Olympiad problems

2009 Today's Calculation Of Integral, 502

(1) For $ 0 < x < 1$, prove that $ (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}.$ (2) Find $ \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}$.

2005 IMC, 5

Tags: calculus , integration , limit

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

2000 Putnam, 4

Tags: trigonometry , calculus , integration , limit , function

Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.

1985 AMC 12/AHSME, 21

Tags: calculus , integration

How many integers $ x$ satisfy the equation \[ (x^2 \minus{} x \minus{} 1)^{x \plus{} 2} \equal{} 1 \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of these}$

2011 AMC 12/AHSME, 19

Tags: calculus , integration , analytic geometry , inequalities

A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? $ \textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25} $

1979 Swedish Mathematical Competition, 4

Tags: algebra , integration , analysis

$f(x)$ 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 Putnam, A6

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

Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

2012 Pre-Preparation Course Examination, 3

Tags: integration , real analysis , function

Consider the set $\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$. Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space?

2007 AIME Problems, 7

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

Let \[N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).\] Find the remainder when N is divided by 1000. (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to x, and $\lceil x \rceil$ denotes the least integer that is greater than or equal to x.)

2010 Today's Calculation Of Integral, 655

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

Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.

2012 Today's Calculation Of Integral, 787

Tags: geometry , calculus , integration , rotation , limit , geometric transformation , trigonometry

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

2010 Today's Calculation Of Integral, 619

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

Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

Today's calculation of integrals, 894

Tags: calculus , integration , calculus computations , geometry , logarithm

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

2005 IMC, 4

Tags: rectangle , trapezoid , integration , calculus , derivative , geometry

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function. Prove that there exists $w \in [-1,1]$ such that \[ \frac{f'''(w)}{6} = \frac{f(1)}{2}-\frac{f(-1)}{2}-f'(0). \]

2010 Today's Calculation Of Integral, 608

Tags: integration , calculus computations , calculus

For $a>0$, find the minimum value of $\int_0^1 \frac{ax^2+(a^2+2a)x+2a^2-2a+4}{(x+a)(x+2)}dx.$ 2010 Gakusyuin University entrance exam/Science

2010 Today's Calculation Of Integral, 532

Tags: calculus , trigonometry , integration , calculus computations , geometry

For a curve $ C: y \equal{} x\sqrt {9 \minus{} x^2}\ (x\geq 0)$, (1) Find the maximum value of the function. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$-axis. (3) Find the volume of the solid by revolution of the figure in (2) around the $ y$-axis. Please find the volume without using cylindrical shells for my students. Last Edited.

Today's calculation of integrals, 863

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

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2024 SEEMOUS, P3

Tags: integration , calculus

For every $n\geq 1$ define $x_n$ by $$x_n=\int_0^1 \ln(1+x+x^2+\dots +x^n)\cdot\ln\frac{1}{1-x}\mathrm dx.$$ a) Show that $x_n$ is finite for every $n\geq 1$ and $\lim_{n\rightarrow\infty}x_n=2$. b) Calculate $\lim_{n\rightarrow\infty}\frac{n}{\ln n}(2-x_n)$.

1979 AMC 12/AHSME, 11

Tags: integration , calculus

Find a positive integral solution to the equation \[\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}\] $\textbf{(A) }110\qquad\textbf{(B) }115\qquad\textbf{(C) }116\qquad\textbf{(D) }231\qquad\textbf{(E) }\text{The equation has no positive integral solutions.}$

PEN S Problems, 38

Tags: integration , function , calculus , trigonometry

The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.

2006 Princeton University Math Competition, 5

Tags: integration , floor function , calculus , function , inequalities

Find the greatest integer less than the number $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$

2006 IMC, 5

Tags: inequalities , function , integration , princeton , college , quadratic , logarithm

Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

1981 Vietnam National Olympiad, 2

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

Consider the polynomials \[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\] \[g(p) = p^3 + 2p + m.\] Find all integral values of $m$ for which $f$ is divisible by $g$.

2010 Today's Calculation Of Integral, 533

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

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$.

PEN M Problems, 13

Tags: function , pigeonhole principle , integration , calculus , recursive sequences , induction

The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.