Olympiad problems

2007 China Team Selection Test, 3

Tags: algebra , polynomial , function , induction , quadratic , limit , integration

Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.

2002 Tournament Of Towns, 2

Tags: limit , algebra

Does there exist points $A,B$ on the curve $y=x^3$ and on $y=x^3+|x|+1$ respectively such that distance between $A,B$ is less than $\frac{1}{100}$ ?

2011 Tokio University Entry Examination, 3

Tags: calculus , integration , analytic geometry , limit , logarithm , geometry , perimeter

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

2000 Romania National Olympiad, 1

Tags: real analysis , continuity , calculus , integration , limit , integral

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

1976 Bulgaria National Olympiad, Problem 6

Tags: analytic geometry , geometry , limit

It is given a plane with a coordinate system with a beginning at the point $O$. $A(n)$, when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$. (a) Find $$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$ (b) For which $\beta$ $(1<\beta<2)$ does the following limit exist? $$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$

2018 VJIMC, 4

Tags: functional equation , decreasing , limit , function

Determine all possible (finite or infinite) values of \[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\] if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying \[f(f(x))^4-f(f(x))+f(x)=1\] for all $x \in \mathbb{R}$.

2009 Brazil National Olympiad, 3

Tags: function , limit , cauchy inequality , inequalities

Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of \[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]

1956 Putnam, A1

Tags: limit , geometric series

Evaluate $$ \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1\slash x},$$ where $a>0$ and $a\ne 1.$

1975 Miklós Schweitzer, 5

Tags: function , integration , limit , real analysis

Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$ [i]J. Szucs[/i]

2003 VJIMC, Problem 3

Tags: nested radical , real analysis , limit

Find the limit $$\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.$$

2011 Putnam, A3

Tags: limit , integration , trigonometry , calculus , ratio , inequalities

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

1982 Putnam, B5

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

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 $x>e^e$.

PEN S Problems, 36

Tags: modular arithmetic , induction , limit

For every natural number $n$, denote $Q(n)$ the sum of the digits in the decimal representation of $n$. Prove that there are infinitely many natural numbers $k$ with $Q(3^{k})>Q(3^{k+1})$.

2020 LIMIT Category 1, 14

Tags: number theory , algebra , limit

Let $(m,n)$ be the pairs of integers satisfying $2(8n^3+m^3)+6(m^2-6n^2)+3(2m+9n)=437$. Find the sum of all possible values of $mn$.

1971 IMO Longlists, 53

Tags: limit , floor function , number theory , logarithm

Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.

2016 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: limit , algebra , sequence

Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that $a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$ $b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$

2010 Today's Calculation Of Integral, 625

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

Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

2019 District Olympiad, 1

Tags: limit , convergence , calculus , sequence

Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$

2012 Today's Calculation Of Integral, 857

Tags: calculus , integration , limit , trigonometry , geometry , geometric transformation , rotation

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2001 VJIMC, Problem 3

Tags: calculus , integration , function , limit

Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.

2024 Mexican University Math Olympiad, 4

Tags: matrices , limit , linear algebra , series

Given $ b > 0 $, consider the following matrix: \[ B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix} \] Denote by $ e_i $ the top left entry of $ B^i $. Prove that the following limit exists and calculate its value: \[ \lim_{i \to \infty} \sqrt[i]{e_i}. \]

2015 BMT Spring, 9

Tags: real analysis , limit

Find $$\lim_{n\to\infty}\frac1{n^3}\left(\sqrt{n^2-1^2}+\sqrt{n^2-2^2}+\ldots+\sqrt{n^2-(n-1)^2}\right).$$

2000 District Olympiad (Hunedoara), 3

Tags: calculus , real analysis , sequence , limit

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 5

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

Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers. Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$

1952 Miklós Schweitzer, 9

Tags: function , limit , integration , real analysis

Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$. (The latter means that $ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$ holds for every integrable function $ h(x)$.)