Olympiad problems

1970 Miklós Schweitzer, 11

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]

2020 LIMIT Category 1, 2

Tags: geometry , limit

In a square $ABCD$ of side $2$ units, $E$ is the midpoint of $AD$ and $F$ on $BE$ such that $CF\perp BE$, then the quadrilateral $CDEF$ has an area of (A)$2$ (B)$2.2$ (C)$\sqrt{5}$ (D)None of these

2020 LIMIT Category 2, 5

Tags: limit , geometry , coordinates

Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is its area? (A)$20\sqrt{3}$ (B)$20\sqrt{2}$ (C)$25\sqrt{3}$ (D)None of these

1997 VJIMC, Problem 2

Tags: limit , real analysis , sequence

Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality $$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.

2006 Moldova National Olympiad, 12.2

Tags: limit , calculus , integration , algebra , floor function , function

Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$

2003 Bulgaria National Olympiad, 3

Tags: polynomial , limit , induction , algebra

Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.

1961 Putnam, A3

Tags: limit , series

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

1997 Flanders Math Olympiad, 3

Tags: geometry , inequalities , trigonometry , ratio , limit , geometric sequence

$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$. Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$, Call the summed area of the first $k$ triangles $A_k$. Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$, going through the points $b_2$ and $a_2$ as shown on the picture. (yes, $cd$ is parallel to $a_1b_1$ there) Show $A_k < S$ for every positive integer $k$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=284[/img]

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 Putnam, A6

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

Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

2003 IMC, 1

Tags: limit

(a) Let $a_1,a_2,...$ be a sequenceof reals with $a_1=1$ and $a_{n+1}>\frac32 a_n$ for all $n$. Prove that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}$ exists. (finite or infinite) (b) Prove that for all $\alpha>1$ there is a sequence $a_1,a_2,...$ with the same properties such that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}=\alpha$

2019 Jozsef Wildt International Math Competition, W. 35

Tags: limit

Calculate$$\lim \limits_{n \to \infty}\frac{n!\left(1+\frac{1}{n}\right)^{n^2+n}}{n^{n+\frac{1}{2}}}$$

2007 Today's Calculation Of Integral, 229

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

Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

1974 IMO Longlists, 25

Tags: trigonometry , limit , algebra

Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$ \[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\] Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$

1987 Traian Lălescu, 1.4

Tags: real analysis , limit , sequence

[b]a)[/b] Determine all sequences of real numbers $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ that satisfy $ x_{n+2}+x_{n+1}=x_n, $ for any nonnegative integer $ n. $ [b]b)[/b] If $ y_k>0 $ and $ y_k^k=y_k+k, $ for all naturals $ k, $ calculate $ \lim_{n\to\infty }\frac{\ln n}{n\left( x_n-1\right)} . $

2006 Stanford Mathematics Tournament, 8

Tags: limit

Evaluate: $\lim_{n\rightarrow\infty}\sum_{k=n^2}^{(n+1)^2} \dfrac{1}{\sqrt{k}}$

2006 Vietnam National Olympiad, 4

Tags: function , calculus , derivative , limit , algebra

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$, where $a$, $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that \[ f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} . \]

1987 Flanders Math Olympiad, 4

Tags: limit , inequalities

Show that for $p>1$ we have \[\lim_{n\rightarrow+\infty}\frac{1^p+2^p+...+(n-1)^p+n^p+(n-1)^p+...+2^p+1^p}{n^2} = +\infty\] Find the limit if $p=1$.

1994 Putnam, 5

Tags: limit

Let $(r_n)_{n\ge 0}$ be a sequence of positive real numbers such that $\lim_{n\to \infty} r_n = 0$. Let $S$ be the set of numbers representable as a sum \[ r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} ,\] with $i_1 < i_2 < \cdots< i_{1994}.$ Show that every nonempty interval $(a, b)$ contains a nonempty subinterval $(c, d)$ that does not intersect $S$.

2013 Stanford Mathematics Tournament, 3

Tags: calculus , limit , trigonometry , analytic geometry

Suppose $a$ and $b$ are real numbers such that \[\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}.\] Determine all possible ordered pairs $(a, b)$.

1968 Miklós Schweitzer, 7

Tags: function , limit , advanced fields

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

2003 Vietnam National Olympiad, 3

Tags: function , induction , limit , algebra

Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.

2012 Today's Calculation Of Integral, 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 $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2009 Today's Calculation Of Integral, 424

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

Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

2020 LIMIT Category 1, 10

Tags: number theory , limit

For natural number $t$, the repeating base-$t$ representation of the (base-ten) rational number $\frac{7}{51}$ is $0.\overline{23}_t=0.232323..._t$. What is $t$ ?