Olympiad problems

2010 Contests, 524

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2005 Moldova Team Selection Test, 3

Tags: calculus , integration , function , number theory

\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\] where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.

2009 Today's Calculation Of Integral, 421

Tags: calculus , integration , limit , calculus computations

Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions. (1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$. (2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal. (3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.

2013 Today's Calculation Of Integral, 896

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

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

2001 USA Team Selection Test, 2

Tags: integration , function , calculus , algebra , binomial theorem , combinatorics

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

2004 Vietnam National Olympiad, 3

Tags: calculus , integration , quadratic , modular arithmetic , number theory

Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.

2013 Today's Calculation Of Integral, 873

Tags: calculus , integration , ellipse , analytic geometry , graphing lines , conic , geometry

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

2012 Today's Calculation Of Integral, 846

Tags: calculus , integration , limit , function , absolute value , calculus computations , logarithm

For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.

2019 Jozsef Wildt International Math Competition, W. 31

Tags: integration , inequalities , calculus

Let $a, b \in \Gamma$, $a < b$ and the differentiable function $f : [a, b] \to \Gamma$, such that $f (a) = a$ and $f (b) = b$. Prove that $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$

1982 Putnam, A3

Tags: integration , calculus

Evaluate $$\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx.$$

2012 IMC, 4

Tags: function , limit , integration

Let $f:\;\mathbb{R}\to\mathbb{R}$ be a continuously differentiable function that satisfies $f'(t)>f(f(t))$ for all $t\in\mathbb{R}$. Prove that $f(f(f(t)))\le0$ for all $t\ge0$. [i]Proposed by Tomáš Bárta, Charles University, Prague.[/i]

2025 Romania EGMO TST, P1

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]

2006 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , integration , logarithm

Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.

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, 577

Tags: calculus , integration , inequalities , calculus computations

Prove the following inequality for any integer $ N\geq 4$. \[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]

1995 National High School Mathematics League, 10

Tags: calculus , integration

The number of integral points satisfy $\begin{cases} y\leq 3x\\ y\geq \frac{x}{3}\\ x+y\geq100 \end{cases}$ on the coordinate plane is________.

2005 Today's Calculation Of Integral, 40

Tags: calculus , integration , calculus computations

Evaluate \[\int_0^1 x^{2005}e^{-x^2}dx\]

Today's calculation of integrals, 861

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

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

2011 Today's Calculation Of Integral, 720

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

1980 Vietnam National Olympiad, 2

Tags: polynomial , calculus , integration , algebra

Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?

2007 Princeton University Math Competition, 5

Tags: integration

Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $.

2011 Today's Calculation Of Integral, 705

Tags: calculus , integration , trigonometry , calculus computations

The parametric equations of a curve are given by $x = 2(1+\cos t)\cos t,\ y =2(1+\cos t)\sin t\ (0\leq t\leq 2\pi)$. (1) Find the maximum and minimum values of $x$. (2) Find the volume of the solid enclosed by the figure of revolution about the $x$-axis.

2010 Today's Calculation Of Integral, 615

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

For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

2013 Waseda University Entrance Examination, 4

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

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

2011 N.N. Mihăileanu Individual, 4

Tags: calculus , integration , real analysis

Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^1 \frac{x^n}{\sqrt{x^{2n} +1}} dx . $ [b]a)[/b] Show that $ \left( I_n \right)_{n\ge 1} $ converges to $ 0. $ [b]b)[/b] Calculate $ \lim_{m\to\infty } m\cdot I_m. $ [b]c)[/b] Prove that the sequence $ \left( n\left( -n\cdot I_n +\lim_{m\to\infty } m\cdot I_m \right) \right)_{n\ge 1} $ is convergent.