Olympiad problems

2009 Today's Calculation Of Integral, 426

Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2. When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.

2004 IMC, 5

Tags: integration , inequalities , calculus , logarithm

Prove that \[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]

1979 Spain Mathematical Olympiad, 5

Tags: trigonometry , integration , integral , calculus

Calculate the definite integral $$\int_2^4 \sin ((x-3)^3) dx$$

Oliforum Contest II 2009, 3

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

Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$. [i](Paolo Leonetti)[/i]

Today's calculation of integrals, 877

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

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

2012 Centers of Excellency of Suceava, 4

Tags: calculus , integration , real analysis , sequence of integrals , definite integral

Let be the sequence $ \left( J_n \right)_{n\ge 1} , $ where $ J_n=\int_{(1+n)^2}^{1+(1+n)^2} \sqrt{\frac{x-1-n-n^2}{x-1}} dx. $ [b]a)[/b] Study its monotony. [b]b)[/b] Calculate $ \lim_{n\to\infty } J_n\sqrt{n} . $ [i]Ion Bursuc[/i]

2012 Today's Calculation Of Integral, 803

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

Answer the following questions: (1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$ (2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$

2009 Today's Calculation Of Integral, 503

Tags: calculus , integration , inequalities , geometry , trapezoid , analytic geometry , graphing lines

Prove the following inequality. \[ \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}\]

2005 Today's Calculation Of Integral, 14

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

Calculate the following indefinite integrals. [1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$ [2] $\int x\log_{10} x dx$ [3] $\int \frac{x}{\sqrt{2x-1}}dx$ [4] $\int (x^2+1)\ln x dx$ [5] $\int e^x\cos x dx$

1956 AMC 12/AHSME, 25

Tags: calculus , integration

The sum of all numbers of the form $ 2k \plus{} 1$, where $ k$ takes on integral values from $ 1$ to $ n$ is: $ \textbf{(A)}\ n^2 \qquad\textbf{(B)}\ n(n \plus{} 1) \qquad\textbf{(C)}\ n(n \plus{} 2) \qquad\textbf{(D)}\ (n \plus{} 1)^2 \qquad\textbf{(E)}\ (n \plus{} 1)(n \plus{} 2)$

2012 VJIMC, Problem 1

Tags: real analysis , calculus , integration

Let $f:[1,\infty)\to(0,\infty)$ be a non-increasing function such that $$\limsup_{n\to\infty}\frac{f(2^{n+1})}{f(2^n)}<\frac12.$$Prove that $$\int^\infty_1f(x)\text dx<\infty.$$

2012 Today's Calculation Of Integral, 836

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\pi} e^{\sin x}\cos ^ 2(\sin x )\cos x\ dx$.

2009 Today's Calculation Of Integral, 488

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

For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality. $ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$

2025 SEEMOUS, P2

Tags: calculus , integration , taylor series

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

1960 AMC 12/AHSME, 28

Tags: calculus , integration

The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has: $ \textbf{(A)}\ \text{infinitely many integral roots} \qquad\textbf{(B)}\ \text{no root} \qquad\textbf{(C)}\ \text{one integral root}\qquad$ $\textbf{(D)}\ \text{two equal integral roots} \qquad\textbf{(E)}\ \text{two equal non-integral roots} $

2013 Today's Calculation Of Integral, 872

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

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

1952 Miklós Schweitzer, 10

Tags: trigonometry , function , integration , real analysis

Let $ n$ be a positive integer. Prove that, for $ 0<x<\frac{\pi}{n\plus{}1}$, $ \sin{x}\minus{}\frac{\sin{2x}}{2}\plus{}\cdots\plus{}(\minus{}1)^{n\plus{}1}\frac{\sin{nx}}{n}\minus{}\frac{x}{2}$ is positive if $ n$ is odd and negative if $ n$ is even.

2009 AIME Problems, 11

Tags: geometry , analytic geometry , vector , calculus , integration , absolute value

Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer.

2011 Putnam, A3

Tags: limit , integration , trigonometry , calculus , ratio , inequalities

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

2010 Today's Calculation Of Integral, 655

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

Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.

2019 Jozsef Wildt International Math Competition, W. 48

Tags: integration , inequalities , convex function , calculus , function

Let $f : (0,+\infty) \to \mathbb{R}$ a convex function and $\alpha, \beta, \gamma > 0$. Then $$\frac{1}{6\alpha}\int \limits_0^{6\alpha}f(x)dx\ +\ \frac{1}{6\beta}\int \limits_0^{6\beta}f(x)dx\ +\ \frac{1}{6\gamma}\int \limits_0^{6\gamma}f(x)dx$$ $$\geq \frac{1}{3\alpha +2\beta +\gamma}\int \limits_0^{3\alpha +2\beta +\gamma}f(x)dx\ +\ \frac{1}{\alpha +3\beta +2\gamma}\int \limits_0^{\alpha +3\beta +2\gamma}f(x)dx\ $$ $$+\ \frac{1}{2\alpha +\beta +3\gamma}\int \limits_0^{2\alpha +\beta +3\gamma}f(x)dx$$

2022 CMIMC Integration Bee, 7

Tags: calculus , integration

\[\int_{-1}^1 \sqrt{\frac{1+x}{1-x}}+\sqrt{\frac{1-x}{1+x}}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2005 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration , function

Let $ f : \mathbf{R} \to \mathbf{R} $ be a continuous function with $ \displaystyle\int_{0}^{1} f(x) f'(x) \, \mathrm{d}x = 0 $ and $ \displaystyle\int_{0}^{1} f(x)^2 f'(x) \, \mathrm{d}x = 18 $. What is $ \displaystyle\int_{0}^{1} f(x)^4 f'(x) \, \mathrm{d} x $?

1980 Swedish Mathematical Competition, 4

Tags: integration , analysis , inequalities

The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that \[ \int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx \]

2010 Today's Calculation Of Integral, 583

Tags: calculus , integration , calculus computations , geometry

Find the values of $ k$ such that the areas of the three parts bounded by the graph of $ y\equal{}\minus{}x^4\plus{}2x^2$ and the line $ y\equal{}k$ are all equal.