Olympiad problems

2007 Today's Calculation Of Integral, 195

Tags: calculus , integration , function , real analysis , algebra , partial fractions , calculus computations

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

Today's calculation of integrals, 879

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

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

2009 All-Russian Olympiad, 4

Tags: calculus , integration , analytic geometry , geometry , geometric transformation , reflection , combinatorics

Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?

2005 Brazil Undergrad MO, 2

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

Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$ Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural. Prove that $(y_n)$ is an increasing and divergent sequence.

2005 Today's Calculation Of Integral, 57

Tags: calculus , integration , trigonometry , calculus computations

Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality. \[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]

PEN G Problems, 27

Tags: irrational number , limit , calculus , function , derivative , integration

Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.

1951 AMC 12/AHSME, 15

Tags: calculus , integration

The largest number by which the expression $ n^3 \minus{} n$ is divisible for all possible integral values of $ n$, is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

2003 Romania Team Selection Test, 5

Tags: algebra , polynomial , calculus , integration , function , complex number , absolute value

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]

2005 Today's Calculation Of Integral, 25

Tags: calculus , integration , trigonometry , calculus computations

Let $|a|<\frac{\pi}{2}$. Evaluate \[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]

2010 Today's Calculation Of Integral, 669

Tags: calculus , integration , function , calculus computations

Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$

2009 Today's Calculation Of Integral, 465

Tags: calculus , integration , calculus computations

Compute $ \int_0^1 x^{2n\plus{}1}e^{\minus{}x^2}dx\ (n\equal{}1,\ 2,\ \cdots)$ , then use this result, prove that $ \sum_{n\equal{}0}^{\infty} \frac{1}{n!}\equal{}e$.

2012 Today's Calculation Of Integral, 781

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

Let $l,\ m$ be the tangent lines passing through the point $A(a,\ a-1)$ on the line $y=x-1$ and touch the parabola $y=x^2$. Note that the slope of $l$ is greater than that of $m$. (1) Exress the slope of $l$ in terms of $a$. (2) Denote $P,\ Q$ be the points of tangency of the lines $l,\ m$ and the parabola $y=x^2$. Find the minimum area of the part bounded by the line segment $PQ$ and the parabola $y=x^2$. (3) Find the minimum distance between the parabola $y=x^2$ and the line $y=x-1$.

2008 AMC 12/AHSME, 21

Tags: probability , analytic geometry , geometry , integration , calculus

Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect? $ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$

2012 Today's Calculation Of Integral, 811

Tags: calculus , integration , trigonometry , calculus computations

Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$

2014 District Olympiad, 1

Tags: function , floor function , limit , integration , real analysis

For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$

2021 CMIMC Integration Bee, 12

Tags: calculus , integration

$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

2012 Grigore Moisil Intercounty, 2

Tags: calculus , real analysis , definite integral , integration

$ \int_0^{\pi^2/4} \frac{dx}{1+\sin\sqrt x +\cos\sqrt x} $

2003 Alexandru Myller, 2

Tags: calculus , integration , trigonometric integral , trigonometry

Calculate $ \int_0^{2\pi }\prod_{i=1}^{2002} cos^i (it) dt. $ [i]Dorin Andrica[/i]

2009 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , logarithm

Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]

2007 ISI B.Stat Entrance Exam, 3

Tags: calculus , integration , function , calculus computations

Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x>1$, \[\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du\]

2024 CMIMC Integration Bee, 11

Tags: calculus , integration

\[\int_1^\infty \frac{\lfloor x^2\rfloor}{x^5}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

2012 Romania National Olympiad, 4

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

[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]

2010 Putnam, A2

Tags: function , integration , real analysis , slope

Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$

2003 Flanders Math Olympiad, 4

Tags: analytic geometry , parameterization , number theory , least common multiple , ratio , calculus , integration

Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points) Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.

2004 IMC, 6

Tags: function , polynomial , calculus , derivative , integration , logarithm , algebra

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]