Olympiad problems

2011 Today's Calculation Of Integral, 747

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 $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2005 IMC, 3

Tags: integration , inequalities

3) $f$ cont diff, $R\rightarrow ]0,+\infty[$, prove $|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}$

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.

2023 Romania National Olympiad, 3

Tags: real analysis , integration , function , riemann integral

Let $a,b \in \mathbb{R}$ with $a < b,$ 2 real numbers. We say that $f: [a,b] \rightarrow \mathbb{R}$ has property $(P)$ if there is an integrable function on $[a,b]$ with property that \[ f(x) - f \left( \frac{x + a}{2} \right) = f \left( \frac{x + b}{2} \right) - f(x) , \forall x \in [a,b]. \] Show that for all real number $t$ there exist a unique function $f:[a,b] \rightarrow \mathbb{R}$ with property $(P),$ such that $\int_{a}^{b} f(x) \text{dx} = t.$

1976 IMO Longlists, 30

Tags: polynomial , induction , integration , algebra

Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.

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 $a+c$.

1979 Swedish Mathematical Competition, 4

Tags: integration , analysis , algebra

$f(x)$ is continuous on the interval $[0, \pi]$ and satisfies \[ \int\limits_0^\pi f(x)dx=0, \qquad \int\limits_0^\pi f(x)\cos x dx=0 \] Show that $f(x)$ has at least two zeros in the interval $(0, \pi)$.

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$.

2009 Jozsef Wildt International Math Competition, W. 14

Tags: integration , inequalities , calculus

If the function $f:[0,1]\to (0.+\infty)$ is increasing and continuous, then for every $a\geq 0$ the following inequality holds: $$\int \limits_0^1 \frac{x^{a+1}}{f(x)}dx \leq \frac{a+1}{a+2} \int \limits_0^1 \frac{x^{a}}{f(x)}dx$$

2005 IMC, 5

Tags: limit , integration , calculus

5) f twice cont diff, $|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1$. prove $\lim_{x\rightarrow +\infty} f(x) = 0$

2009 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , function , integration , logarithm

Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]

1975 Canada National Olympiad, 4

Tags: calculus , integration , ratio , floor function , quadratic , geometric sequence , algebra

For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

2011 Today's Calculation Of Integral, 679

Tags: calculus , integration , calculus computations

Find $\sum_{k=1}^{3n} \frac{1}{\int_0^1 x(1-x)^k\ dx}$. [i]2011 Hosei University entrance exam/Design and Enginerring[/i]

1998 Romania National Olympiad, 1

Tags: calculus , integration , function , real analysis

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 $\forall x\geq 0 , f(x)=0$

2011 Today's Calculation Of Integral, 768

Tags: calculus , integration , limit , geometry , 3d geometry , sphere , symmetry

Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

2007 Romania National Olympiad, 1

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

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]