Olympiad problems

Today's calculation of integrals, 891

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

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.

2007 Today's Calculation Of Integral, 249

Tags: logarithm , calculus , integration , calculus computations

Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.

2012 Today's Calculation Of Integral, 770

Tags: calculus computations , calculus , integration

Find the value of $a$ such that : \[101a=6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]

1970 IMO Longlists, 25

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

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

2022 JHMT HS, 7

Tags: integration , calculus

Let $a$ be the unique real number $x$ satisfying $xe^x = 2$. Find a closed-form expression for \[ \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. \] You may express your answer in terms of elementary operations, functions, and constants.

2024 CMIMC Integration Bee, 6

Tags: integration , calculus

\[\int_1^2 \frac{\sqrt{1-\frac 1x}}{x^2-1}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2009 Moldova National Olympiad, 12.1

Tags: definite integral , cosinus , integration , calculus

Calculate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{cos(x)^7}{e^x+1} dx$.

2009 Today's Calculation Of Integral, 464

Tags: logarithm , calculus , integration , calculus computations

Evaluate $ \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx$.

Today's calculation of integrals, 858

Tags: perimeter , calculus computations , calculus , integration , geometry

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

1991 Arnold's Trivium, 50

Tags: calculus , integration

Calculate \[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx\]

2009 Today's Calculation Of Integral, 434

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

Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.

2013 Bogdan Stan, 3

Tags: indefinite integral , antiderivative , calculus , integration

$ \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx $ [i]Ion Nedelcu[/i] and [i]Lucian Tutescu[/i]

2009 Today's Calculation Of Integral, 468

Tags: calculus , function , calculus computations , integration

Evaluate $ \int_{\minus{}\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\{(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\plus{}(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

2003 China Western Mathematical Olympiad, 1

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

The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.

2013 Kosovo National Mathematical Olympiad, 2

Tags: calculus , algebra , polynomial , integration

Find all integer $n$ such that $n-5$ divide $n^2+n-27$.

2009 Olympic Revenge, 2

Tags: real analysis , integration , trigonometry

Prove that $\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}$

1998 VJIMC, Problem 4-M

Tags: riemann sum , calculus , inequalities , integration

Prove the inequality $$\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4$$for every integer $n\ge2$.

2012 Today's Calculation Of Integral, 819

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

For real numbers $a,\ b$ with $0\leq a\leq \pi,\ a<b$, let $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$ Determine the value of $a$ such that $\lim_{b\rightarrow \infty} I(a,\ b)=0.$

2010 Today's Calculation Of Integral, 654

Tags: function , limit , calculus , calculus computations , integration

A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$. Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$. [i]1997 Hokkaido University entrance exam/Science[/i]

2011 Math Prize For Girls Problems, 18

Tags: relatively prime , quadratic , calculus , integration , polynomial , algebra , number theory

The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?

2014 Indonesia MO Shortlist, N4

Tags: calculus , integration , number theory

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

2008 India National Olympiad, 4

Tags: calculus , combinatorics , integration , analytic geometry

All the points with integer coordinates in the $ xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $ (0,0)$ is red and the point $ (0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.

2013 Today's Calculation Of Integral, 877

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

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 $\int_0^{\frac{\pi}2} f(x)\ dx.$

2011 Today's Calculation Of Integral, 766

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

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

1964 Miklós Schweitzer, 6

Tags: function , real analysis , integration

Let $ y_1(x)$ be an arbitrary, continuous, positive function on $ [0,A]$, where $ A$ is an arbitrary positive number. Let \[ y_{n+1}=2 \int_0^x \sqrt{y_n(t)}dt \;(n=1,2,...)\ .\] Prove that the functions $ y_n(x)$ converge to the function $ y=x^2$ uniformly on $ [0,A]$.