Olympiad problems

Today's calculation of integrals, 890

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

1999 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , derivative , trigonometry

Evaluate $\dfrac{d}{dx}\left(\sin x - \dfrac{4}{3}\sin^3 x\right)$ when $x=15$.

1977 IMO Longlists, 52

Tags: calculus , geometry

Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R.$ Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from $P$ to the center of the circle is $kR.$

1950 AMC 12/AHSME, 17

Tags: calculus , derivative , quadratic , algebra , system of equations , arithmetic series

The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is: \[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline \end{tabular}\] $\textbf{(A)}\ y=100-10x \qquad \textbf{(B)}\ y=100-5x^2 \qquad \textbf{(C)}\ y=100-5x-5x^2 \qquad\\ \textbf{(D)}\ y=20-x-x^2 \qquad \textbf{(E)}\ \text{None of these}$

2006 Czech-Polish-Slovak Match, 5

Tags: calculus , integration , algebra

Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.

2011 Singapore MO Open, 4

Tags: algebra , polynomial , calculus , number theory

Find all polynomials $P(x)$ with real coefficients such that \[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]

2007 Romania Team Selection Test, 1

Tags: induction , calculus , derivative , logarithm , inequalities

If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\] then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.

2010 N.N. Mihăileanu Individual, 2

Tags: function , calculus , integration , real analysis

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that there exists a continuous and bounded function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that verifies the equality $$ f(x)=\int_0^x f(\xi )g(\xi )d\xi , $$ for any real number $ x. $ Prove that $ f=0. $ [i]Nelu Chichirim[/i]

2008 Brazil Team Selection Test, 3

Tags: inequalities , function , algebra , calculus

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2007 Today's Calculation Of Integral, 234

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

For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$. Evaluate $ \int_0^{100\pi } f(x)\ dx.$

2025 ISI Entrance UGB, 1

Tags: calculus , fixed point

Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.

2003 Vietnam National Olympiad, 1

Tags: function , trigonometry , calculus , algebra

Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$

Today's calculation of integrals, 882

Tags: calculus , integration , limit , calculus computations , riemann sum , logarithm

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2000 USA Team Selection Test, 4

Tags: induction , function , calculus , integration

Let $n$ be a positive integer. Prove that \[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]

1998 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , analytic geometry , graphing lines , slope , derivative

Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

2009 Today's Calculation Of Integral, 462

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

Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.

2013 District Olympiad, 4

Tags: function , limit , calculus , calculus computations

Let$f:\mathbb{R}\to \mathbb{R}$be a monotone function. a) Prove that$f$ have side limits in each point ${{x}_{0}}\in \mathbb{R}$. b) We define the function $g:\mathbb{R}\to \mathbb{R}$, $g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)$( $g\left( x \right)$ with limit at at left in $x$). Prove that if the $g$ function is continuous, than the function $f$ is continuous.

2010 Today's Calculation Of Integral, 551

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

In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.

2010 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , algebra , polynomial , vieta

Let $p$ be a monic cubic polynomial such that $p(0)=1$ and such that all the zeroes of $p^\prime (x)$ are also zeroes of $p(x)$. Find $p$. Note: monic means that the leading coefficient is $1$.

2014 Dutch IMO TST, 5

Tags: polynomial , calculus , integration , ratio , arithmetic sequence , algebra

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$