Olympiad problems

1958 November Putnam, B4

Let $C$ be a real number, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function such that $$ \lim_{x \to \infty} f(x)=C, \;\; \; \lim_{x \to \infty} f'''(x)=0.$$ Prove that $$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$

2005 Croatia National Olympiad, 1

Tags: limit , algebra

A sequence $(a_{n})$ is deﬁned by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \geq 2.$ Find the smallest real number $M$ such that $\sum_{n=1}^{m}\frac{1}{a_{n}}<M\; \forall m\in\mathbb{N}$.

1950 Miklós Schweitzer, 4

Tags: matrix , limit , vector , linear algebra

Put $ M\equal{}\begin{pmatrix}p&q&r\\ r&p&q\\q&r&p\end{pmatrix}$ where $ p,q,r>0$ and $ p\plus{}q\plus{}r\equal{}1$. Prove that $ \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\ \frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}$

2018 Brazil Undergrad MO, 10

Tags: limit , college contest , calculus

How many ordered pairs of real numbers $ (a, b) $ satisfy equality $\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}$?

2014 Paenza, 1

Tags: limit , induction , logarithm

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

1994 Putnam, 4

Tags: linear algebra , matrix , limit , induction

For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where \[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\] Show that $\lim_{n\to \infty}d_n=\infty$.

2004 Czech-Polish-Slovak Match, 2

Tags: limit , number theory

Show that for each natural number $k$ there exist only finitely many triples $(p, q, r)$ of distinct primes for which $p$ divides $qr-k$, $q$ divides $pr-k$, and $r$ divides $pq - k$.

2010 Iran MO (3rd Round), 1

Tags: polynomial , limit , algebra

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

2007 Moldova National Olympiad, 11.2

Tags: limit , calculus , logarithm , calculus computations

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

2001 Romania National Olympiad, 3

Tags: function , limit , real analysis

Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $|f(x)-f(y)|\le |x-y|$ for every $x,y\in\mathbb{R}$. Show that: a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$. b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.

1998 IberoAmerican Olympiad For University Students, 6

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

Take the following differential equation: \[3(3+x^2)\frac{dx}{dt}=2(1+x^2)^2e^{-t^2}\] If $x(0)\leq 1$, prove that there exists $M>0$ such that $|x(t)|<M$ for all $t\geq 0$.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3

Tags: function , limit , calculus , calculus computations

Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value.

2006 IMS, 5

Tags: limit , function , algebra

Suppose that $a_{1},a_{2},\dots,a_{k}\in\mathbb C$ that for each $1\leq i\leq k$ we know that $|a_{k}|=1$. Suppose that \[\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.\] Prove that $c=k$ and $a_{i}=1$ for each $i$.

2020 LIMIT Category 2, 18

Tags: trigonometry , sum , limit , calculus

Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$ (A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$ (C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$ (D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$

2010 Putnam, B5

Tags: function , limit , integration , induction , inequalities , logarithm

Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$

1965 Putnam, A3

Tags: limit

Show that, for any sequence $a_1,a_2,\ldots$ of real numbers, the two conditions \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_n)}}n = \alpha \] and \[ \lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_{n^2})}}{n^2} = \alpha \] are equivalent.

1997 Traian Lălescu, 4

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

Compute the limit: \[ \lim_{n\to\infty} \frac{1}{n^2}\sum\limits_{1\leq i <j\leq n}\sin \frac{i+j}{n}\].

Today's calculation of integrals, 899

Tags: calculus , integration , limit , calculus computations

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2005 Morocco TST, 1

Tags: function , calculus , derivative , induction , limit , algebra

Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying : $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$

2024 ISI Entrance UGB, P4

Tags: function , calculus , limit

Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows: $$g(x) = \begin{cases} f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\ 0 &\text{if} ~ x = 0. \end{cases}$$ Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]

1999 VJIMC, Problem 1

Tags: limit , real analysis

Find the limit $$\lim_{n\to\infty}\left(\prod_{k=1}^n\frac k{k+n}\right)^{e^{\frac{1999}n}-1}.$$

2014 BMT Spring, 5

Tags: limit , real analysis

Determine $$\lim_{x\to\infty}\frac{\sqrt{x+2014}}{\sqrt x+\sqrt{x+2014}}$$

2004 Alexandru Myller, 2

Tags: limit , analysis , two-indexed , ln

$\lim_{n\to\infty } \sum_{1\le i\le j\le n} \frac{\ln (1+i/n)\cdot\ln (1+j/n)}{\sqrt{n^4+i^2+j^2}} $ [i]Gabriel Mîrșanu[/i] and [i]Andrei Nedelcu[/i]

2001 Austrian-Polish Competition, 2

Tags: limit , system of equations , pigeonhole principle , algebra

Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.

2020 Jozsef Wildt International Math Competition, W10

Tags: limit , real analysis , sequence

Let there be $(a_n)_{n\ge1},(b_n)_{n\ge1},a_n,b_n\in\mathbb R^*_+=(0,\infty)$ such that $\lim_{n\to\infty}a_n=a\in\mathbb R^*_+$ and $(b_n)_{n\ge1}$ is a bounded sequence. If $(x_n)_{n\ge1}$, $x_n=\prod_{k=1}^n(ka_h+b_h)$ find: $$\lim_{n\to\infty}\left(\sqrt[n+1]{x_{n+1}}-\sqrt[n]{x_n}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]