Found problems: 2215
2014 Putnam, 4
Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.
2014 Contests, 3
Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.
2013 Today's Calculation Of Integral, 886
Find the functions $f(x),\ g(x)$ such that
$f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$
$g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$
2005 Today's Calculation Of Integral, 15
Calculate the following indefinite integrals.
[1] $\int \frac{(x^2-1)^2}{x^4}dx$
[2] $\int \frac{e^{3x}}{\sqrt{e^x+1}}dx$
[3] $\int \sin 2x\cos 3xdx$
[4] $\int x\ln (x+1)dx$
[5] $\int \frac{x}{(x+3)^2}dx$
2009 Serbia Team Selection Test, 1
Let $ \alpha$ and $ \beta$ be the angles of a non-isosceles triangle $ ABC$ at points $ A$ and $ B$, respectively. Let the bisectors of these angles intersect opposing sides of the triangle in $ D$ and $ E$, respectively. Prove that the acute angle between the lines $ DE$ and $ AB$ isn't greater than $ \frac{|\alpha\minus{}\beta|}3$.
2024 CMIMC Integration Bee, 5
\[\int_1^e \frac{2x^2+1}{x^3+x\log(x)}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2010 Today's Calculation Of Integral, 531
(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that
\[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\]
Explain the fact by using graph. Note that you don't need to prove the statement.
(2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$,
Prove that there exists $ \theta$ such that
\[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]
1990 Putnam, B1
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]
2005 Today's Calculation Of Integral, 28
Evaluate
\[\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx\]
2003 IMC, 2
Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)
2004 USAMO, 5
Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.
Today's calculation of integrals, 852
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows.
(1) $g_n(x)=(1+x)^n$
(2) $g_n(x)=\sin n\pi x$
(3) $g_n(x)=e^{nx}$
2009 Today's Calculation Of Integral, 420
Let $ K$ be the figure bounded by the curve $ y\equal{}e^x$ and 3 lines $ x\equal{}0,\ x\equal{}1,\ y\equal{}0$ in the $ xy$ plane.
(1) Find the volume of the solid formed by revolving $ K$ about the $ x$ axis.
(2) Find the volume of the solid formed by revolving $ K$ about the $ y$ axis.
1981 Spain Mathematical Olympiad, 5
Given a nonzero natural number $n$, let $f_n$ be the function of the closed interval $[0, 1]$ in $R$ defined like this:
$$f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}$$
a) Represent the function graphically.
b) Calculate $A_n =\int_0^1 f_n(x) dx$.
c) Find, if it exists, $\lim_{n\to \infty} A_n$ .
2024 CMIMC Integration Bee, 7
\[\int_1^2 \frac{\sqrt{1-\frac 1x}}{x-1}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
Today's calculation of integrals, 873
Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$
(1) Find the condition for which $C_1$ is inscribed in $C_2$.
(2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$.
(3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$.
60 point
1999 IberoAmerican, 1
Let $B$ be an integer greater than 10 such that everyone of its digits belongs to the set $\{1,3,7,9\}$. Show that $B$ has a [b]prime divisor[/b] greater than or equal to 11.
2012 India National Olympiad, 4
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a given triangle $ABC$.
2007 F = Ma, 33
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$.
Find the maximum value of $\beta$.
$ \textbf{(A)}\ 1$
$ \textbf{(B)}\ \sqrt{2}$
$ \textbf{(C)}\ 1/\sqrt{2}$
$ \textbf{(D)}\ \beta \text{ does not attain a maximum value}$
$ \textbf{(E)}\ \text{none of the above}$
1993 AMC 12/AHSME, 19
How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1$?
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $
2008 Harvard-MIT Mathematics Tournament, 9
([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty}
n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)}
\left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.
2010 German National Olympiad, 5
The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$
2010 Today's Calculation Of Integral, 641
Evaluate
\[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\]
Own
2007 IberoAmerican Olympiad For University Students, 3
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
2011 Today's Calculation Of Integral, 763
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$