Found problems: 1687
1956 AMC 12/AHSME, 43
The number of scalene triangles having all sides of integral lengths, and perimeter less than $ 13$ is:
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$
2009 Today's Calculation Of Integral, 444
Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$
2024 CMIMC Integration Bee, 12
\[\int_1^\infty \frac{\sec^{-1}(x^{2})-\sec^{-1}(\sqrt x)}{x\log(x)}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2011 Today's Calculation Of Integral, 709
Evaluate $ \int_0^1 \frac{x}{1\plus{}x}\sqrt{1\minus{}x^2}\ dx$.
2010 Today's Calculation Of Integral, 543
Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$.
(1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies.
(2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.
2011 Today's Calculation Of Integral, 701
Evaluate
\[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]
2010 Today's Calculation Of Integral, 627
Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(2\sin \theta +1)\cos ^ 3 \theta}{(\sin ^ 2 \theta +1)^2}d\theta .$
[i]Proposed by kunny[/i]
2012 NIMO Problems, 2
For which positive integer $n$ is the quantity $\frac{n}{3} + \frac{40}{n}$ minimized?
[i]Proposed by Eugene Chen[/i]
1949 Miklós Schweitzer, 2
Compute
$ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .
2003 All-Russian Olympiad, 1
Let $\alpha , \beta , \gamma , \delta$ be positive numbers such that for all $x$, $\sin{\alpha x}+\sin {\beta x}=\sin {\gamma x}+\sin {\delta x}$. Prove that $\alpha =\gamma$ or $\alpha=\delta$.
2012 Kyoto University Entry Examination, 1B
Let $n\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\ X_2,\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once a card taken out, it is not taken a back.
2013 Today's Calculation Of Integral, 876
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition :
1) $f(-1)\geq f(1).$
2) $x+f(x)$ is non decreasing function.
3) $\int_{-1}^ 1 f(x)\ dx=0.$
Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$
2024 Romania National Olympiad, 3
Let $f:[0,1] \to \mathbb{R}$ be a continuous function with $f(1)=0.$ Prove that the limit $$\lim_{t \nearrow 1} \left( \frac{1}{1-t} \int\limits_0^1x(f(tx)-f(x)) \mathrm{d}x\right)$$ exists and find its value.
2007 Putnam, 2
Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$
Prove that for every $ \alpha\in(0,1),$
\[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]
2010 Today's Calculation Of Integral, 534
Find the indefinite integral $ \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx$.
2023 CMIMC Integration Bee, 6
\[\int_0^2 e^x(x^4+8x^3+18x^2+16x+5)\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2011 Today's Calculation Of Integral, 722
Find the continuous function $f(x)$ such that :
\[\int_0^x f(t)\left(\int_0^t f(t)dt\right)dt=f(x)+\frac 12\]
2007 Today's Calculation Of Integral, 199
Let $m,\ n$ be non negative integers.
Calculate
\[\sum_{k=0}^{n}(-1)^{k}\frac{n+m+1}{k+m+1}\ nC_{k}. \]
where $_{i}C_{j}$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
2009 Putnam, B5
Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that
\[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\]
Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$
1997 USAMO, 1
Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define
\[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \]
where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.
2012 Today's Calculation Of Integral, 797
In the $xyz$-space take four points $P(0,\ 0,\ 2),\ A(0,\ 2,\ 0),\ B(\sqrt{3},-1,\ 0),\ C(-\sqrt{3},-1,\ 0)$.
Find the volume of the part satifying $x^2+y^2\geq 1$ in the tetrahedron $PABC$.
50 points
2010 Today's Calculation Of Integral, 643
Evaluate
\[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\]
Own
2005 Today's Calculation Of Integral, 19
Calculate the following indefinite integrals.
[1] $\int \tan ^ 3 x dx$
[2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$
[3] $\int \cos ^ 5 x dx$
[4] $\int \sin ^ 2 x\cos ^ 3 x dx$
[5]$ \int \frac{dx}{\sin x}$
2009 Today's Calculation Of Integral, 469
Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.
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 $\frac{11}{3}.$