Found problems: 1687
2008 Harvard-MIT Mathematics Tournament, 1
How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?
2012 Today's Calculation Of Integral, 853
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
2014 Tuymaada Olympiad, 8
Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that
\[ g(a, b, c)\ge \sqrt{2abc}\]
[i](M. Ivanov)[/i]
[hide="Remarks (containing spoilers!)"]
1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$.
2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$.
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2023 AIME, 6
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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2001 Romania National Olympiad, 3
Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that:
a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$.
b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.
2007 F = Ma, 6
At time $t = 0$ a drag racer starts from rest at the origin and moves along a straight line with velocity given by $v = 5t^2$, where $v$ is in $\text{m/s}$ and $t$ in $\text{s}$. The expression for the displacement of the car from $t = 0$ to time $t$ is
$ \textbf{(A)}\ 5t^3 \qquad\textbf{(B)}\ 5t^3/3\qquad\textbf{(C)}\ 10t \qquad\textbf{(D)}\ 15t^2 \qquad\textbf{(E)}\ 5t/2 $
1981 Putnam, B6
Let $C$ be a fixed unit circle in the cartesian plane. For any convex polygon $P$ , each of whose sides is tangent to $C$, let $N( P, h, k)$ be the number of points common to $P$ and the unit circle with center at $(h, k).$ Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P).$ Find the smallest number $u$ with
$$ \frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u$$
for all polygons $P$, where the double integral is taken over $H(P).$
2009 Today's Calculation Of Integral, 448
Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$
2010 Today's Calculation Of Integral, 619
Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized.
Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$
[i]2010 Saitama University entrance exam/Mathematics[/i]
Last Edited
2005 Today's Calculation Of Integral, 32
Evaluate
\[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]
2010 Today's Calculation Of Integral, 652
Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$
Evaluate
\[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\]
[i]1981 Tokyo University, Master Course[/i]
2020 Jozsef Wildt International Math Competition, W33
Let $p\in\mathbb N,f:[0,1]\to(0,\infty)$ be a continuous function and
$$a_n=\int^1_0x^p\sqrt[n]{f(x)}dx,n\in\mathbb N,n\ge2.$$
Demonstrate that:
a) $\lim_{n\to\infty}a_n=\frac1{p+1}$
b) $\lim_{n\to\infty}((p+1)a_n)^n=\exp\left((p+1)\int^1_0x^p\ln f(x)dx\right)$
[i]Proposed by Nicolae Papacu[/i]
2007 AIME Problems, 7
Let \[N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).\] Find the remainder when N is divided by 1000. (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to x, and $\lceil x \rceil$ denotes the least integer that is greater than or equal to x.)
2005 ISI B.Math Entrance Exam, 6
Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that
$\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$
Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$
2024 CMIMC Integration Bee, 14
\[\int_1^\infty \frac{\lfloor x \rfloor}{9x^3-x}\mathrm dx\]
[i]Proposed by Robert Trosten and Connor Gordon[/i]
2015 District Olympiad, 2
[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $
[b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $
[i]Florin Stănescu[/i]
2011 Today's Calculation Of Integral, 755
Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane.
Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.
2009 Today's Calculation Of Integral, 397
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis
2007 Stanford Mathematics Tournament, 15
Evaluate $\int_{0}^{\infty}\frac{\tan^{-1}(\pi x)-\tan^{-1}x}{x}dx$
2007 Today's Calculation Of Integral, 207
Evaluate the following definite integral.
\[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]
2007 Today's Calculation Of Integral, 204
Evaluate
\[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]
Today's calculation of integrals, 882
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.
2007 Today's Calculation Of Integral, 243
A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.
2021 CMIMC Integration Bee, 11
$$\int_0^\frac{\pi}{2}\frac{1}{4-3\cos^2(x)}\,dx$$
[i]Proposed by Connor Gordon[/i]
2011 Today's Calculation Of Integral, 726
Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.