Olympiad problems

2010 Today's Calculation Of Integral, 533

Tags: calculus , integration , function , trigonometry , analytic geometry , parameterization , algebra

Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1,\ 0)$ and put the other end point $ P$ on the point $ P_0(1,\ 2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.

2009 Harvard-MIT Mathematics Tournament, 6

Tags: calculus , algebra , polynomial , function

Let $p_0(x),p_1(x),p_2(x),\ldots$ be polynomials such that $p_0(x)=x$ and for all positive integers $n$, $\dfrac{d}{dx}p_n(x)=p_{n-1}(x)$. Define the function $p(x):[0,\infty)\to\mathbb{R}$ by $p(x)=p_n(x)$ for all $x\in [n,n+1)$. Given that $p(x)$ is continuous on $[0,\infty)$, compute \[\sum_{n=0}^\infty p_n(2009).\]

2010 Today's Calculation Of Integral, 614

Tags: calculus , integration , ratio , trigonometry , calculus computations

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2014 AIME Problems, 7

Tags: trigonometry , calculus , derivative , vector , ratio , analytic geometry , graphing lines

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

2000 Putnam, 3

Tags: trigonometry , limit , calculus , derivative , algebra , polynomial

Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$. Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]

2007 Germany Team Selection Test, 1

Tags: calculus , floor function , number theory , sequence , summation

We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by \[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right), \] where $\lfloor x\rfloor$ denotes the integer part of $x$. [b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often. [b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often. [i]Proposed by Johan Meyer, South Africa[/i]

2018 PUMaC Live Round, Calculus 3

Tags: calculus

Let $\mathcal{R}(f(x))$ denote the number of distinct real roots of $f(x)$. Compute $$\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).$$

2011 Today's Calculation Of Integral, 760

Tags: calculus , integration , trigonometry , calculus computations

Prove that there exists a positive integer $n$ such that $\int_0^1 x\sin\ (x^2-x+1)dx\geq \frac {n}{n+1}\sin \frac{n+2}{n+3}.$

2013 Today's Calculation Of Integral, 893

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

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2013 Today's Calculation Of Integral, 867

Tags: calculus , integration , quadratic , function , derivative , algebra , linear equation

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2013 SEEMOUS, Problem 3

Tags: calculus , integration , inequalities

Find the maximum value of $$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$\int^1_0|f'(x)|^2dx\le1.$$

2013 Today's Calculation Of Integral, 875

Tags: calculus , integration , trigonometry , ratio , calculus computations

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

1964 Spain Mathematical Olympiad, 2

Tags: calculus

The RTP tax is a function $f(x)$, where $x$ is the total of the annual profits (in pesetas). Knowing that: a) $f(x)$ is a continuous function b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$; and when $x>P$ is constant and equal 0.14. c) $f(0)=0$ and $f(140000)=14000$. Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$.

2011 ISI B.Stat Entrance Exam, 8

Tags: integration , trigonometry , calculus , calculus computations

Let \[I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} \, dx , \ \ \ \ n=1,2,3,4\] Arrange $I_1, I_2, I_3, I_4$ in increasing order of magnitude. Justify your answer.

2011 Today's Calculation Of Integral, 708

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

Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.

2012 Today's Calculation Of Integral, 819

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

For real numbers $a,\ b$ with $0\leq a\leq \pi,\ a<b$, let $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$ Determine the value of $a$ such that $\lim_{b\rightarrow \infty} I(a,\ b)=0.$

2006 Swedish Mathematical Competition, 3

Tags: analytic geometry , graphing lines , slope , calculus , derivative , algebra

A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.

2022 CMIMC Integration Bee, 10

Tags: calculus , integration

\[\int_0^1 \frac{(x+1)\log(x)}{x^3-1}\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

1996 IMC, 2

Tags: calculus , integration , real analysis

Evaluate the definite integral $$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$ where $n$ is a natural number.

Today's calculation of integrals, 875

Tags: calculus , integration , trigonometry , ratio , calculus computations

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

2012 ISI Entrance Examination, 6

Tags: geometry , perimeter , calculus , ellipse , inequalities , perpendicular bisector , conic

[b]i)[/b] Let $0<a<b$.Prove that amongst all triangles having base $a$ and perimeter $a+b$ the triangle having two sides(other than the base) equal to $\frac {b}{2}$ has the maximum area. [b]ii)[/b]Using $i)$ or otherwise, prove that amongst all quadrilateral having give perimeter the square has the maximum area.

2016 ISI Entrance Examination, 7

Tags: integration , calculus , real analysis

$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of $$\int_0^1(x-f(x))^{2016}dx$$

2010 Today's Calculation Of Integral, 560

Tags: calculus , integration , function , geometry , geometric transformation , rotation , logarithm

Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$. (1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis. (2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis. Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.

2004 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , trigonometry , algebra , polynomial , induction

Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.

1993 AMC 12/AHSME, 19

Tags: calculus , integration , sfft , special factorizations

How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $