Olympiad problems

2008 Moldova MO 11-12, 8

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

2007 Today's Calculation Of Integral, 201

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

Evaluate the following definite integral. \[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]

2007 Germany Team Selection Test, 1

Tags: integration , inequalities , algebra , sequence , summation , calculus

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

2014 BMT Spring, 9

Tags: calculus , integration , limit

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.

2004 District Olympiad, 4

Tags: integral , real analysis , calculus , integration

Let $ a,b\in (0,1) $ and a continuous function $ f:[0,1]\longrightarrow\mathbb{R} $ with the property that $$ \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt,\quad\forall x\in [0,1] . $$ [b]a)[/b] Show that if $ a+b<1, $ then $ f=0. $ [b]b)[/b] Show that if $ a+b=1, $ then $ f $ is constant.

2011 Today's Calculation Of Integral, 736

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

Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

1948 Putnam, A4

Tags: integration

Let $D$ be a plane region bounded by a circle of radius $r.$ Let $(x,y)$ be a point of $D$ and consider a circle of radius $\delta$ and center at $(x,y).$ Denote by $l(x,y)$ the length of that arc of the circle which is outside $D.$ Find $$\lim_{\delta \to 0} \frac{1}{\delta^{2}} \int_{D} l(x,y)\; dx\; dy.$$

2001 Italy TST, 4

Tags: induction , calculus , integration , combinatorics

We are given $2001$ balloons and a positive integer $k$. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most $k$ balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of $k$ such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.

2008 Grigore Moisil Intercounty, 3

Tags: function , calculus , integration , inequalities

Let $ f[0,\infty )\longrightarrow\mathbb{R} $ be a convex and differentiable function with $ f(0)=0. $ [b]a)[/b] Prove that $ \int_0^x f(t)dt\le \frac{x^2}{2}f'(x) , $ for any nonnegative $ x. $ [b]b)[/b] Determine $ f $ if the above inequality is actually an equality. [i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]

2016 Danube Mathematical Olympiad, 3

Tags: inequalities , calculus , integration

3. Let n > 1 be an integer and $a_1, a_2, . . . , a_n$ be positive reals with sum 1. a) Show that there exists a constant c ≥ 1/2 so that $\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$, where $a_0 = 0$. b) Show that ’the best’ value of c is at least $\frac{\pi}{4}$.

2014 India Regional Mathematical Olympiad, 2

Tags: calculus , integration , arithmetic sequence , algebra

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

2007 Today's Calculation Of Integral, 220

Tags: calculus , integration , inequalities , calculus computations

Prove that $ \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx$.

2010 Today's Calculation Of Integral, 543

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

Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$. (1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies. (2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.

2009 Putnam, B2

Tags: integration , function , calculus , algebra , polynomial , limit

A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$

2010 Today's Calculation Of Integral, 544

Tags: calculus , integration , function , calculus computations

(1) Evaluate $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx$. (2) If a linear function $ f(x)$ satifies $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)f(x)dx\equal{}3\sqrt{3}$, then we have $ f(x)\equal{}\boxed{\ A\ }(x\minus{}1)\plus{}\boxed{\ B\ }(x\plus{}1)$, thus we have $ f(x)\equal{}\boxed{\ C\ }$.

1975 Canada National Olympiad, 4

Tags: calculus , integration , ratio , floor function , quadratic , geometric sequence , algebra

For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

2013 Today's Calculation Of Integral, 881

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

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

2005 Today's Calculation Of Integral, 45

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

Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

2005 Today's Calculation Of Integral, 59

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]

Today's calculation of integrals, 899

Tags: calculus , integration , limit , calculus computations

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2006 China Girls Math Olympiad, 5

Tags: calculus , integration , analytic geometry , combinatorics

The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral points on segment $PQ$. Find the least value of the number of elements of $T$.

2010 Today's Calculation Of Integral, 590

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac{\pi}{8}} \frac{(\cos \theta \plus{}\sin \theta)^{\frac{3}{2}}\minus{}(\cos \theta \minus{}\sin \theta)^{\frac{3}{2}}}{\sqrt{\cos 2\theta}}\ d\theta$.

2005 Today's Calculation Of Integral, 38

Tags: calculus , integration , calculus computations , logarithm

Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis. If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.

2021 CMIMC Integration Bee, 11

Tags: calculus , integration

$$\int_0^\frac{\pi}{2}\frac{1}{4-3\cos^2(x)}\,dx$$ [i]Proposed by Connor Gordon[/i]

2007 Today's Calculation Of Integral, 182

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

Find the area of the domain of the system of inequality \[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]