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

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 837

2006 Moldova National Olympiad, 11.1
Tags: limit , trigonometry , algebra
Let $n\in\mathbb{N}^*$. Prove that \[ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}. \]

2009 Albania Team Selection Test, 2
Tags: function , algebra , polynomial , induction , calculus , derivative , limit
Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

2010 Paenza, 3
Tags: limit , trigonometry , real analysis
Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

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}$.

2007 Princeton University Math Competition, 4
Tags: limit
Let $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$ and $a_0 = 1$, $a_1 = 2$. Find $\lim_{n \to \infty} a_n$.

1954 Putnam, B7
Tags: limit , exponential
Let $a>0$. Show that $$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$ lies between $e^a$ and $e^{a+1}.$

2020 Jozsef Wildt International Math Competition, W21
Tags: real analysis , limit
Evaluate $$\lim_{n\to\infty}\left(\frac{1+\frac13+\ldots+\frac1{2n+1}}{\ln\sqrt n}\right)^{\ln\sqrt n}$$ [i]Proposed by Ángel Plaza[/i]

2005 Iran MO (3rd Round), 1
Tags: rotation , limit , geometry
An airplane wants to go from a point on the equator, and at each moment it will go to the northeast with speed $v$. Suppose the radius of earth is $R$. a) Will the airplane reach to the north pole? If yes how long it will take to reach the north pole? b) Will the airplne rotate finitely many times around the north pole? If yes how many times?

1958 November Putnam, B4
Tags: function , derivative , limit
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 $$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$

2020 LIMIT Category 1, 16
Tags: combinatorics , limit
A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that atleast $15$ balls of a single colour will be drawn?

2013 Today's Calculation Of Integral, 863
Tags: calculus , integration , trigonometry , limit , derivative , inequalities , calculus computations
For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

Today's calculation of integrals, 877
Tags: calculus , integration , trigonometry , calculus computations , definite integral , limit
Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

2011 Today's Calculation Of Integral, 768
Tags: calculus , integration , limit , 3d geometry , sphere , symmetry , geometry
Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2004 Nicolae Coculescu, 4
Tags: function , real analysis , limit , primitive
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a continuous function having a primitive $ F $ having the property that $ f-F $ is positive globally. Calculate $ \lim_{x\to\infty } f(x) . $ [i]Florian Dumitrel[/i]

2012 Pre-Preparation Course Examination, 2
Tags: limit , real analysis
Suppose that $\lim_{n\to \infty} a_n=a$ and $\lim_{n\to \infty} b_n=b$. Prove that $\lim_{n\to \infty}\frac{1}{n}(a_1b_n+a_2b_{n-1}+...+a_nb_1)=ab$.

2005 Today's Calculation Of Integral, 90
Tags: calculus , integration , limit , factorial , function , ratio , logarithm
Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$ where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

2003 Romania National Olympiad, 4
Tags: abstract algebra , group theory , ring theory , limit
$ i(L) $ denotes the number of multiplicative binary operations over the set of elements of the finite additive group $ L $ such that the set of elements of $ L, $ along with these additive and multiplicative operations, form a ring. Prove that [b]a)[/b] $ i\left( \mathbb{Z}_{12} \right) =4. $ [b]b)[/b] $ i(A\times B)\ge i(A)i(B) , $ for any two finite commutative groups $ B $ and $ A. $ [b]c)[/b] there exist two sequences $ \left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1} $ of finite commutative groups such that $$ \lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0 $$ and $$ \lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty. $$ [i]Barbu Berceanu[/i]

2020 LIMIT Category 2, 16
Tags: limit , derivative , calculus
The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $. Let $f(x)=\frac{e^x}{x}$. Suppose $f$ is differentiable infinitely many times in $(0,\infty) $. Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$

2007 Today's Calculation Of Integral, 183
Tags: calculus , integration , limit , trigonometry , calculus computations
Let $n\geq 2$ be integer. On a plane there are $n+2$ points $O,\ P_{0},\ P_{1},\ \cdots P_{n}$ which satisfy the following conditions as follows. [1] $\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).$ [2] $\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$

1966 Swedish Mathematical Competition, 5
Tags: limit , lattice , combinatorial geometry
Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin. Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$. Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?

1997 VJIMC, Problem 4-M
Tags: real analysis , number theory , digit , limit
Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

2006 ISI B.Stat Entrance Exam, 6
Tags: function , limit , calculus , calculus computations
(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$. (b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.

2019 Mathematical Talent Reward Programme, MCQ: P 6
Tags: limit
Find the limit $\lim \limits_{n \to \infty} \sin{n!}$ [list=1] [*] 1 [*] 0 [*] $\frac{\pi}{4}$ [*] None of the above [/list]

2010 Today's Calculation Of Integral, 598
Tags: calculus , integration , trigonometry , limit , calculus computations
For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam

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 $S$.