Olympiad problems

1996 National High School Mathematics League, 12

Tags: calculus , integration

The number of integral points on the circle with center $(199,0)$, radius of $199$ is________.

2008 Grigore Moisil Intercounty, 2

Tags: real analysis , integration , function , calculus , inequalities

Let $ n\in \mathbb{N^*}$ and $ f: [0,1]\rightarrow \mathbb{R}$ a continuos function with the prop. $ \int_{0}^{1}(1\minus{}x^n)f(x)dx\equal{}0$. Prove that $ \int_{0}^{1}f^2(x)dx \geq 2(n\plus{}1)\left(\int_{0}^{1}f(x)dx\right)^2$

2005 Today's Calculation Of Integral, 8

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

Calculate the following indefinite integrals. [1] $\int x(x^2+3)^2 dx$ [2] $\int \ln (x+2) dx$ [3] $\int x\cos x dx$ [4] $\int \frac{dx}{(x+2)^2}dx$ [5] $\int \frac{x-1}{x^2-2x+3}dx$

2003 Moldova National Olympiad, 12.1

Tags: limit , calculus , calculus computations , logarithm

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

2021 The Chinese Mathematics Competition, Problem 6

Tags: calculus

Let $x_1=2021$, $x_n^2-2(x_n+1)x_{n+1}+2021=0$ ($n\geq1$). Prove that the sequence ${x_n}$ converges. Find the limit $\lim_{n \to \infty} x_n$.

2006 ISI B.Stat Entrance Exam, 6

Tags: calculus computations , limit , function , calculus

(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.

2011 Gheorghe Vranceanu, 2

Tags: algebra , squeezing , am-gm , caluclus , limit , calculus

$ a>0,\quad\lim_{n\to\infty }\sum_{i=1}^n \frac{1}{n+a^i} $

2011 Today's Calculation Of Integral, 707

Tags: calculus , calculus computations , integration

In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that \[\left\{ \begin{array}{ll} x^2+y^2\leq 4 &\quad \\ 0\leq z\leq 4 &\quad \end{array} \right.\] Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying \[\left\{ \begin{array}{ll} z\leq (x-2)^2 &\quad \\ z\leq y^2 &\quad \end{array} \right.\] Find the volume of the solid $V$.

2010 Today's Calculation Of Integral, 562

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

(1) Show the following inequality for every natural number $ k$. \[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\] (2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$. \[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]

PEN Q Problems, 2

Tags: algebra , integration , calculus , pigeonhole principle , polynomial

Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

2009 Today's Calculation Of Integral, 457

Tags: calculus , integration , trigonometry , logarithm , calculus computations

Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$

2014 ELMO Shortlist, 1

Tags: trigonometry , inequalities , topology , three variable inequality , calculus

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2010 Today's Calculation Of Integral, 626

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

Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$ [i]2010 Nara Medical University entrance exam[/i]

2005 Romania Team Selection Test, 2

Tags: calculus , integration , number theory

Let $m,n$ be co-prime integers, such that $m$ is even and $n$ is odd. Prove that the following expression does not depend on the values of $m$ and $n$: \[ \frac 1{2n} + \sum^{n-1}_{k=1} (-1)^{\left[ \frac{mk}n \right]} \left\{ \frac {mk}n \right\} . \] [i]Bogdan Enescu[/i]

2012 Math Prize For Girls Problems, 19

Tags: derivative , induction , calculus , function

Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by \[ a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr), \] where there are $n$ iterations of $L$. For example, \[ a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr). \] As $n$ approaches infinity, what value does $n a_n$ approach?

2008 ISI B.Stat Entrance Exam, 3

Tags: calculus , function , derivative , geometry , 3d geometry , calculus computations

Study the derivatives of the function \[y=\sqrt{x^3-4x}\] and sketch its graph on the real line.

2011 Today's Calculation Of Integral, 674

Tags: integration , calculus , logarithm , calculus computations

Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$ [i]2011 Doshisya University entrance exam/Science and Technology[/i]

2003 Alexandru Myller, 4

Tags: function , integration , calculus

[b]a)[/b] Prove that the function $ 1\le t\mapsto\int_{1}^t\frac{\sin x}{x^n} dx $ has an horizontal asymptote, for any natural number $ n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty }\lim_{t\to\infty }\int_{1}^t\frac{\sin x}{x^n} . $ [i]Mihai Piticari[/i]

2007 Today's Calculation Of Integral, 172

Tags: trigonometry , integration , calculus computations , euler , calculus

Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$

1991 Arnold's Trivium, 39

Tags: calculus , trigonometry , gauss , integration

Calculate the Gauss integral \[\oint\frac{(d\overrightarrow{A},d\overrightarrow{B},\overrightarrow{A}-\overrightarrow{B})}{|\overrightarrow{A}-\overrightarrow{B}|^3}\] where $\overrightarrow{A}$ runs along the curve $x=\cos\alpha$, $y=\sin\alpha$, $z=0$, and $\overrightarrow{B}$ along the curve $x=2\cos^2\beta$, $y=\frac12\sin\beta$, $z=\sin2\beta$. Note: that $\oint$ was supposed to be oiint (i.e. $\iint$ with a circle) but the command does not work on AoPS.

2013 Today's Calculation Of Integral, 894

Tags: geometry , integration , calculus computations , calculus , logarithm

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

2011 Iran MO (3rd Round), 2

Tags: function , modular arithmetic , integration , induction , calculus , polynomial , algebra

Let $n$ and $k$ be two natural numbers such that $k$ is even and for each prime $p$ if $p|n$ then $p-1|k$. let $\{a_1,....,a_{\phi(n)}\}$ be all the numbers coprime to $n$. What's the remainder of the number $a_1^k+.....+a_{\phi(n)}^k$ when it's divided by $n$? [i]proposed by Yahya Motevassel[/i]

2008 ISI B.Stat Entrance Exam, 6

Tags: logarithm , function , calculus , integration , limit , real analysis , calculus computations

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

2010 Today's Calculation Of Integral, 650

Tags: algebra , calculus computations , calculus , polynomial , integration

Find the values of $p,\ q,\ r\ (-1<p<q<r<1)$ such that for any polynomials with degree$\leq 2$, the following equation holds: \[\int_{-1}^p f(x)\ dx-\int_p^q f(x)\ dx+\int_q^r f(x)\ dx-\int_r^1 f(x)\ dx=0.\] [i]1995 Hitotsubashi University entrance exam/Law, Economics etc.[/i]

2007 Today's Calculation Of Integral, 183

Tags: limit , integration , calculus , 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 $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$