Olympiad problems

2006 Moldova National Olympiad, 12.2

Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$

2009 Today's Calculation Of Integral, 483

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

Let $ n\geq 2$ be natural number. Answer the following questions. (1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$ (2) Prove the following inequality. $ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$ (3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$

1981 AMC 12/AHSME, 11

Tags: arithmetic sequence , integration , calculus

The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length $\text{(A)}\ 22 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 81 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 361$

2010 Today's Calculation Of Integral, 611

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

Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$. Evaluate $\int_0^2 g(t)dt$. [i]2010 Kumamoto University entrance exam/Science[/i]

2006 Turkey Junior National Olympiad, 2

Tags: integration , calculus

Find all integer triples $(x,y,z)$ such that \[ \begin{array}{rcl} x-yz &=& 11 \\ xz+y &=& 13. \end{array}\]

PEN A Problems, 100

Tags: integration , divisibility theory , modular arithmetic , calculus

Find all positive integers $n$ such that $n$ has exactly $6$ positive divisors $1<d_{1}<d_{2}<d_{3}<d_{4}<n$ and $1+n=5(d_{1}+d_{2}+d_{3}+d_{4})$.

2007 Today's Calculation Of Integral, 234

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

For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$. Evaluate $ \int_0^{100\pi } f(x)\ dx.$

2012 Today's Calculation Of Integral, 822

Tags: integration , calculus computations , calculus , limit

For $n=0,\ 1,\ 2,\ \cdots$, let $a_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}(x-n)\}\ dx,$ $b_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}\}\ dx.$ Find $\lim_{n\to\infty} \sum_{k=0}^n (a_k-b_k).$

2023 CMIMC Integration Bee, 8

Tags: integration , calculus

\[\int_{-10}^{10}|4-|3-|2-|1-|x|||||\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2009 Today's Calculation Of Integral, 490

Tags: logarithm , calculus computations , inequalities , integration , calculus

For a positive real number $ a > 1$, prove the following inequality. $ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: integral , integration , algebra , calculus

Find some antiderivative of the function $y = 1/x^3$, the graph of which has exactly three common points with the graph of the function $y = |x|$.

2010 Today's Calculation Of Integral, 643

Tags: integration , trigonometry , calculus computations , calculus

Evaluate \[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\] Own

1982 Putnam, A3

Tags: integration , calculus

Evaluate $$\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx.$$

2011 All-Russian Olympiad, 3

Tags: polynomial , number theory , integration , modular arithmetic , algebra , calculus

Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$. [i]A. Golovanov[/i]

1999 Vietnam Team Selection Test, 1

Tags: algebra , inequalities , integration , calculus , limit

Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$, let $k_n$ be the least positive integer satisfying: \[\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.\] Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.

2009 Princeton University Math Competition, 3

Tags: logarithm , ceiling function , floor function , induction , integration , calculus

It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?

2005 Today's Calculation Of Integral, 69

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

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 $\lim_{n\to\infty} f_n(x)$

2012 Today's Calculation Of Integral, 804

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

For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$

1981 Spain Mathematical Olympiad, 4

Tags: integration , integral , calculus

Calculate the integral $$\int \frac{dx}{\sin (x - 1) \sin (x - 2)} .$$ Hint: Change $\tan x = t$ .

Today's calculation of integrals, 895

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

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

2013 Online Math Open Problems, 25

Tags: modular arithmetic , calculus , integration

Positive integers $x,y,z \le 100$ satisfy \begin{align*} 1099x+901y+1110z &= 59800 \\ 109x+991y+101z &= 44556 \end{align*} Compute $10000x+100y+z$. [i]Evan Chen[/i]

2007 Today's Calculation Of Integral, 188

Tags: integration , calculus , calculus computations

Find the volume of the solid obtained by revolving the region bounded by the graphs of $y=xe^{1-x}$ and $y=x$ around the $x$ axis.

1960 AMC 12/AHSME, 23

Tags: integration , calculus

The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. The volume $V = \pi R^2H$ is to be increased by the same fixed positive amount when $R$ is increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by: $ \textbf{(A)}\ \text{no real value of} \text{ } x\qquad$ $\textbf{(B)}\ \text{one integral value of} \text{ } x\qquad$ $\textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad$ $\textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad$ $\textbf{(E)}\ \text{two real values of} \text{ } x $

2011 Today's Calculation Of Integral, 767

Tags: integration , calculus computations , trigonometry , calculus

For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

2009 Today's Calculation Of Integral, 488

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

For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality. $ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$