Olympiad problems

2012 SEEMOUS, Problem 4

Tags: calculus , integration

a) Compute $$\lim_{n\to\infty}n\int^1_0\left(\frac{1-x}{1+x}\right)^ndx.$$ b) Let $k\ge1$ be an integer. Compute $$\lim_{n\to\infty}n^{k+1}\int^1_0\left(\frac{1-x}{1+x}\right)^nx^kdx.$$

2015 Kyoto University Entry Examination, 5

Tags: calculus , integration

5. Let $a,b,c,d,e$ be positive rational numbers. Consider integral expressions $f(x)=ax^2+bx+c$ $g(x)=dx+e$ Put $\frac{f(n)}{g(n)}$ an integer for all positive integers $n$. Then, show that $f(x)$ is dividible by $g(x)$.

2009 Greece National Olympiad, 3

Tags: calculus , inequalities

Let $ x,y,z$ be nonnegative real numbers such that $ x \plus{} y \plus{} z \equal{} 2$. Prove that $ x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1$. When does the equality occur?

2007 AIME Problems, 7

Tags: integration , geometry , floor function , 3d geometry , calculus

Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, n_{70}$ such that $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$ Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$

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]

2011 Tokyo Instutute Of Technology Entrance Examination, 2

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

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

2004 Poland - First Round, 3

Tags: inequalities , geometry , vector , derivative , calculus , trigonometry

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$

2011 India IMO Training Camp, 2

Tags: algebra , integration , calculus

Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$ such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.

2009 Vietnam Team Selection Test, 1

Tags: function , quadratic , logarithm , inequalities , calculus

Let $ a,b,c$ be positive numbers.Find $ k$ such that: $ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$

2013 Stanford Mathematics Tournament, 6

Tags: trigonometry , geometric series , integration , calculus

Compute $\sum_{k=0}^{\infty}\int_{0}^{\frac{\pi}{3}}\sin^{2k} x \, dx$.

Today's calculation of integrals, 889

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

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

2002 Iran Team Selection Test, 9

Tags: function , integration , calculus , inequalities , number theory

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

1989 IMO Longlists, 61

Tags: function , inequalities , calculus , derivative

Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds: \[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]

2011 Romania National Olympiad, 2

Tags: real analysis , calculus , function , derivative , abstract algebra , inequalities

[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]

2009 Today's Calculation Of Integral, 442

Tags: integration , trigonometry , calculus , calculus computations

Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$

2005 Alexandru Myller, 2

Tags: real analysis , calculus , integration , function

Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$. [i]Mihai Piticari[/i]

2007 Today's Calculation Of Integral, 173

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

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

1980 Vietnam National Olympiad, 2

Tags: algebra , polynomial , integration , calculus

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

2010 Today's Calculation Of Integral, 647

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

Evaluate \[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\] Own

Today's calculation of integrals, 878

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

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2019 Romania National Olympiad, 3

Tags: real analysis , differentiability , calculus

$\textbf{a)}$ Prove that there exists a differentiable function $f:(0, \infty) \to (0, \infty)$ such that $f(f'(x)) = x, \: \forall x>0.$ $\textbf{b)}$ Prove that there is no differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f'(x)) = x, \: \forall x \in \mathbb{R}.$

2017 VJIMC, 4

Tags: integration , real analysis , calculus

Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that \[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]

2013 Today's Calculation Of Integral, 883

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

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

1994 IMC, 3

Tags: real analysis , derivative , calculus , function

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that $$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$ there is a number $c$ in the open interval $(a,b)$ for which $$f^{(n+1)}(c)=f(c)$$

2006 Putnam, B5

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

For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$