Olympiad problems

2010 Today's Calculation Of Integral, 632

Tags: calculus , integration , limit , trigonometry , floor function , ceiling function , calculus computations

Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$ [i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]

Today's calculation of integrals, 853

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

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

2024 CIIM, 4

Tags: limit , coordinate geometry , area , geometry

Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit: \[ \lim_{n \to \infty} \frac{S(n)}{n^3}. \]

2006 ISI B.Stat Entrance Exam, 8

Tags: limit , algebra , logarithm

Show that there exists a positive real number $x\neq 2$ such that $\log_2x=\frac{x}{2}$. Hence obtain the set of real numbers $c$ such that \[\frac{\log_2x}{x}=c\] has only one real solution.

2011 VTRMC, Problem 5

Tags: real analysis , limit

Find $\lim_{x\to\infty}\left((2x)^{1+\frac1{2x}}-x^{1+\frac1x}-x\right)$.

2010 Today's Calculation Of Integral, 665

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

Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$. [i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]

2005 Romania National Olympiad, 2

Tags: function , trigonometry , algebra , domain , limit , real analysis

Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function. a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto; b) Give an example of such a function.

1970 Miklós Schweitzer, 11

Tags: limit , probability and stats

Let $ \xi_1,\xi_2,...$ be independent random variables such that $ E\xi_n=m>0$ and $ \textrm{Var}(\xi_n)=\sigma^2 < \infty \;(n=1,2,...)\ .$ Let $ \{a_n \}$ be a sequence of positive numbers such that $ a_n\rightarrow 0$ and $ \sum_{n=1}^{\infty} a_n= \infty$. Prove that \[ P \left( \lim_{n\rightarrow \infty} %Error. "diaplaymath" is a bad command. \sum_{k=1}^n a_k \xi_k =\infty \right)=1.\] [i]P. Revesz[/i]

2009 Vietnam National Olympiad, 2

Tags: limit , algebra

The sequence $ \{x_n\}$ is defined by \[ \left\{ \begin{array}{l}x_1 \equal{} \frac{1}{2} \\x_n \equal{} \frac{{\sqrt {x_{n \minus{} 1} ^2 \plus{} 4x_{n \minus{} 1} } \plus{} x_{n \minus{} 1} }}{2} \\\end{array} \right.\] Prove that the sequence $ \{y_n\}$, where $ y_n\equal{}\sum_{i\equal{}1}^{n}\frac{1}{{{x}_{i}}^{2}}$, has a finite limit and find that limit.

2012 Today's Calculation Of Integral, 831

Tags: calculus , integration , limit , induction , calculus computations

Let $n$ be a positive integer. Answer the following questions. (1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$. (2) Show that $\lim_{x\to\infty} f_n(x)=0$. (3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.

2010 IMC, 3

Tags: limit , function , real analysis

Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.

1975 Canada National Olympiad, 7

Tags: function , trigonometry , quadratic , calculus , derivative , limit , algebra

A function $ f(x)$ is [i]periodic[/i] if there is a positive number $ p$ such that $ f(x\plus{}p) \equal{} f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.

2012 Grigore Moisil Intercounty, 3

Tags: riemann sum , definite integral , claculus , real analysis , limit

$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $

2009 Putnam, B5

Tags: function , limit , integration , algebra , domain , geometry

Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$

2004 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , trigonometry , limit

Let $f(x)=\sin(\sin(x))$. Evaluate \[ \lim_{h \to 0} \dfrac {f(x+h)-f(h)}{x} \] at $x=\pi$.

2001 Czech-Polish-Slovak Match, 5

Tags: function , limit , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy \[f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.\]

2010 Today's Calculation Of Integral, 626

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

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]

1994 IMC, 6

Tags: real analysis , limit

Find $$\lim_{N\to\infty}\frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}$$

2013 BMT Spring, 5

Tags: real analysis , sequence , limit

Suppose that $c_n=(-1)^n(n+1)$. While the sum $\sum_{n=0}^\infty c_n$ is divergent, we can still attempt to assign a value to the sum using other methods. The Abel Summation of a sequence, $a_n$, is $\operatorname{Abel}(a_n)=\lim_{x\to1^-}\sum_{n=0}^\infty a_nx^n$. Find $\operatorname{Abel}(c_n)$.

2010 Today's Calculation Of Integral, 568

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

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

1954 Putnam, B2

Tags: series , harmonic series , limit

Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$ Assume as known that this series converges as well and denote its sum by $S$. Denote by $s_k, S_k$ respectively the $k$-th partial sums of both series. Prove that $$ \!\!\!\! \text{i})\; S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$ $$ \text{ii}) \; S\ne s.$$

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 $

1970 IMO Longlists, 49

Tags: limit , number theory

For $n \in \mathbb N$, let $f(n)$ be the number of positive integers $k \leq n$ that do not contain the digit $9$. Does there exist a positive real number $p$ such that $\frac{f(n)}{n} \geq p$ for all positive integers $n$?

1968 Miklós Schweitzer, 9

Tags: function , limit , real analysis

Let $ f(x)$ be a real function such that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1\] and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1.\] [i]P. Erdos[/i]

2020 LIMIT Category 1, 9

Tags: number theory , counting , limit

What is the sum of all two-digit positive integer $n<50$ for which the sum of the squares of first $n$ positive integers is not a divisor of $(2n)!$ ?