Olympiad problems

PEN G Problems, 7

Tags: integration , trigonometry , limit , algebra , polynomial , calculus , derivative

Show that $ \pi$ is irrational.

$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$. Find $\lim_{n \to \infty} \sqrt[n]{a_n}$.

2011 VTRMC, Problem 5

Tags: limit , real analysis

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

1997 VJIMC, Problem 4-M

Tags: limit , trigonometry , summation , real analysis , factorial

Prove that $$\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).$$

2001 IMC, 3

Tags: limit , integration , real analysis

Find $\lim_{t\rightarrow 1^-} (1-t) \sum_{n=1}^{\infty}\frac{t^n}{1+t^n}$.

2023 District Olympiad, P3

Tags: real analysis , integral , limit

Let $f:[0,1]\to\mathbb{R}$ be a continuous function. Prove that \[\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).\]Furthermore, if $f(0)=0$ and $f$ is right-differentiable in $0{}$, prove that the limits \[\lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx\quad\text{and}\quad\lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)\]exist, are finite and are equal.

2012 Today's Calculation Of Integral, 842

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

Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty} nS_nS_{n+1}.$

2012 Kyoto University Entry Examination, 1

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

2006 Romania National Olympiad, 3

Tags: inequalities , limit , complex number , real analysis

We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]

2007 AMC 12/AHSME, 11

Tags: limit , probability , geometry , geometric transformation , homothety

A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $ 247,$ $ 275,$ and $ 756$ and end with the term $ 824.$ Let $ \mathcal{S}$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $ \mathcal{S}?$ $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$

1973 Poland - Second Round, 3

Tags: algebra , limit , calculus

Let $ f:\mathbb{R} \to \mathbb{R} $ be an increasing function satisfying the following conditions: 1. $ f(x+1) = f(x) + 1 $ for each $ x \in \mathbb{R} $, 2. there exists an integer p such that $ f(f(f(O))) = p $. Prove that for every real number $ x $ $$ \lim_{n\to \infty} \frac{x_n}{n} = \frac{p}{3}.$$ where $ x_1 = x $ and $ x_n =f(x_{n-1}) $ for $ n = 2, 3, \ldots $.

1969 Canada National Olympiad, 2

Tags: limit , inequalities , calculus , function

Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.

2016 Korea USCM, 5

Tags: limit , real analysis , lp norm , function

For $f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)$, find the value of $$\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} $$

2020 LIMIT Category 1, 11

Tags: limit , geometry

In $\triangle ABC$, $\angle A=30^{\circ}$, $BC=13$. Given $2$ circles $\gamma_1, \gamma_2$ ith radius $r_1,r_2$ contain $A$ and touch $BC$ at $B$ and $C$ respectively. Find $r_1r_2$.

2013 IPhOO, 9

Tags: limit , geometry , 3d geometry , sphere , buoyancy

Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$. [i](B. Dejean, 6 points)[/i]

2010 ISI B.Math Entrance Exam, 8

Tags: function , limit , calculus , calculus computations

Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.

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]

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 $n$?

1965 AMC 12/AHSME, 10

Tags: quadratic , inequalities , limit

The statement $ x^2 \minus{} x \minus{} 6 < 0$ is equivalent to the statement: $ \textbf{(A)}\ \minus{} 2 < x < 3 \qquad \textbf{(B)}\ x > \minus{} 2 \qquad \textbf{(C)}\ x < 3$ $ \textbf{(D)}\ x > 3 \text{ and }x < \minus{} 2 \qquad \textbf{(E)}\ x > 3 \text{ and }x < \minus{} 2$