Found problems: 2215
2009 Today's Calculation Of Integral, 484
Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$.
(1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively.
(2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.
2021 CMIMC Integration Bee, 12
$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$
[i]Proposed by Vlad Oleksenko[/i]
2010 Today's Calculation Of Integral, 632
Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$
[i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]
2006 IberoAmerican Olympiad For University Students, 7
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.
2003 AMC 12-AHSME, 25
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?
$ \textbf{(A)}\ \frac{1}{36} \qquad
\textbf{(B)}\ \frac{1}{24} \qquad
\textbf{(C)}\ \frac{1}{18} \qquad
\textbf{(D)}\ \frac{1}{12} \qquad
\textbf{(E)}\ \frac{1}{9}$
1992 India National Olympiad, 6
Let $f(x)$ be a polynomial in $x$ with integer coefficients and suppose that for five distinct integers $a_1, \ldots, a_5$ one has $f(a_1) = f(a_2) = \ldots = f(a_5) = 2$. Show that there does not exist an integer $b$ such that $f(b) = 9$.
1960 AMC 12/AHSME, 28
The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has:
$ \textbf{(A)}\ \text{infinitely many integral roots} \qquad\textbf{(B)}\ \text{no root} \qquad\textbf{(C)}\ \text{one integral root}\qquad$
$\textbf{(D)}\ \text{two equal integral roots} \qquad\textbf{(E)}\ \text{two equal non-integral roots} $
2009 Today's Calculation Of Integral, 460
$ \int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ dx$.
2010 China National Olympiad, 3
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.
2010 Contests, 2
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
2011 Albania Team Selection Test, 2
The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.
2006 District Olympiad, 1
Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]
2013 Today's Calculation Of Integral, 887
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows.
Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find $f(\sqrt{3})$
(2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$
(3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.
2007 Harvard-MIT Mathematics Tournament, 6
The elliptic curve $y^2=x^3+1$ is tangent to a circle centered at $(4,0)$ at the point $(x_0,y_0)$. Determine the sum of all possible values of $x_0$.
2004 Iran MO (3rd Round), 15
This problem is easy but nobody solved it.
point $A$ moves in a line with speed $v$ and $B$ moves also with speed $v'$ that at every time the direction of move of $B$ goes from $A$.We know $v \geq v'$.If we know the point of beginning of path of $A$, then $B$ must be where at first that $B$ can catch $A$.
2022 JHMT HS, 8
Let $P = (-4, 0)$ and $Q = (4, 0)$ be two points on the $x$-axis of the Cartesian coordinate plane, and let $X$ and $Y$ be points on the $x$-axis and $y$-axis, respectively, such that over all $Z$ on line $\overleftrightarrow{XY}$, the perimeter of $\triangle ZPQ$ has a minimum value of $25$. What is the smallest possible value of $XY^2$?
2009 Canada National Olympiad, 3
Define $f(x,y,z)=\frac{(xy+yz+zx)(x+y+z)}{(x+y)(y+z)(z+x)}$.
Determine the set of real numbers $r$ for which there exists a triplet of positive real numbers satisfying $f(x,y,z)=r$.
2008 Harvard-MIT Mathematics Tournament, 4
([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.
2007 Estonia Math Open Junior Contests, 4
Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers.
(a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter.
(b) Let K be an arbitrary integral disguisable triangle for which no smaller integral
disguisable triangle similar to it exists. Prove that at least two side lengths of K are
perfect squares.
2012 Today's Calculation Of Integral, 783
Define a sequence $a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)$.
(1) Find $a_n$.
(2) Let ${b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})$ for $k=1,\ 2,\ 3,\ \cdots$.
Prove that $\sum_{k=1}^n b_k<\sqrt{2}-1$ for each $n$.
Last Edited
2011 Moldova Team Selection Test, 1
Find all real numbers $x, y$ such that:
$y+3\sqrt{x+2}=\frac{23}2+y^2-\sqrt{49-16x}$
2009 Today's Calculation Of Integral, 471
Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.
2022 JHMT HS, 4
Consider the rectangle in the coordinate plane with corners $(0, 0)$, $(16, 0)$, $(16, 4)$, and $(0, 4)$. For a constant $x_0 \in [0, 16]$, the curves
\[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \]
partition this rectangle into four 2D regions. Over all choices of $x_0$, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition.
(The bottom-left region is $\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}$, and the top-right region is $\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}$.)
2012 Today's Calculation Of Integral, 796
Answer the following questions:
(1) Let $a$ be non-zero constant. Find $\int x^2 \cos (a\ln x)dx.$
(2) Find the volume of the solid generated by a rotation of the figures enclosed by the curve $y=x\cos (\ln x)$, the $x$-axis and
the lines $x=1,\ x=e^{\frac{\pi}{4}}$ about the $x$-axis.
2005 Today's Calculation Of Integral, 37
Evaluate
\[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]