Found problems: 2215
2022 CMIMC Integration Bee, 6
\[\int_0^{2022} \{x\lfloor x \rfloor\}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
1999 Putnam, 1
Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]
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)$.
1992 Vietnam National Olympiad, 3
Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.
2021 The Chinese Mathematics Competition, Problem 5
Let $D=\{ (x,y)|x^2+y^2\le \pi \}$. Find $\iint\limits_D(sin x^2cosx^2+x\sqrt{x^2+y^2})dxdy$.
2007 Today's Calculation Of Integral, 185
Evaluate the following integrals.
(1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$
(2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$
2013 ELMO Shortlist, 5
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]
2014 NIMO Problems, 8
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Proposed by Akshaj[/i]
2011 AMC 10, 13
Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?
$ \textbf{(A)}\ \frac{1}{9} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{4}{9} \qquad
\textbf{(D)}\ \frac{5}{9} \qquad
\textbf{(E)}\ \frac{2}{3} $
2010 Contests, 2
Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram.
Show that $\angle BPC > \angle BAC$.
2007 Germany Team Selection Test, 1
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]
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 $C$.
2011 Singapore MO Open, 4
Find all polynomials $P(x)$ with real coefficients such that
\[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]
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 $\lim_{n\to\infty} S_n.$
2021 JHMT HS, 3
There is a unique ordered triple of real numbers $(a, b, c)$ that makes the piecewise function
\begin{align*}
f(x) = \begin{cases}
(x - a)^2 + b & \text{if } x \geq c \\
x^3 - x & \text{if } x < c
\end{cases}
\end{align*}
twice continuously differentiable for all real $x.$ The value of $a + b + c$ can be expressed as a common fraction $p/q.$ Compute $p + q.$
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 $\sin \frac{-3 \pi}{14}.$
2009 AIME Problems, 4
A group of children held a grape-eating contest. When the contest was over, the winner had eaten $ n$ grapes, and the child in $ k$th place had eaten $ n\plus{}2\minus{}2k$ grapes. The total number of grapes eaten in the contest was $ 2009$. Find the smallest possible value of $ n$.
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)$.
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.$
2001 China Team Selection Test, 3
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
1957 AMC 12/AHSME, 36
If $ x \plus{} y \equal{} 1$, then the largest value of $ xy$ is:
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 0.5\qquad
\textbf{(C)}\ \text{an irrational number about }{0.4}\qquad
\textbf{(D)}\ 0.25\qquad
\textbf{(E)}\ 0$
2005 Morocco TST, 1
Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying :
$(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$