Olympiad problems

2011 N.N. Mihăileanu Individual, 4

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.

2024 CMIMC Integration Bee, 13

Tags: integration , calculus

\[\int_0^{2\pi} \frac{1}{3+2 \sqrt{3} \cos x + \cos^2 x}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

2009 Stanford Mathematics Tournament, 5

Tags: algebra , calculus

Compute $\int_{0}^{\infty} t^5e^{-t}dt$

2011 Today's Calculation Of Integral, 748

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

Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

1969 German National Olympiad, 6

Tags: analysis , calculus , polynomial , algebra

Let $n$ be a positive integer, $h$ a real number and $f(x)$ a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial $$F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)$$ has no real zeros.

2013 Today's Calculation Of Integral, 893

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

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2003 District Olympiad, 2

Tags: function , calculus , sequence , integral , ratio , integration

Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as $$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$ [b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $ [b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.

2007 Harvard-MIT Mathematics Tournament, 8

Tags: algebra , polynomial , vieta , binomial theorem , calculus

Suppose that $\omega$ is a primitive $2007^{\text{th}}$ root of unity. Find $\left(2^{2007}-1\right)\displaystyle\sum_{j=1}^{2006}\dfrac{1}{2-\omega^j}$.

2010 Today's Calculation Of Integral, 669

Tags: function , integration , calculus , calculus computations

Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$

1998 IberoAmerican Olympiad For University Students, 1

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

The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$. Prove that there is a real number $c$ such that \[f(c)+g(c)\leq 2\]

2007 Today's Calculation Of Integral, 243

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

A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.

2009 Putnam, A2

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

Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions \begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*} Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$

2001 India Regional Mathematical Olympiad, 3

Tags: floor function , calculus , integration

Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]

2008 Brazil Team Selection Test, 3

Tags: inequalities , algebra , function , 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]

2010 Today's Calculation Of Integral, 595

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

Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$ 2009 Kumamoto University entrance exam/Medicine

1990 AIME Problems, 7

Tags: graphing lines , trigonometry , analytic geometry , slope , ratio , calculus , integration

A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.

2004 IMC, 5

Tags: integration , logarithm , inequalities , calculus

Prove that \[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]

2007 Romania National Olympiad, 2

Tags: real analysis , integration , calculus

Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function. a) Show that for any integer $n\geq 1$, there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$ holds for all $k=0,1,\ldots,n-1$. b) For each $n$, consider the $a_{i}$ above (that depend on $n$) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$. Show that the sequence $(b_{n})$ is convergent and compute it's limit.

2001 District Olympiad, 4

Tags: logarithm , real analysis , limit , integration , calculus

a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$. b)Let $a>0$. Prove that: \[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\] [i]***[/i]

2007 Harvard-MIT Mathematics Tournament, 7

Tags: calculus

Compute \[\sum_{n=1}^\infty \dfrac{1}{n\cdot(n+1)\cdot(n+1)!}.\]

2007 Princeton University Math Competition, 2

Tags: calculus , integration

Find the biggest non-integer $x$ such that $(x+2)^2 + (x+3)^3 + (x+4)^4 = 2$.

2009 Today's Calculation Of Integral, 509

Tags: algebra , calculus , integration , partial fractions , calculus computations , logarithm , trigonometry

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

2018 PUMaC Live Round, Calculus 1

Tags: calculus

Freddy the king of flavortext has an infinite chest of coins. For each number $p$ in the interval $[0, 1]$, Freddy has a coin that has probability $p$ of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coin again. If the probability that the coin comes up heads on this 11th flip is $\frac{p}{q}$ for two integers $p, q$, find $p + q$. Note: flavortext is made up

2010 Today's Calculation Of Integral, 614

Tags: calculus , integration , ratio , trigonometry , calculus computations

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2019 ISI Entrance Examination, 2

Tags: calculus

Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$ [b](a)[/b] Show that $f$ has exactly one point of discontinuity. [b](b)[/b] Evaluate $f$ at its point of discontinuity.