Olympiad problems

2007 Today's Calculation Of Integral, 256

Find the value of $ a$ for which $ \int_0^{\pi} \{ax(\pi ^ 2 \minus{} x^2) \minus{} \sin x\}^2dx$ is minimized.

2012 Today's Calculation Of Integral, 816

Tags: calculus , integration , geometric transformation , rotation , combinatorial geometry , calculus computations , geometry

Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.

2009 Today's Calculation Of Integral, 504

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

Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.

1960 AMC 12/AHSME, 23

Tags: calculus , integration

The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. The volume $V = \pi R^2H$ is to be increased by the same fixed positive amount when $R$ is increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by: $ \textbf{(A)}\ \text{no real value of} \text{ } x\qquad$ $\textbf{(B)}\ \text{one integral value of} \text{ } x\qquad$ $\textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad$ $\textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad$ $\textbf{(E)}\ \text{two real values of} \text{ } x $

Today's calculation of integrals, 870

Tags: calculus , integration , ellipse , hyperbola , geometry , inequalities , conic

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

1983 Miklós Schweitzer, 7

Tags: function , search , induction , integration , calculus , real analysis

Prove that if the function $ f : \mathbb{R}^2 \rightarrow [0,1]$ is continuous and its average on every circle of radius $ 1$ equals the function value at the center of the circle, then $ f$ is constant. [i]V. Totik[/i]

2009 Today's Calculation Of Integral, 489

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

Find the following limit. $ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.

1981 Miklós Schweitzer, 6

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

Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: \[ 1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .\] [i]Zs. Pales[/i]

2023 Romania National Olympiad, 4

Tags: real analysis , integration , monotone functions , differentiable function

Let $f:[0,1] \rightarrow \mathbb{R}$ a non-decreasing function, $f \in C^1,$ for which $f(0) = 0.$ Let $g:[0,1] \rightarrow \mathbb{R}$ a function defined by \[ g(x) = f(x) + (x - 1) f'(x), \forall x \in [0,1]. \] a) Show that \[ \int_{0}^{1} g(x) \text{dx} = 0. \] b) Prove that for all functions $\phi :[0,1] \rightarrow [0,1],$ convex and differentiable with $\phi(0) = 0$ and $\phi(1) = 1,$ the inequality holds \[ \int_{0}^{1} g( \phi(t)) \text{dt} \leq 0. \]

1955 AMC 12/AHSME, 30

Tags: calculus , integration

Each of the equations $ 3x^2\minus{}2\equal{}25$, $ (2x\minus{}1)^2\equal{}(x\minus{}1)^2$, $ \sqrt{x^2\minus{}7}\equal{}\sqrt{x\minus{}1}$ has: $ \textbf{(A)}\ \text{two integral roots} \qquad \textbf{(B)}\ \text{no root greater than 3} \qquad \textbf{(C)}\ \text{no root zero} \\ \textbf{(D)}\ \text{only one root} \qquad \textbf{(E)}\ \text{one negative root and one positive root}$

2014 BMT Spring, 9

Tags: limit , calculus , integration

Find $\alpha$ such that $$\lim_{x\to0^+}x^\alpha I(x)=a\enspace\text{given}\enspace I(x)=\int^\infty_0\sqrt{1+t}\cdot e^{-xt}dt$$ where $a$ is a nonzero real number.

2005 Today's Calculation Of Integral, 84

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

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

2015 VTRMC, Problem 5

Tags: calculus , integration

Evaluate $\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx$ (where $0\le\operatorname{arctan}(x)<\frac\pi2$ for $0\le x<\infty$).

2013 Today's Calculation Of Integral, 878

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

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2013 Today's Calculation Of Integral, 879

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

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

1989 Putnam, A2

Tags: integration

Evaluate $\int^{a}_{0}{\int^{b}_{0}{e^{max(b^{2}x^{2},a^{2}y^{2})}dy dx}}$

1987 Swedish Mathematical Competition, 4

Tags: integration , analysis , algebra

A differentiable function $f$ with $f(0) = f(1) = 0$ is defined on the interval $[0,1]$. Prove that there exists a point $y \in [0,1]$ such that $| f' (y)| = 4 \int _0^1 | f(x)|dx$.

2001 Vietnam National Olympiad, 3

Tags: trigonometry , integration , calculus , calculus computations

For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.

2021 CMIMC Integration Bee, 7

Tags: calculus , integration

$$\int_0^\infty \frac{1}{(x^2+4)^{5/2}}\,dx$$ [i]Proposed by Connor Gordon[/i]

2012 Hanoi Open Mathematics Competitions, 7

Tags: calculus , integration

[b]Q7.[/b] Find all integers $n$ such that $60+2n-n^2$ is a perfect square.

2005 Today's Calculation Of Integral, 6

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

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

1991 Arnold's Trivium, 74

Tags: integration , trigonometry

Sketch the graph of $u(x, 1)$, if $0 \le x\le1$, \[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]

2004 Olympic Revenge, 5

Tags: calculus , integration , induction , number theory

$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n. prove that $a_n$ is one integer for all positive integers n.

Today's calculation of integrals, 876

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

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

2012 Today's Calculation Of Integral, 843

Tags: calculus , integration , function , inequalities , special factorizations , calculus computations

Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.