Olympiad problems

Today's calculation of integrals, 881

Tags: calculus , integration , trigonometry , calculus computations , definite integral

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2003 Vietnam Team Selection Test, 3

Tags: floor function , calculus , integration , induction , algebra

Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and \[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\] for every natural number $m > 0$ and for every integer $n$. Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$

2009 Today's Calculation Of Integral, 419

Tags: calculus , integration , parabola , geometry , calculus computations , conic

In the $ xy$ plane, the line $ l$ touches to 2 parabolas $ y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax$, where $ a$ is positive constant. (1) Find the equation of $ l$. (2) Find the area $ S$ bounded by the parabolas and the tangent line $ l$.

2016 Israel Team Selection Test, 1

Tags: inequalities , calculus , lagrange

Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.

2009 Today's Calculation Of Integral, 514

Tags: calculus , integration , inequalities , trigonometry , function , algebra , domain

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

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.

2003 Tuymaada Olympiad, 4

Tags: algebra , polynomial , calculus , integration , number theory

Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite. Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$ [i]Proposed by F. Petrov[/i] [hide="For those of you who liked this problem."] Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread[/url] out.[/hide]

1984 Iran MO (2nd round), 4

Tags: calculus , integration , combinatorics

Find number of terms when we expand $(a+b+c)^{99}$ (in the general case).

2007 Today's Calculation Of Integral, 168

Tags: calculus , integration , trigonometry , calculus computations

Prove that $\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx}$ diverge for $x>0.$

2010 Today's Calculation Of Integral, 626

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

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

2007 ISI B.Stat Entrance Exam, 2

Tags: calculus , function , trigonometry , derivative , calculus computations

Use calculus to find the behaviour of the function \[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\] and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.

2014 PUMaC Number Theory B, 8

Tags: floor function , number theory , greatest common divisor , calculus , integration , modular arithmetic

Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

2019 Jozsef Wildt International Math Competition, W. 66

Tags: integration , inequalities , trigonometry , inverse trigonometric function , calculus

If $0 < a \leq b$ then$$\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)$$

2011 Today's Calculation Of Integral, 716

Tags: calculus , integration , calculus computations , logarithm

Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]

2023 IMC, 7

Tags: calculus , function

Let $V$ be the set of all continuous functions $f\colon [0,1]\to \mathbb{R}$, differentiable on $(0,1)$, with the property that $f(0)=0$ and $f(1)=1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f\in V$, there exists some $\xi \in (0,1)$ such that \[f(\xi)+\alpha = f'(\xi)\]

2010 Today's Calculation Of Integral, 608

Tags: calculus , integration , calculus computations

For $a>0$, find the minimum value of $\int_0^1 \frac{ax^2+(a^2+2a)x+2a^2-2a+4}{(x+a)(x+2)}dx.$ 2010 Gakusyuin University entrance exam/Science

1991 Baltic Way, 8

Tags: algebra , polynomial , calculus , derivative

Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation \[(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)\] \[+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)\] \[+(x - b)(x - c)(x - d)(x - e) = 0\] has four distinct real solutions.

2010 Today's Calculation Of Integral, 641

Tags: calculus , integration , calculus computations , logarithm

Evaluate \[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\] Own

1993 Balkan MO, 2

Tags: calculus , integration , ceiling function , floor function , inequalities , combinatorics

A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called [i]monotone[/i] if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.

2009 Today's Calculation Of Integral, 478

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac{\pi}{4}} \{(x\sqrt{\sin x}\plus{}2\sqrt{\cos x})\sqrt{\tan x}\plus{}(x\sqrt{\cos x}\minus{}2\sqrt{\sin x})\sqrt{\cot x}\}\ dx.$

2003 Alexandru Myller, 4

Tags: function , calculus , integration

[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]

2001 VJIMC, Problem 3

Tags: limit , calculus , integration , function

Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.

2004 Poland - First Round, 3

Tags: trigonometry , vector , inequalities , calculus , derivative , geometry

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$

1950 AMC 12/AHSME, 17

Tags: calculus , derivative , quadratic , algebra , system of equations , arithmetic series

The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is: \[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline \end{tabular}\] $\textbf{(A)}\ y=100-10x \qquad \textbf{(B)}\ y=100-5x^2 \qquad \textbf{(C)}\ y=100-5x-5x^2 \qquad\\ \textbf{(D)}\ y=20-x-x^2 \qquad \textbf{(E)}\ \text{None of these}$

2010 Romania National Olympiad, 4

Tags: function , calculus , derivative , integration , limit , real analysis

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]