Olympiad problems

1989 Balkan MO, 2

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

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.

2010 Today's Calculation Of Integral, 612

Tags: calculus , integration , conic , parabola , quadratic , calculus computations

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

2013 Today's Calculation Of Integral, 886

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

Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

2009 Today's Calculation Of Integral, 452

Tags: calculus , integration , calculus computations , logarithm

Let $ a,\ b$ are postive constant numbers. (1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$ (2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$, evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$

2009 Germany Team Selection Test, 3

Tags: function , algebra , polynomial , calculus

Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If \[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\] then \[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]

2012 Today's Calculation Of Integral, 853

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

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2007 QEDMO 5th, 5

Tags: induction , algebra , function , domain , calculus , integration , polynomial

Let $ a$, $ b$, $ c$ be three integers. Prove that there exist six integers $ x$, $ y$, $ z$, $ x^{\prime}$, $ y^{\prime}$, $ z^{\prime}$ such that $ a\equal{}yz^{\prime}\minus{}zy^{\prime};\ \ \ \ \ \ \ \ \ \ b\equal{}zx^{\prime}\minus{}xz^{\prime};\ \ \ \ \ \ \ \ \ \ c\equal{}xy^{\prime}\minus{}yx^{\prime}$.

2013 Today's Calculation Of Integral, 865

Tags: calculus , integration , geometric transformation , rotation , analytic geometry , rectangle , geometry

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

1967 IMO Shortlist, 4

Tags: parameterization , calculus , trigonometry , algebra , trigonometric identities

Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

2014 Taiwan TST Round 2, 1

Tags: inequalities , calculus , derivative , function , n-variable inequality

Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$, \[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]

2007 German National Olympiad, 6

Tags: inequalities , calculus , derivative , quadratic , algebra

For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

2007 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration

Let $a$ be a positive real number. Find the value of $a$ such that the definite integral \[\int_a^{a^2} \dfrac{dx}{x+\sqrt{x}}\] achieves its smallest possible value.

2022 Brazil Undergrad MO, 1

Tags: function , functional analysis , calculus

Let $0<a<1$. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$

2001 District Olympiad, 4

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

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]

1991 Arnold's Trivium, 1

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

Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.

2014 ISI Entrance Examination, 7

Tags: calculus , integration , inequalities , rearrangement inequality , real analysis , integral inequality

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}

2013 F = Ma, 16

Tags: calculus , derivative

A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? (The constant $0.5427$ is actually $\sqrt{\frac{3 \pi}{32}}$.) $\textbf{(A) } \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\ \textbf{(B) } \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\ \textbf{(C) } \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\ \textbf{(D) } \frac{0.5427}{\sqrt{G \rho_0}}\\ \\ \textbf{(E) } \frac{0.5427}{\sqrt{G \rho_0}}r_0$

2009 Today's Calculation Of Integral, 497

Tags: calculus , integration , trigonometry , calculus computations , geometry

Consider a parameterized curve $ C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right).$ (1) Find the length $ L$ of $ C$. (2) Find the area $ S$ of the region bounded by $ C$, the $ x$ axis and $ y$ axis. You may not use the formula $ \boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }$ here.

2005 Korea National Olympiad, 4

Tags: function , induction , calculus , derivative , functional equation , algebra

Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]

2005 Today's Calculation Of Integral, 86

Tags: calculus , integration , trigonometry , calculus computations

Prove \[\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}\]

2013 Today's Calculation Of Integral, 880

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

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

1962 AMC 12/AHSME, 26

Tags: parabola , calculus , conic

For any real value of $ x$ the maximum value of $ 8x \minus{} 3x^2$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac83 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \frac{16}{3}$

1979 IMO Shortlist, 11

Tags: inequality , maximization , calculus , optimization , lagrange

Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.

2007 Today's Calculation Of Integral, 197

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

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

2010 Today's Calculation Of Integral, 599

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\frac{\pi}{6}} \frac{e^x(\sin x+\cos x+\cos 3x)}{\cos^ 2 {2x}}\ dx$. created by kunny