Olympiad problems

1986 Traian Lălescu, 2.3

Tags: function , inequalities , calculus , derivative , ftc , rolle theorem , integral

Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

2007 Hungary-Israel Binational, 3

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

Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.

1998 Harvard-MIT Mathematics Tournament, 4

Tags: derivative , polynomial , calculus , function , algebra , trigonometry

Find the range of $ f(A)=\frac{\sin A(3\cos^{2}A+\cos^{4}A+3\sin^{2}A+\sin^{2}A\cos^{2}A)}{\tan A (\sec A-\sin A\tan A)} $ if $A\neq \dfrac{n\pi}{2}$.

1998 IberoAmerican Olympiad For University Students, 6

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

Take the following differential equation: \[3(3+x^2)\frac{dx}{dt}=2(1+x^2)^2e^{-t^2}\] If $x(0)\leq 1$, prove that there exists $M>0$ such that $|x(t)|<M$ for all $t\geq 0$.

1946 Putnam, A4

Tags: function , derivative , calculus

Let $g(x)$ be a function that has a continuous first derivative $g'(x)$. Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.

1991 Arnold's Trivium, 4

Tags: function , calculus , derivative

Calculate the $100$th derivative of the function \[\frac{x^2+1}{x^3-x}\]

2009 ISI B.Stat Entrance Exam, 2

Tags: trigonometry , derivative , calculus , function , integration , analytic geometry , graphing lines

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that \[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]

2005 Today's Calculation Of Integral, 87

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

Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized. \[ \int_0^{\pi} |\sin x-ax|\ dx \]

2011 South East Mathematical Olympiad, 1

Tags: algebra , function , derivative , calculus

If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$.

1975 Miklós Schweitzer, 7

Tags: calculus , real analysis , analytic geometry , function , derivative , graphing lines , slope

Let $ a<a'<b<b'$ be real numbers and let the real function $ f$ be continuous on the interval $ [a,b']$ and differentiable in its interior. Prove that there exist $ c \in (a,b), c'\in (a',b')$ such that \[ f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a),\] \[ f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'),\] and $ c<c'$. [i]B. Szokefalvi Nagy[/i]

1978 AMC 12/AHSME, 11

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

If $r$ is positive and the line whose equation is $x + y = r$ is tangen to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$

2013 Today's Calculation Of Integral, 867

Tags: calculus , derivative , integration , function , quadratic , linear equation , algebra

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2004 Romania National Olympiad, 1

Tags: functional equation , calculus , derivative , function , integration , limit , algebra

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \] [i]Mihai Piticari[/i]

PEN G Problems, 7

Tags: trigonometry , algebra , polynomial , derivative , limit , integration , calculus

Show that $ \pi$ is irrational.

2002 VJIMC, Problem 1

Tags: calculus , function , linear algebra , derivative , linear independence

Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.

1984 Iran MO (2nd round), 3

Tags: function , algebra , geometry , calculus , derivative , geometric transformation

Let $f : \mathbb R \to \mathbb R$ be a function such that \[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\] Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$

2014 AIME Problems, 7

Tags: vector , analytic geometry , trigonometry , derivative , graphing lines , calculus , ratio

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

2010 Moldova Team Selection Test, 2

Tags: trigonometry , inequalities , calculus , derivative , algebra

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

2010 Today's Calculation Of Integral, 555

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

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

1991 Arnold's Trivium, 87

Tags: derivative , function , calculus , quadratic , vector , linear algebra , matrix

Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

PEN Q Problems, 6

Tags: polynomial , calculus , derivative , binomial theorem , binomial coefficient , algebra

Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.

2009 Albania Team Selection Test, 2

Tags: function , polynomial , induction , calculus , derivative , limit , algebra

Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

2008 Putnam, B5

Tags: derivative , integration , greatest common divisor , function , number theory , calculus

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)