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

2014 District Olympiad, 1
Tags: function , floor function , limit , integration , real analysis
For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$

1984 Vietnam National Olympiad, 2
Tags: limit , trigonometry , algebra
The sequence $(u_n)$ is defined by $u_1 = 1, u_2 = 2$ and $u_{n+1} = 3u_n - u_{n-1}$ for $n \ge 2$. Set $v_n =\sum_{k=1}^n \text{arccot }u_k$. Compute $\lim_{n\to\infty} v_n$.

2021 Belarusian National Olympiad, 10.1
Tags: limit , sequence , algebra
An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$ Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$

2004 Harvard-MIT Mathematics Tournament, 9
Tags: calculus , limit , ratio , algebra , polynomial , logarithm , absolute value
Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.

PEN E Problems, 14
Tags: algebra , polynomial , limit , function , gauss , number theory , logarithm
Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

2013 BMT Spring, 10
Tags: calculus , limit
Let the class of functions $f_n$ be defined such that $f_1(x)=|x^3-x^2|$ and $f_{k+1}(x)=|f_k(x)-x^3|$ for all $k\ge1$. Denote by $S_n$ the sum of all $y$-values of $f_n(x)$'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms, $$\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}$$

2024 CIIM, 4
Tags: geometry , limit , coordinate geometry , area
Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit: \[ \lim_{n \to \infty} \frac{S(n)}{n^3}. \]

2013 India IMO Training Camp, 1
Tags: function , induction , limit , algebra , cauchy equation
Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

2014 Contests, 2
Tags: function , limit , functional equation , algebra
Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

1970 Putnam, A4
Tags: sequence , limit
Given a sequence $(x_n )$ such that $\lim_{n\to \infty} x_n - x_{n-2}=0,$ prove that $$\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.$$

2014 JBMO Shortlist, 3
Tags: floor function , modular arithmetic , limit , combinatorics , logarithm
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

2012 Today's Calculation Of Integral, 831
Tags: calculus , integration , limit , induction , calculus computations
Let $n$ be a positive integer. Answer the following questions. (1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$. (2) Show that $\lim_{x\to\infty} f_n(x)=0$. (3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.

2004 Vietnam National Olympiad, 1
Tags: trigonometry , limit , algebra
The sequence $ (x_n)^{\infty}_{n\equal{}1}$ is defined by $ x_1 \equal{} 1$ and $ x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}$, for all $ n \in\mathbb{N}$, where $ \alpha$ is a given real parameter. Find all values of $ \alpha$ for which the sequence $ (y_n)$ given by $ y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1}$ has a finite limit when $ n \to \plus{}\infty$ and find that limit.

1993 Flanders Math Olympiad, 4
Tags: trigonometry , limit
Define the sequence $oa_n$ as follows: $oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)$. Find $\lim\limits_{n\rightarrow+\infty} oa_n$.

1989 Greece National Olympiad, 3
Tags: limit , real analysis , sequence
Find the limit of the sequence $x_n$ defined by recurrence relation $$x_{n+2}=\frac{1}{12}x_{n+1}+\frac{1}{2}x_{n}+1$$ where $n=0,1,2,...$ for any initial values $x_2,x_1$.

1964 Putnam, B3
Tags: function , limit , search , topology , real analysis
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?

2019 Brazil Undergrad MO, Problem 5
Tags: combinatorics , counting , limit , number theory
Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$

1995 VJIMC, Problem 4
Tags: limit , real analysis , trigonometry
Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it.

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 LIMIT Category 1, 17
Tags: algebra , limit
The sum of $k$ consecutive integers is $90$. Then the sum of all possible values of $k$ is? (A)$89$ (B)$179$ (C)$168$ (D)$119$

2014 Contests, 3
Tags: limit , induction , inequalities , algebra , difference of squares
Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2012 Today's Calculation Of Integral, 812
Tags: calculus , integration , trigonometry , limit , calculus computations , logarithm
Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$, evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$

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.

2013 Today's Calculation Of Integral, 877
Tags: limit , calculus , integration , trigonometry , calculus computations , definite integral
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 $\int_0^{\frac{\pi}2} f(x)\ dx.$

2010 Today's Calculation Of Integral, 657
Tags: calculus , integration , trigonometry , limit , function , linear algebra , calculus computations
A sequence $a_n$ is defined by $\int_{a_n}^{a_{n+1}} (1+|\sin x|)dx=(n+1)^2\ (n=1,\ 2,\ \cdots),\ a_1=0$. Find $\lim_{n\to\infty} \frac{a_n}{n^3}$.