This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 837

2007 Today's Calculation Of Integral, 176
Tags: calculus , integration , trigonometry , limit , induction , symmetry , calculus computations
Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$ Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$

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]

1982 Putnam, A6
Tags: real analysis , limit
Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties: $(\text i)$ $|x_n|$ is a strictly decreasing function of $n$; $(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$; $(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$. Prove or disprove that these conditions imply that $$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$

1967 Swedish Mathematical Competition, 4
Tags: algebra , sum , limit
The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges. Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.

2011 Today's Calculation Of Integral, 695
Tags: calculus , integration , limit , calculus computations , logarithm
For a positive integer $n$, let \[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\] Answer the following questions: (1) Show the following inequality. \[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\] (2) Express $T_n-2S_n$ in terms of $n$. (3) Find the limit $\lim_{n\to\infty} T_n.$

1949 Miklós Schweitzer, 1
Tags: limit , real analysis
Let an infinite sequence of measurable sets be given on the interval $ (0,1)$ the measures of which are $ \geq \alpha>0$. Show that there exists a point of $ (0,1)$ which belongs to infinitely many terms of the sequence.

2006 Moldova National Olympiad, 11.2
Tags: function , limit , real analysis
Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

1978 Miklós Schweitzer, 4
Tags: function , limit , real analysis
Let $ \mathbb{Q}$ and $ \mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $ f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $ h \in \mathbb{Q} , \;x_0 \in \mathbb{R}$, \[ f(x\plus{}h)\minus{}f(x) \rightarrow 0\] as $ x \in \mathbb{Q}$ tends to $ x_0$. Does it follow that $ f$ is bounded on some interval? [i]M. Laczkovich[/i]

2009 Today's Calculation Of Integral, 484
Tags: calculus , integration , geometry , limit , trapezoid , calculus computations , logarithm
Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

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

2010 Today's Calculation Of Integral, 605
Tags: calculus , integration , limit , calculus computations
Let $f(x)$ be a differentiable function. Find the following limit value: \[\lim_{n\to\infty} \dbinom{n}{k}\left\{f\left(\frac{x}{n}\right)-f(0)\right\}^k.\] Especially, for $f(x)=(x-\alpha)(x-\beta)$ find the limit value above. 1956 Tokyo Institute of Technology entrance exam

2000 Putnam, 4
Tags: calculus , integration , limit , trigonometry , function
Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.

1960 Putnam, A6
Tags: probability , game , limit
A player repeatedly throwing a die is to play until their score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n,$ and find the value of $\lim_{n \to \infty} p(n).$

2009 Jozsef Wildt International Math Competition, W. 9
Tags: series , limit
Let the series $$s(n,x)=\sum \limits_{k= 0}^n \frac{(1-x)(1-2x)(1-3x)\cdots(1-nx)}{n!}$$ Find a real set on which this series is convergent, and then compute its sum. Find also $$\lim \limits_{(n,x)\to (\infty ,0)} s(n,x)$$

1971 Miklós Schweitzer, 6
Tags: function , limit , integration , real analysis
Let $ a(x)$ and $ r(x)$ be positive continuous functions defined on the interval $ [0,\infty)$, and let \[ \liminf_{x \rightarrow \infty} (x-r(x)) >0.\] Assume that $ y(x)$ is a continuous function on the whole real line, that it is differentiable on $ [0, \infty)$, and that it satisfies \[ y'(x)=a(x)y(x-r(x))\] on $ [0, \infty)$. Prove that the limit \[ \lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command. \int_0^x a(u)du \right \}\] exists and is finite. [i]I. Gyori[/i]

2010 Today's Calculation Of Integral, 656
Tags: calculus , integration , limit , trigonometry , function , calculus computations
Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

2010 Today's Calculation Of Integral, 584
Tags: calculus , integration , limit , calculus computations , logarithm
Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

1980 IMO, 4
Tags: limit , real analysis
Given a real number $x>1$, prove that there exists a real number $y >0$ such that \[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]

2019 Jozsef Wildt International Math Competition, W. 43
Tags: limit , integration , polynomial , sequence , algebra
Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$

2011 Today's Calculation Of Integral, 753
Tags: calculus , integration , limit , calculus computations , logarithm
Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$

1992 Putnam, A4
Tags: function , calculus , derivative , limit , trigonometry , search
Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$

Today's calculation of integrals, 871
Tags: calculus , integration , trigonometry , limit , calculus computations , definite integral , logarithm
Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2010 Today's Calculation Of Integral, 586
Tags: calculus , integration , algebra , polynomial , trigonometry , limit , calculus computations
Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

2005 Today's Calculation Of Integral, 48
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

Today's calculation of integrals, 765
Tags: calculus , integration , function , limit , calculus computations
Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.