Olympiad problems

1998 Irish Math Olympiad, 1

Tags: integration , inequalities

Prove that if $ x \not\equal{} 0$ is a real number, then: $ x^8\minus{}x^5\minus{}\frac{1}{x}\plus{}\frac{1}{x^4} \ge 0$.

1972 Canada National Olympiad, 10

Tags: ratio , calculus , integration , geometric sequence , algebra

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

2013 Romanian Master of Mathematics, 4

Tags: ratio , geometric transformation , reflection , integration , function , homothety , geometry

Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?

2009 Today's Calculation Of Integral, 429

Tags: calculus , integration , trigonometry , calculus computations

Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.

2010 Today's Calculation Of Integral, 664

Tags: calculus , integration , trigonometry , calculus computations

For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$. Find $I_1+I_2+I_3+I_4$. [i]1992 University of Fukui entrance exam/Medicine[/i]

2010 Today's Calculation Of Integral, 575

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

For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions. (1) Find $ f'(x)$. (2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 3

Tags: calculus , integration , trigonometry , real analysis

Let $a$ be a positive real number. Evaluate $I=\int_0^{+\infty} \frac{\sin x\cos x}{x(x^2+a^2)}dx.$

2004 Romania National Olympiad, 1

Tags: function , integration , calculus , derivative , limit , algebra , functional equation

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \] [i]Mihai Piticari[/i]

2007 Today's Calculation Of Integral, 234

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

For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$. Evaluate $ \int_0^{100\pi } f(x)\ dx.$

2007 ITest, 40

Tags: floor function , calculus , integration , desargues involution theorem

Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and \[\{x^2\}=\{x\}^2.\] Compute $\lfloor S\rfloor$.

2010 Today's Calculation Of Integral, 584

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

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

1960 AMC 12/AHSME, 24

Tags: geometry , 3d geometry , calculus , integration , function , irrational number , logarithm

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$ $\textbf{(C)}\ \text{An irrational number} \qquad$ $\textbf{(D)}\ \text{A perfect square}\qquad$ $\textbf{(E)}\ \text{A perfect cube} $

1985 ITAMO, 10

Tags: floor function , calculus , integration , function

How many of the first 1000 positive integers can be expressed in the form \[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \] where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

2008 ISI B.Stat Entrance Exam, 6

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

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

2006 Romania National Olympiad, 2

Tags: limit , integration , real analysis

Prove that \[ \lim_{n \to \infty} n \left( \frac{\pi}{4} - n \int_0^1 \frac{x^n}{1+x^{2n}} \, dx \right) = \int_0^1 f(x) \, dx , \] where $f(x) = \frac{\arctan x}{x}$ if $x \in \left( 0,1 \right]$ and $f(0)=1$. [i]Dorin Andrica, Mihai Piticari[/i]

2020 Jozsef Wildt International Math Competition, W60

Tags: integration , calculus

Compute $$\int\frac{(\sin x+\cos x)(4-2\sin2x-\sin^22x)e^x}{\sin^32x}dx$$ where $x\in\left(0,\frac\pi2\right)$. [i]Proposed by Mihály Bencze[/i]

2006 Mathematics for Its Sake, 2

Tags: integration , calculus

Calculate: [b]a)[/b] $ \int \frac{1-x^2-x^6+x^8}{1+x^{10}} dx $ [b]b)[/b] $ \int\frac{x^{n-1}+x^{5n-1}}{1+x^{6n}} dx $ [i]Dumitru Acu[/i]

2009 Princeton University Math Competition, 3

Tags: calculus , integration , floor function , induction , ceiling function , logarithm

It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?

2014 International Zhautykov Olympiad, 1

Tags: polynomial , calculus , integration , algebra

Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $? [i]Proposed by Alexander S. Golovanov, Russia[/i]

PEN Q Problems, 7

Tags: algebra , polynomial , calculus , integration , number theory , relatively prime , rational root theorem

Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

2008 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , integration , calculus computations , logarithm

([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.

1998 Vietnam Team Selection Test, 1

Tags: algebra , polynomial , calculus , integration , rational root theorem , number theory

Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

2009 Putnam, B4

Tags: algebra , polynomial , vector , calculus , integration , complex analysis

Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$

2010 Today's Calculation Of Integral, 601

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

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

Today's calculation of integrals, 891

Tags: calculus , integration , geometry , 3d geometry , hyperbola , rotation , conic

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.