Found problems: 1687
1979 USAMO, 1
Determine all non-negative integral solutions $ (n_{1},n_{2},\dots , n_{14}) $ if any, apart from permutations, of the Diophantine Equation \[n_{1}^{4}+n_{2}^{4}+\cdots+n_{14}^{4}=1,599.\]
1991 Arnold's Trivium, 13
Calculate with $5\%$ relative error
\[\int_1^{10}x^xdx\]
2012 Today's Calculation Of Integral, 837
Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$
Find $\lim_{n\to\infty} f_n(1).$
2013 Today's Calculation Of Integral, 862
Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$
1994 China Team Selection Test, 2
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.
2022 CMIMC Integration Bee, 6
\[\int_0^{2022} \{x\lfloor x \rfloor\}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2009 Today's Calculation Of Integral, 429
Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.
2007 Today's Calculation Of Integral, 185
Evaluate the following integrals.
(1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$
(2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$
2017 Korea USCM, 5
Evaluate the following limit.
\[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]
Today's calculation of integrals, 868
In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation.
(1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$.
(2) Find the volume of the common part of $V_1$ and $V_2$.
2012 Today's Calculation Of Integral, 817
Define two functions $f(t)=\frac 12\left(t+\frac{1}{t}\right),\ g(t)=t^2-2\ln t$. When real number $t$ moves in the range of $t>0$, denote by $C$ the curve by which the point $(f(t),\ g(t))$ draws on the $xy$-plane.
Let $a>1$, find the area of the part bounded by the line $x=\frac 12\left(a+\frac{1}{a}\right)$ and the curve $C$.
2013 Today's Calculation Of Integral, 872
Let $n$ be a positive integer.
(1) For a positive integer $k$ such that $1\leq k\leq n$, Show that :
\[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\]
(2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$
If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$
(3) Find $\lim_{n\to\infty} S_n.$
1993 Irish Math Olympiad, 4
Let $x$ be a real number with $0<x<\pi $.Prove that, for all natural number $n$ ,\[sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.\]
2004 District Olympiad, 2
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous function such that
$$ \int_0^1 f(x)g(x)dx =\int_0^1 f(x)dx\cdot\int_0^1 g(x)dx , $$
for all functions $ g:[0,1]\longrightarrow\mathbb{R} $ that are continuous and non-differentiable.
Prove that $ f $ is constant.
1974 IMO Longlists, 49
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
2015 AMC 10, 25
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
2011 Today's Calculation Of Integral, 708
Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.
1980 Putnam, A5
Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations
$$ \int_{0}^{x} P(t)\sin t \, dt =0, \;\;\;\; \int_{0}^{x} P(t) \cos t \, dt =0 $$
has only finitely many solutions $x.$
2009 Putnam, A6
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
2008 Moldova MO 11-12, 6
Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.
1969 AMC 12/AHSME, 27
A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is:
$\textbf{(A) }\dfrac2{n-1}\qquad
\textbf{(B) }\dfrac{n-1}2\qquad
\textbf{(C) }\dfrac2n\qquad
\textbf{(D) }2n\qquad
\textbf{(E) }2(n-1)$
2010 Today's Calculation Of Integral, 630
Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$
2004 Romania National Olympiad, 3
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \]
[i]Ion Rasa[/i]
2009 Today's Calculation Of Integral, 402
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.
2012 Today's Calculation Of Integral, 832
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]