Olympiad problems

1996 Romania National Olympiad, 4

Tags: limit , function , real analysis , calculus , monotone functions , integral

Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.

2012 VJIMC, Problem 1

Tags: function , calculus

Let $f:[0,1]\to[0,1]$ be a differentiable function such that $|f'(x)|\ne1$ for all $x\in[0,1]$. Prove that there exist unique $\alpha,\beta\in[0,1]$ such that $f(\alpha)=\alpha$ and $f(\beta)=1-\beta$.

2016 District Olympiad, 4

Tags: function , calculus , integration

Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence $$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$ is convergent and calculate its limit.

2009 Today's Calculation Of Integral, 443

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_1^{e^2} \frac{(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)\plus{}(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)}{\sqrt{x}}\ dx.$

2011 BMO TST, 2

Tags: perimeter , calculus , integration , inradius , trigonometry , algorithm , geometry

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

1991 Arnold's Trivium, 60

Tags: calculus , integration

Is there a solution of the Cauchy problem \[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\] in a neighbourhood of the point $(x_0,0)$ of the $x$-axis? Is it unique?

2019 Jozsef Wildt International Math Competition, W. 12

Tags: integration , inequalities , calculus

If $0 < a < b$ then: $$\frac{\int \limits^{\frac{a+b}{2}}_{a}\left(\tan^{-1}t\right)dt}{\int \limits_{a}^{b}\left(\tan^{-1}t\right)dt}<\frac{1}{2}$$

2006 China Girls Math Olympiad, 5

Tags: calculus , integration , analytic geometry , combinatorics

The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral points on segment $PQ$. Find the least value of the number of elements of $T$.

2009 Today's Calculation Of Integral, 435

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_{\frac{\pi}{4}}^{\frac {\pi}{2}} \frac {1}{(\sin x \plus{} \cos x \plus{} 2\sqrt {\sin x\cos x})\sqrt {\sin x\cos x}}dx$.

2005 Taiwan National Olympiad, 3

Tags: quadratic , calculus , integration , algebra

If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.

1989 Balkan MO, 2

Tags: polynomial , gauss , calculus , derivative , combinatorial geometry , absolute value , algebra

Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.

1986 IMO Shortlist, 7

Tags: algebra , polynomial , system of equations , calculus

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

2010 German National Olympiad, 5

Tags: algebra , polynomial , calculus , derivative , combinatorics

The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$

2009 Today's Calculation Of Integral, 428

Tags: calculus , integration , algebra , polynomial , calculus computations

Let $ f(x)$ be a polynomial and $ C$ be a real number. Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.

2007 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , limit

Compute: \[\lim_{x\to 0}\text{ }\dfrac{x^2}{1-\cos(x)}\]

2009 Today's Calculation Of Integral, 414

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{2(2\plus{}\sqrt{3})} \frac{16}{(x^2\plus{}4)^2}\ dx$.

1989 IMO Longlists, 84

Tags: maximization , minimization , optimization , algebra , calculus

Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which \[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\] where $ x_0, x_1, \ldots , x_n$ are real variables.

2019 Jozsef Wildt International Math Competition, W. 52

Tags: integration , calculus , periodic function

Let $f : \mathbb{R} \to \mathbb{R}$ a periodic and continue function with period $T$ and $F : \mathbb{R} \to \mathbb{R}$ antiderivative of $f$. Then $$\int \limits_0^T \left[F(nx)-F(x)-f(x)\frac{(n-1)T}{2}\right]dx=0$$

PEN H Problems, 46

Tags: diophantine equation , calculus , integration

Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.

2010 Today's Calculation Of Integral, 522

Tags: calculus , integration , limit , function , geometry , derivative , logarithm

Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.

2004 Nicolae Coculescu, 1

Tags: limit , series , harmonic series , calculus , sequence

Calculate $ \lim_{n\to\infty } \left( e^{1+1/2+1/3+\cdots +1/n+1/(n+1)} -e^{1+1/2+1/3+\cdots +1/n} \right) . $

2008 Alexandru Myller, 1

Tags: integration , squeeze theorem , calculus

$ \lim_{n\to\infty} n2^n\int_1^n \frac{dx}{\left( 1+x^2\right)^n} $ [i][i]Bogdan Enescu[/i][/i]

PEN S Problems, 19

Tags: floor function , calculus , integration , induction

Determine all pairs $(a, b)$ of real numbers such that $a\lfloor bn\rfloor =b\lfloor an\rfloor$ for all positive integer $n$.

2012 Today's Calculation Of Integral, 857

Tags: calculus , integration , limit , trigonometry , geometry , geometric transformation , rotation

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2013 Stanford Mathematics Tournament, 6

Tags: calculus , integration , trigonometry , geometric series

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