Olympiad problems

2009 Putnam, A2

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

Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions \begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*} Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$

2001 India Regional Mathematical Olympiad, 3

Tags: floor function , calculus , integration

Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]

2010 Today's Calculation Of Integral, 595

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

Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$ 2009 Kumamoto University entrance exam/Medicine

1990 AIME Problems, 7

Tags: graphing lines , trigonometry , analytic geometry , slope , ratio , calculus , integration

A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.

2004 IMC, 5

Tags: integration , logarithm , inequalities , calculus

Prove that \[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]

2007 Romania National Olympiad, 2

Tags: real analysis , integration , calculus

Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function. a) Show that for any integer $n\geq 1$, there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$ holds for all $k=0,1,\ldots,n-1$. b) For each $n$, consider the $a_{i}$ above (that depend on $n$) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$. Show that the sequence $(b_{n})$ is convergent and compute it's limit.

2001 District Olympiad, 4

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

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]

2007 Princeton University Math Competition, 2

Tags: calculus , integration

Find the biggest non-integer $x$ such that $(x+2)^2 + (x+3)^3 + (x+4)^4 = 2$.

2009 Today's Calculation Of Integral, 509

Tags: algebra , calculus , integration , partial fractions , calculus computations , logarithm , trigonometry

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

2010 Laurențiu Panaitopol, Tulcea, 2

Tags: real analysis , function , primitivableness , integration

Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that $$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$ for any real numbers $ a\neq b. $ Prove that $ f'(c)=0. $ [i]Florin Rotaru[/i]

2010 Today's Calculation Of Integral, 614

Tags: calculus , integration , ratio , trigonometry , calculus computations

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2005 Today's Calculation Of Integral, 79

Tags: geometry , integration , algebra , function , factorial , domain , calculus

Find the area of the domain expressed by the following system inequalities. \[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]

2005 Today's Calculation Of Integral, 11

Tags: integration , domain , complex number , function , algebra , logarithm , calculus

Calculate the following indefinite integrals. [1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$ [2] $\int \frac{e^x}{e^x+e^{a-x}}dx$ [3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$ [4] $\int x\ln (x^2-1)dx$ [5] $\int \frac{2(x+2)}{x^2+4x+1}dx$

2011 Today's Calculation Of Integral, 710

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

Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2010 Today's Calculation Of Integral, 569

Tags: analytic geometry , graphing lines , geometry , calculus , integration , conic , slope

In the coordinate plane, denote by $ S(a)$ the area of the region bounded by the line passing through the point $ (1,\ 2)$ with the slope $ a$ and the parabola $ y\equal{}x^2$. When $ a$ varies in the range of $ 0\leq a\leq 6$, find the value of $ a$ such that $ S(a)$ is minimized.

2019 Jozsef Wildt International Math Competition, W. 56

Tags: integration , function , calculus

Let $f$, $g$, $h : [a, b] \to \mathbb{R}$, three integrable functions such that:$$\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1$$Then$$\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1$$

2012 Kyoto University Entry Examination, 1

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2007 Today's Calculation Of Integral, 245

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

A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma .$ created by kunny

2009 Today's Calculation Of Integral, 503

Tags: calculus , graphing lines , inequalities , integration , trapezoid , analytic geometry , geometry

Prove the following inequality. \[ \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}\]

2012 VJIMC, Problem 1

Tags: integration , real analysis , calculus

Let $f:[1,\infty)\to(0,\infty)$ be a non-increasing function such that $$\limsup_{n\to\infty}\frac{f(2^{n+1})}{f(2^n)}<\frac12.$$Prove that $$\int^\infty_1f(x)\text dx<\infty.$$

2001 VJIMC, Problem 3

Tags: function , limit , calculus , integration

Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.

2021 JHMT HS, 8

Tags: logarithm , integration , calculus

Find the unique integer $a > 1$ that satisfies \[ \int_{a}^{a^2} \left(\frac{1}{\ln x} - \frac{2}{(\ln x)^3}\right) dx = \frac{a}{\ln a}. \]

2009 Putnam, B4

Tags: polynomial , calculus , integration , vector , complex analysis , algebra

Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$

2020 Jozsef Wildt International Math Competition, W22

Tags: calculus , integration , complex analysis , complex number

Prove that $$\operatorname{Re}\left(\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)+\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)=\frac{7\pi^2}{72}-\frac{\ln^23}8$$ where as usual $$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$ [i]Proposed by Paolo Perfetti[/i]

2005 Today's Calculation Of Integral, 47

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

Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$. \[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]