Olympiad problems

2022 CMIMC Integration Bee, 7

Tags: calculus , integration

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

2005 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration , function

Let $ f : \mathbf{R} \to \mathbf{R} $ be a continuous function with $ \displaystyle\int_{0}^{1} f(x) f'(x) \, \mathrm{d}x = 0 $ and $ \displaystyle\int_{0}^{1} f(x)^2 f'(x) \, \mathrm{d}x = 18 $. What is $ \displaystyle\int_{0}^{1} f(x)^4 f'(x) \, \mathrm{d} x $?

2003 Greece National Olympiad, 1

Tags: function , calculus , derivative , inequalities

If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]

2010 Today's Calculation Of Integral, 583

Tags: calculus , integration , calculus computations , geometry

Find the values of $ k$ such that the areas of the three parts bounded by the graph of $ y\equal{}\minus{}x^4\plus{}2x^2$ and the line $ y\equal{}k$ are all equal.

2011 Today's Calculation Of Integral, 747

Tags: calculus , integration , trigonometry , inequalities , calculus computations

Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$

Today's calculation of integrals, 883

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

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2015 CIIM, Problem 1

Tags: calculus , integration

Find the real number $a$ such that the integral $$\int_a^{a+8}e^{-x}e^{-x^2}dx$$ attain its maximum.

2005 Today's Calculation Of Integral, 69

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

Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$. Find $\lim_{n\to\infty} f_n(x)$

Today's calculation of integrals, 878

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

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2008 Harvard-MIT Mathematics Tournament, 1

Tags: algebra , polynomial , function , calculus , calculus computations

Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.

2012 Today's Calculation Of Integral, 799

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

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

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$

Today's calculation of integrals, 860

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

For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below. (a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$. (b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.

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 AIME Problems, 8

Tags: modular arithmetic , real analysis , calculus , derivative , geometric series

Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.

2015 BMT Spring, 10

Tags: integration , calculus

Evaluate $$\int^{\pi/2}_0\ln(4\sin x)dx.$$

2011 Today's Calculation Of Integral, 719

Tags: calculus , integration , trigonometry , calculus computations

Compute $\int_0^x \sin t\cos t\sin (2\pi\cos t)\ dt$.

1990 AIME Problems, 7

Tags: ratio , trigonometry , analytic geometry , graphing lines , slope , 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 $a+c$.

2010 Today's Calculation Of Integral, 549

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

Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.