Olympiad problems

2011 VTRMC, Problem 1

Tags: calculus , integration

Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$

PEN M Problems, 13

Tags: function , induction , pigeonhole principle , calculus , integration , recursive sequences

The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.

2008 All-Russian Olympiad, 5

Tags: calculus , integration , inequalities , number theory

The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?

1994 China Team Selection Test, 2

Tags: algebra , polynomial , calculus , integration , gauss , number theory , prime number

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1969 Putnam, A4

Tags: integration , summation

Show that $$ \int_{0}^{1} x^{x} \, dx = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^n }.$$

2011 Federal Competition For Advanced Students, Part 1, 1

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

Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

2002 VJIMC, Problem 3

Tags: calculus , integration , inequalities

Let $E$ be the set of all continuous functions $u:[0,1]\to\mathbb R$ satisfying $$u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].$$Let $\varphi:E\to\mathbb R$ be defined by $$\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.$$Prove that $\varphi$ has a maximum value and find it.

1979 Miklós Schweitzer, 8

Tags: function , integration , real analysis

Let $ K_n(n=1,2,\ldots)$ be periodical continuous functions of period $ 2 \pi$, and write \[ k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .\] Prove that the following statements are equivalent: (i) $ \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $ f \in \mathcal{L}_1[0,2 \pi]$. (ii) $ k_n(f;0) \rightarrow f(0)$ for all continuous, $ 2 \pi$-periodic functions $ f$. [i]V. Totik[/i]

2010 Today's Calculation Of Integral, 538

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: calculus , algebra , integration , integral

Find some antiderivative of the function $y = 1/x^3$, the graph of which has exactly three common points with the graph of the function $y = |x|$.

2009 Today's Calculation Of Integral, 472

Tags: calculus , integration , parabola , geometry , analytic geometry , calculus computations , conic

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

2005 Today's Calculation Of Integral, 85

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

Evaluate \[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\] where $ [x] $ is the integer equal to $ x $ or less than $ x $.

1987 AIME Problems, 3

Tags: calculus , integration , geometry , 3d geometry , tetrahedron

By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?

2002 Iran Team Selection Test, 9

Tags: function , calculus , integration , 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 $\pi(n)|n$?

2011 Today's Calculation Of Integral, 713

Tags: calculus , integration , limit , calculus computations

If a positive sequence $\{a_n\}_{n\geq 1}$ satisfies $\int_0^{a_n} x^{n}\ dx=2$, then find $\lim_{n\to\infty} a_n.$

2009 Today's Calculation Of Integral, 502

Tags: calculus , integration , limit , calculus computations

(1) For $ 0 < x < 1$, prove that $ (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}.$ (2) Find $ \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}$.

2024 CMIMC Integration Bee, 9

Tags: calculus , integration

\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2009 Today's Calculation Of Integral, 416

Tags: calculus , integration , trigonometry , calculus computations

Answer the following questions. (1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$. (2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$. Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$. (3) Find $ f_n(x)$ for all $ n$. (4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.

2010 Today's Calculation Of Integral, 523

Tags: calculus , integration , calculus computations , logarithm

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

2006 Turkey Junior National Olympiad, 2

Tags: calculus , integration

Find all integer triples $(x,y,z)$ such that \[ \begin{array}{rcl} x-yz &=& 11 \\ xz+y &=& 13. \end{array}\]

2011 Today's Calculation Of Integral, 745

Tags: calculus , integration , trigonometry , calculus computations

When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$

2009 Today's Calculation Of Integral, 445

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

1997 IberoAmerican, 1

Tags: floor function , calculus , integration , ratio , number theory

Let $r\geq1$ be areal number that holds with the property that for each pair of positive integer numbers $m$ and $n$, with $n$ a multiple of $m$, it is true that $\lfloor{nr}\rfloor$ is multiple of $\lfloor{mr}\rfloor$. Show that $r$ has to be an integer number. [b]Note: [/b][i]If $x$ is a real number, $\lfloor{x}\rfloor$ is the greatest integer lower than or equal to $x$}.[/i]

2012 Today's Calculation Of Integral, 771

Tags: calculus , integration , parabola , geometry , analytic geometry , calculus computations , conic

(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis. (2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.

2019 Jozsef Wildt International Math Competition, W. 27

Tags: integration , functional equation , function

Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$