Olympiad problems

1995 National High School Mathematics League, 10

Tags: calculus , integration

The number of integral points satisfy $\begin{cases} y\leq 3x\\ y\geq \frac{x}{3}\\ x+y\geq100 \end{cases}$ on the coordinate plane is________.

2005 Today's Calculation Of Integral, 40

Tags: calculus , integration , calculus computations

Evaluate \[\int_0^1 x^{2005}e^{-x^2}dx\]

Today's calculation of integrals, 861

Tags: calculus , integration , geometry , function , analytic geometry , calculus computations

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

2011 Today's Calculation Of Integral, 720

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

1980 Vietnam National Olympiad, 2

Tags: polynomial , calculus , integration , algebra

Can the equation $x^3-2x^2-2x+m = 0$ have three different rational roots?

2011 Today's Calculation Of Integral, 705

Tags: calculus , integration , trigonometry , calculus computations

The parametric equations of a curve are given by $x = 2(1+\cos t)\cos t,\ y =2(1+\cos t)\sin t\ (0\leq t\leq 2\pi)$. (1) Find the maximum and minimum values of $x$. (2) Find the volume of the solid enclosed by the figure of revolution about the $x$-axis.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4

Tags: calculus , derivative , topology

Define mapping $F : \mathbb{R}^4\rightarrow \mathbb{R}^4$ as $F(x,\ y,\ z,\ w)=(xy,\ y,\ z,\ w)$ and let mapping $f : S^3\rightarrow \mathbb{R}^4$ be restriction of $F$ to 3 dimensional ball $S^3=\{(x,\ y,\ z,\ w)\in{\mathbb{R}^4} | x^2+y^2+z^2+w^2=1\}$. Find the rank of $df_p$, or the differentiation of $f$ at every point $p$ in $S^3$.

2010 Today's Calculation Of Integral, 615

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

For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

2013 Waseda University Entrance Examination, 4

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

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

2011 N.N. Mihăileanu Individual, 4

Tags: calculus , integration , real analysis

Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^1 \frac{x^n}{\sqrt{x^{2n} +1}} dx . $ [b]a)[/b] Show that $ \left( I_n \right)_{n\ge 1} $ converges to $ 0. $ [b]b)[/b] Calculate $ \lim_{m\to\infty } m\cdot I_m. $ [b]c)[/b] Prove that the sequence $ \left( n\left( -n\cdot I_n +\lim_{m\to\infty } m\cdot I_m \right) \right)_{n\ge 1} $ is convergent.

2000 Finnish National High School Mathematics Competition, 2

Tags: calculus , integration , number theory

Prove that the integral part of the decimal representation of the number $(3+\sqrt{5})^n$ is odd, for every positive integer $n.$

2009 Indonesia TST, 3

Tags: calculus , derivative , 3d geometry , function , inequalities , geometry

Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

2013 Today's Calculation Of Integral, 898

Tags: calculus , integration , calculus computations , logarithm

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

2012 Today's Calculation Of Integral, 845

Tags: calculus , integration , calculus computations

Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$ Find the minimum value of $I(t)\ (t>1).$

2009 Today's Calculation Of Integral, 461

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

Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.

2000 CentroAmerican, 1

Tags: calculus , integration

Find all three-digit numbers $ abc$ (with $ a \neq 0$) such that $ a^{2}+b^{2}+c^{2}$ is a divisor of 26.

2006 Romania Team Selection Test, 4

Tags: inequalities , calculus , algebra , three variable inequality

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]

2007 Today's Calculation Of Integral, 191

Tags: calculus , integration , limit , ratio , geometric series , calculus computations , geometry

(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$ (2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$

2009 Moldova National Olympiad, 12.1

Tags: definite integral , cosinus , calculus , integration

Calculate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{cos(x)^7}{e^x+1} dx$.

2010 Today's Calculation Of Integral, 594

Tags: calculus , integration , calculus computations , geometry

In the $x$-$y$ plane, two variable points $P,\ Q$ stay in $P(2t,\ -2t^2+2t),\ Q(t+2,-3t+2)$ at the time $t$. Let denote $t_0$ as the time such that $\overline{PQ}=0$. When $t$ varies in the range of $0\leq t\leq t_0$, find the area of the region swept by the line segment $PQ$ in the $x$-$y$ plane.

2009 Today's Calculation Of Integral, 402

Tags: calculus , integration , geometry , rectangle , rotation , 3d geometry , sphere

Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.

2011 Today's Calculation Of Integral, 711

Tags: calculus , integration , calculus computations , logarithm

Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$

1983 USAMO, 2

Tags: calculus , derivative , quadratic , algebra , marvio

Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.

2007 AMC 12/AHSME, 19

Tags: geometry , analytic geometry , vector , trigonometry , trapezoid , calculus , integration

Triangles $ ABC$ and $ ADE$ have areas $ 2007$ and $ 7002,$ respectively, with $ B \equal{} (0,0),$ $ C \equal{} (223,0),$ $ D \equal{} (680,380),$ and $ E \equal{} (689,389).$ What is the sum of all possible x-coordinates of $ A?$ $ \textbf{(A)}\ 282 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 900 \qquad \textbf{(E)}\ 1200$

2024 CMIMC Integration Bee, 2

Tags: calculus , integration

\[\int_0^2 |\sin(\pi x)|+|\cos(\pi x)|\mathrm dx\] [i]Proposed by Anagh Sangavarapu[/i]