Olympiad problems

PEN Q Problems, 2

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$

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]

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 $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$

2025 SEEMOUS, P2

Tags: integration , calculus , taylor series

Calculate $$\lim_{n\rightarrow\infty}n\int_0^{\infty} e^{-x}\sqrt[n]{e^x - 1 -\frac{x}{1!} - \frac{x^2}{2!} - \dots -\frac{x^n}{n!}}\,dx.$$

2005 Today's Calculation Of Integral, 23

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

Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

2005 Today's Calculation Of Integral, 50

Tags: logarithm , limit , derivative , integration , polynomial , algebra , calculus

Let $a,b$ be real numbers such that $a<b$. Evaluate \[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].

1986 AMC 12/AHSME, 29

Tags: inequalities , calculus , triangle inequality , integration

Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $

1999 Romania Team Selection Test, 7

Tags: arithmetic sequence , integration , algebra , calculus , geometric sequence , geometric series , binomial theorem

Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that \[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \] Give an example of two such progressions having at least five terms. [i]Mihai Baluna[/i]

1998 IMC, 6

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

Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$. (a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$. (b) Find such a funtion for which equality occurs.

Today's calculation of integrals, 768

Tags: limit , symmetry , geometry , integration , 3d geometry , calculus , sphere

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 $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2011 Today's Calculation Of Integral, 758

Tags: conic , geometry , analytic geometry , graphing lines , slope , calculus , integration

Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$

2007 Tournament Of Towns, 2

Tags: integration , calculus

Initially, the number $1$ and a non-integral number $x$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write $x^2$ on the blackboard in a finite number of moves?

2005 Moldova Team Selection Test, 3

Tags: calculus , integration , function , number theory

\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\] where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.

2005 Today's Calculation Of Integral, 84

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

Evaluate \[\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx\]

2001 Vietnam Team Selection Test, 1

Tags: algebra , calculus , floor function , integration

Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by \[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\] for all $n \in \mathbb{N}^{*}$. Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$. (Here $[x]$ denotes the integral part of real number $x$).