags: limit

Start with an angle of $60^\circ$ and bisect it, then bisect the lower $30^\circ$ angle, then the upper $15^\circ$ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original $60^\circ$ angle into two angles. Find the measure (degrees) of the smaller angle.

2007 Gheorghe Vranceanu, 3

Tags: limit , binom , calculus

$ \lim_{n\to\infty } \sqrt[n]{\sum_{i=0}^n\binom{n}{i}^2} $

2001 Romania National Olympiad, 4

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

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

2005 VTRMC, Problem 5

Tags: limit , real analysis

Define $f(x,y)=\frac{xy}{x^2+y^2\ln(x^2)^2}$ if $x\ne0$, and $f(0,y)=0$ if $y\ne0$. Determine whether $\lim_{(x,y)\to(0,0)}f(x,y)$ exists, and find its value is if the limit does exist.

2023 SEEMOUS, P2

Tags: real analysis , limit , sequence

For the sequence \[S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},\]find the limit \[\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).\]

2008 Vietnam National Olympiad, 4

Tags: limit , algebra

he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit.

2012 Today's Calculation Of Integral, 792

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

Today's calculation of integrals, 855

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

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2013 Today's Calculation Of Integral, 864

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

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2005 Grigore Moisil Urziceni, 3

Tags: limit , real analysis , sequence

Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality $$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$ for all natural numbers $ n. $ [b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $ [b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $

1977 All Soviet Union Mathematical Olympiad, 239

Tags: limit , algebra , sequence

Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.

2005 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , limit , logarithm

Calculate \[ \lim_{x \to 0^+} \left( x^{x^x} - x^x \right). \]

2001 Romania National Olympiad, 4

Tags: function , limit , real analysis

The continuous function $f:[0,1]\rightarrow\mathbb{R}$ has the property: \[\lim_{x\rightarrow\infty}\ n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)=0 \] for every $x\in [0,1)$. Show that: a) For every $\epsilon >0$ and $\lambda\in (0,1)$, we have: \[ \sup\ \{x\in[0,\lambda )\mid |f(x)-f(0)|\le \epsilon x \}=\lambda \] b) $f$ is a constant function.

1982 IMO, 3

Tags: limit , geometric sequence , geometric series , inequality , minimization

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

Today's calculation of integrals, 764

Tags: calculus , integration , limit , trigonometry , derivative , real analysis , function

Find $\lim_{n\to\infty} \int_0^{\pi} e^{x}|\sin nx|dx.$

2020 LIMIT Category 2, 8

Tags: limit , probability , set

Let $S$ be a finite set of size $s\geq 1$ defined with a uniform probability $\mathbb{P}$( i.e. for any subset $X\subset S$ of size $x$, $\mathbb{P}(x)=\frac{x}{s}$). Suppose $A$ and $B$ are subsets of $S$. They are said to be independent iff $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$. Which if these is sufficient for independence? (A)$|A\cup B|=|A|+|B|$ (B)$|A\cap B|=|A|+|B|$ (C)$|A\cup B|=|A|\cdot |B|$ (D)$|A\cap B|=|A|\cdot |B|$

2012 Today's Calculation Of Integral, 808

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

For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\lim_{n\to\infty} a_n$.

2007 Moldova National Olympiad, 12.7

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

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

2012 Putnam, 3

Tags: function , limit , algebra , functional equation , topology

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that (i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$ (ii) $ f(0)=1,$ and (iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite. Prove that $f$ is unique, and express $f(x)$ in closed form.

2012 Pre-Preparation Course Examination, 6

Tags: limit , real analysis

Suppose that $a_{ij}$ are real numbers in such a way that for each $i$, the series $\sum_{j=1}^{\infty}a_{ij}$ is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence $\{b_j\}_j$ we have $\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0$. Prove that $\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0$.

1952 AMC 12/AHSME, 36

Tags: limit

To be continuous at $ x \equal{} \minus{} 1$, the value of $ \frac {x^3 \plus{} 1}{x^2 \minus{} 1}$ is taken to be: $ \textbf{(A)}\ \minus{} 2 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \infty \qquad\textbf{(E)}\ \minus{} \frac {3}{2}$

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

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

For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$