Olympiad problems

2010 Today's Calculation Of Integral, 554

Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.

2010 SEEMOUS, Problem 1

Tags: function , convergence , integration , calculus , sequence

Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by $$f_n(x)=\int^x_0f_{n-1}(t)dt$$ for all integers $n\ge1$. a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$. b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.

2010 Today's Calculation Of Integral, 603

Tags: calculus , integration , derivative , calculus computations

Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$. Please solve the problem without using partial differentiation for those who don't learn it. 1961 Waseda University entrance exam/Science and Technology

2000 Moldova National Olympiad, Problem 6

Tags: calculus , integration

Show that there is a positive number $p$ such that $\int^\pi_0x^p\sin xdx=\sqrt[10]{2000}$.

2010 Contests, 523

Tags: calculus , integration , calculus computations , logarithm

Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2008 Putnam, B5

Tags: function , number theory , greatest common divisor , calculus , derivative , integration

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)

2010 Purple Comet Problems, 12

Tags: geometry , calculus , integration

A good approximation of $\pi$ is $3.14.$ Find the least positive integer $d$ such that if the area of a circle with diameter $d$ is calculated using the approximation $3.14,$ the error will exceed $1.$

2010 Today's Calculation Of Integral, 558

Tags: calculus , integration , quadratic , function , inequalities , algebra , calculus computations

For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$. Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$

1940 Putnam, A1

Tags: algebra , polynomial , calculus , integration

Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.

2010 Today's Calculation Of Integral, 564

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

In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$. (1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$. Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal. (2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.

MIPT Undergraduate Contest 2019, 2.2

Tags: inequalities , probability and stats , expectation , calculus , integration , probability , function

Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?

2005 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , analytic geometry , graphing lines , slope

Let $ f(x) = x^3 + ax + b $, with $ a \ne b $, and suppose the tangent lines to the graph of $f$ at $x=a$ and $x=b$ are parallel. Find $f(1)$.

2011 Today's Calculation Of Integral, 757

Tags: calculus , integration , calculus computations , logarithm

Evaluate \[\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.\]

1997 Romania National Olympiad, 4

Tags: calculus , integration , function , real analysis

Suppose that $(f_n)_{n\in N}$ be the sequence from all functions $f_n:[0,1]\rightarrow \mathbb{R^+}$ s.t. $f_0$ be the continuous function and $\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt$. Prove that for every $x\in [0,1]$ the sequence of $(f_n(x))_{n\in N}$ be the convergent sequence and calculate the limitation.

2014 India National Olympiad, 4

Tags: quadratic , polynomial , calculus , integration , probability , analytic geometry , algebra

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

2011 Iran MO (3rd Round), 2

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

Let $n$ and $k$ be two natural numbers such that $k$ is even and for each prime $p$ if $p|n$ then $p-1|k$. let $\{a_1,....,a_{\phi(n)}\}$ be all the numbers coprime to $n$. What's the remainder of the number $a_1^k+.....+a_{\phi(n)}^k$ when it's divided by $n$? [i]proposed by Yahya Motevassel[/i]

2024 CMIMC Integration Bee, 3

Tags: calculus , integration

\[\int_0^1 \frac{\log(x)}{\sqrt x}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

Today's calculation of integrals, 852

Tags: calculus , integration , trigonometry , polynomial , absolute value , calculus computations , algebra

Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$

2019 Jozsef Wildt International Math Competition, W. 53

Tags: limit , summation , calculus

Compute $$\lim \limits_{n \to \infty}\frac{1}{n}\sum \limits_{k=1}^n\frac{\sqrt[n+k+1]{n+1}-\sqrt[n+k]{n}}{\sqrt[n+k]{n+1}-\sqrt[n+k]{n}}$$

1972 Miklós Schweitzer, 3

Tags: advanced fields , limit , calculus , integration

Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\] [i]G. Halasz[/i]

2012 Today's Calculation Of Integral, 832

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

Find the limit \[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]

2007 Today's Calculation Of Integral, 181

Tags: calculus , integration , trigonometry , function , derivative , analytic geometry , logarithm

For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$

2019 Korea USCM, 2

Tags: volume , calculus , linear algebra

Matrices $A$, $B$ are given as follows. \[A=\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 12\end{pmatrix}\] Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$.

2007 Today's Calculation Of Integral, 194

Tags: calculus , integration , calculus computations

Evaluate \[\sum_{n=0}^{2006}\int_{0}^{1}\frac{dx}{2(x+n+1)\sqrt{(x+n)(x+n+1)}}\]

1998 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , integration , function , ratio , geometric series , logarithm

Let $f(x)=1+\dfrac{x}{2}+\dfrac{x^2}{4}+\dfrac{x^3}{8}+\cdots,$ for $-1\leq x \leq 1$. Find $\sqrt{e^{\int\limits_0^1 f(x)dx}}$.