Olympiad problems

1989 India National Olympiad, 1

Tags: calculus , quadratic , integration , modular arithmetic , polynomial , algebra

Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.

2009 Today's Calculation Of Integral, 441

Tags: logarithm , integration , calculus , calculus computations

Evaluate $ \int_1^e \frac{(x^2\ln x\minus{}1)e^x}{x}\ dx.$

2009 USA Team Selection Test, 9

Tags: function , inequalities , derivative , search , calculus , polynomial , algebra

Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]

2006 Harvard-MIT Mathematics Tournament, 7

Tags: calculus

Find all positive real numbers $c$ such that the graph of $f\text{ : }\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3-cx$ has the property that the circle of curvature at any local extremum is centered at a point on the $x$-axis.

2007 Romania National Olympiad, 1

Tags: calculus , inequalities , derivative , logarithm , integration , trigonometry , function

Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''. [/hide]

2007 Today's Calculation Of Integral, 206

Tags: calculus , integration , calculus computations

Calculate $\int \frac{x^{3}}{(x-1)^{3}(x-2)}\ dx$

Today's calculation of integrals, 854

Tags: ellipse , conic , integration , limit , geometry , calculus , analytic geometry

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

2012 Online Math Open Problems, 49

Tags: polynomial , induction , derivative , complex number , algebra , calculus

Find the magnitude of the product of all complex numbers $c$ such that the recurrence defined by $x_1 = 1$, $x_2 = c^2 - 4c + 7$, and $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$ also satisfies $x_{1006} = 2011$. [i]Author: Alex Zhu[/i]

2009 Today's Calculation Of Integral, 514

Tags: domain , algebra , trigonometry , integration , calculus , inequalities , function

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

1998 Romania National Olympiad, 1

Tags: real analysis , function , calculus , integration

Suppose that $a,b\in\mathbb{R}^+$ which $a+b<1$ and $f:[0,+\infty) \rightarrow [0,+\infty) $ be the increasing function s.t. $\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt$. Prove that $\forall x\geq 0 , f(x)=0$

2021 Alibaba Global Math Competition, 5

Tags: complex analysis , ode , differential equation , calculus

For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$. (a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$. (b) For any integer $k$, let $f_k(x)=(1+x^2)^{-1-k}$. Assume $k\geq 1$, find constant $c_1$, $c_2$ such that the function $y=(Sf_k)(x)$ satisfies the ODE with second order: $xy''+c_1y'+c_2xy=0$.

2010 Today's Calculation Of Integral, 564

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

In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$. (1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$. Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal. (2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.

1985 IMO Shortlist, 17

Tags: recurrence relation , fixed point , limit , function , calculus

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

2009 Today's Calculation Of Integral, 504

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

Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.

2012 Today's Calculation Of Integral, 844

Tags: calculus computations , integration , limit , calculus

Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$

2011 Today's Calculation Of Integral, 729

Tags: logarithm , integration , calculus , calculus computations

Evaluate $\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.$

2012 Today's Calculation Of Integral, 842

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

Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty} nS_nS_{n+1}.$

2011 Today's Calculation Of Integral, 761

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

Find $\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.$

2013 SEEMOUS, Problem 1

Tags: calculus , function , integration

Find all continuous functions $f:[1,8]\to\mathbb R$, such that $$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$

2016 District Olympiad, 2

Tags: function , integration , calculus

Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that $$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$ with equality if and only if $ 1 $ is also a period of $ f. $