Olympiad problems

2010 Today's Calculation Of Integral, 567

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

Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $x$-axis and denote $T$ by the area of the figure bounded by the curves $y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $x$-axis. Find the value of $a$ such that $ S: T=3: 1$.

2012 Kyoto University Entry Examination, 1A

Tags: calculus , integration , geometry , symmetry , calculus computations

Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.

2010 Today's Calculation Of Integral, 527

Tags: calculus , integration , calculus computations

Let $ n,\ m$ be positive integers and $ \alpha ,\ \beta$ be real numbers. Prove the following equations. (1) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3$ (2) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2}$ (3) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}$

2002 District Olympiad, 4

Tags: calculus , integration , breaking integral , periodic function , periodicity

Let be a continuous and periodic function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ of period $ 1. $ Show: [b]a)[/b] $ a\in\mathbb{R}\implies\int_a^{a+1} f(x)dx =\int_0^1 f(x) dx . $ [b]b)[/b] $ \lim_{n\to\infty} \int_0^1 f(x)f(nx) dx=\left( \int_0^1 f(x) dx \right)^2 . $ [i]C. Mortici[/i]

2007 Grigore Moisil Intercounty, 1

Tags: calculus , integration , find all functions

Find all functions $ f:[0,1]\longrightarrow \mathbb{R} $ that are continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow [0,1] , $ the following equality holds. $$ \int_0^1 f\left( g(x) \right) dx =\int_0^1 g(x) dx $$

2018 Romania National Olympiad, 2

Tags: function , calculus , integration , real analysis

Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$ For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$ Determine $\min_{f \in \mathcal{F}}I(f).$ [i]Liviu Vlaicu[/i]

2005 Today's Calculation Of Integral, 40

Tags: calculus , integration , calculus computations

Evaluate \[\int_0^1 x^{2005}e^{-x^2}dx\]

2021 Alibaba Global Math Competition, 15

Tags: calculus , integration , topology , analysis , manifold , differential geometry

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

1972 Canada National Olympiad, 10

Tags: ratio , calculus , integration , geometric sequence , algebra

What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?

2014 Turkey MO (2nd round), 5

Tags: integration , algebra

Find all natural numbers $n$ for which there exist non-zero and distinct real numbers $a_1, a_2, \ldots, a_n$ satisfying \[ \left\{a_i+\dfrac{(-1)^i}{a_i} \, \Big | \, 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}. \]

2023 CMIMC Integration Bee, 11

Tags: calculus , integration

\[\int_{-1}^1 \frac{1}{(8 + x^2)\sqrt{1-x^2}} \,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2005 Today's Calculation Of Integral, 8

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

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

2009 Today's Calculation Of Integral, 518

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

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

2008 Grigore Moisil Intercounty, 2

Tags: calculus , integration , inequalities , function , real analysis

Let $ n\in \mathbb{N^*}$ and $ f: [0,1]\rightarrow \mathbb{R}$ a continuos function with the prop. $ \int_{0}^{1}(1\minus{}x^n)f(x)dx\equal{}0$. Prove that $ \int_{0}^{1}f^2(x)dx \geq 2(n\plus{}1)\left(\int_{0}^{1}f(x)dx\right)^2$

2012 Today's Calculation Of Integral, 855

Tags: calculus , integration , function , limit , calculus computations

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2010 Today's Calculation Of Integral, 639

Tags: calculus , integration , calculus computations

Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$

MathLinks Contest 7th, 1.2

Tags: algebra , polynomial , number theory , greatest common divisor , quadratic , calculus , integration

Let $ a,b,c,d$ be four distinct positive integers in arithmetic progression. Prove that $ abcd$ is not a perfect square.

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$.

1969 Miklós Schweitzer, 5

Tags: function , integration , calculus , real analysis

Find all continuous real functions $ f,g$ and $ h$ defined on the set of positive real numbers and satisfying the relation \[ f(x\plus{}y)\plus{}g(xy)\equal{}h(x)\plus{}h(y)\] for all $ x>0$ and $ y>0$. [i]Z. Daroczy[/i]

2007 Today's Calculation Of Integral, 243

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

A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with 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$.

2006 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , trigonometry , logarithm

Compute $\displaystyle\int_0^{\pi/3}x\tan^2(x)dx$.

2012 Romania National Olympiad, 1

Tags: function , integration , real analysis

[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]

2019 Jozsef Wildt International Math Competition, W. 29

Tags: integration , summation

Prove that $$\int \limits_0^{\infty} e^{3t}\frac{4e^{4t}(3t - 1) + 2e^{2t}(15t - 17) + 18(1 - t)}{\left(1 + e^{4t} - e^{2t}\right)^2}=12\sum \limits_{k=0}^{\infty}\frac{(-1)^k}{(2k + 1)^2}-10$$

2009 Today's Calculation Of Integral, 400

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

(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$. (2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$. Remark: [color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.

2005 Brazil Undergrad MO, 6

Tags: linear algebra , matrix , calculus , integration , college

Prove that for any natural numbers $0 \leq i_1 < i_2 < \cdots < i_k$ and $0 \leq j_1 < j_2 < \cdots < j_k$, the matrix $A = (a_{rs})_{1\leq r,s\leq k}$, $a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}$ ($1\leq r,s\leq k$) is nonsingular.