Olympiad problems

2007 Today's Calculation Of Integral, 224

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

1993 Irish Math Olympiad, 4

Tags: integration , trigonometry

Let $x$ be a real number with $0<x<\pi $.Prove that, for all natural number $n$ ,\[sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.\]

2012 Today's Calculation Of Integral, 823

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

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.

2008 Putnam, A1

Tags: function , abstract algebra , calculus , limit , integration , algebra

Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$

2012 Today's Calculation Of Integral, 856

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

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

2007 Today's Calculation Of Integral, 216

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

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

2011 Today's Calculation Of Integral, 691

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

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

Today's calculation of integrals, 880

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

For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows. (1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$. (2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$ (3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.

2007 Princeton University Math Competition, 3

Tags: geometry , analytic geometry , quadratic , calculus , integration , algebra , quadratic formula

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

2011 Today's Calculation Of Integral, 675

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

In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$. [i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]

2014 Online Math Open Problems, 30

Tags: modular arithmetic , integration , algebra , binomial theorem

Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$. [i]Proposed by Yang Liu[/i]

2005 Today's Calculation Of Integral, 58

Tags: calculus , integration , calculus computations

Let $f(x)=\frac{e^x}{e^x+1}$ Prove the following equation. \[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]

2013 Putnam, 3

Tags: algebra , polynomial , function , integration

Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]

Today's calculation of integrals, 849

Tags: calculus , integration , function , calculus computations

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

2007 Estonia Math Open Junior Contests, 4

Tags: calculus , integration , ratio , perimeter , number theory , geometry

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

2010 Today's Calculation Of Integral, 649

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

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

2005 Today's Calculation Of Integral, 77

Tags: calculus , integration , geometry , calculus computations

Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

2007 Today's Calculation Of Integral, 171

Tags: calculus , integration , calculus computations

Evaluate $\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.$

1989 Putnam, B3

Tags: calculus , integration , de , differential equation

Let $f:[0,\infty)\to\mathbb R$ be differentiable and satisfy $$f'(x)=-3f(x)+6f(2x)$$for $x>0$. Assume that $|f(x)|\le e^{-\sqrt x}$ for $x\ge0$. For $n\in\mathbb N$, define $$\mu_n=\int^\infty_0x^nf(x)dx.$$ $a.$ Express $\mu_n$ in terms of $\mu_0$. $b.$ Prove that the sequence $\frac{3^n\mu_n}{n!}$ always converges, and the the limit is $0$ only if $\mu_0$.

2009 Today's Calculation Of Integral, 467

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

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

2007 Today's Calculation Of Integral, 197

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

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

2022 CMIMC Integration Bee, 1

Tags: calculus , integration

\[\int_0^{\pi/1011}\sin^2(2022x)+\cos^2(2022x)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2022 SEEMOUS, 2

Tags: functional equation , continuity , palic , differentiation , integration , analysis

Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

2005 Today's Calculation Of Integral, 13

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

Calculate the following integarls. [1] $\int x\cos ^ 2 x dx$ [2] $\int \frac{x-1}{(3x-1)^2}dx$ [3] $\int \frac{x^3}{(2-x^2)^4}dx$ [4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$ [5] $\int (\ln x)^2 dx$

1984 Vietnam National Olympiad, 1

Tags: polynomial , calculus , integration , algebra

$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root. $(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.