Found problems: 2215
2010 ISI B.Math Entrance Exam, 8
Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that
$\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.
1999 Harvard-MIT Mathematics Tournament, 4
$f$ is a continuous real-valued function such that $f(x+y)=f(x)f(y)$ for all real $x$, $y$. If $f(2)=5$, find $f(5)$.
PEN O Problems, 1
Suppose all the pairs of a positive integers from a finite collection \[A=\{a_{1}, a_{2}, \cdots \}\] are added together to form a new collection \[A^{*}=\{a_{i}+a_{j}\;\; \vert \; 1 \le i < j \le n \}.\] For example, $A=\{ 2, 3, 4, 7 \}$ would yield $A^{*}=\{ 5, 6, 7, 9, 10, 11 \}$ and $B=\{ 1, 4, 5, 6 \}$ would give $B^{*}=\{ 5, 6, 7, 9, 10, 11 \}$. These examples show that it's possible for different collections $A$ and $B$ to generate the same collections $A^{*}$ and $B^{*}$. Show that if $A^{*}=B^{*}$ for different sets $A$ and $B$, then $|A|=|B|$ and $|A|=|B|$ must be a power of $2$.
1998 Miklós Schweitzer, 6
Let U be the union of a finite number (not necessarily connected and not necessarily disjoint) of closed unit squares lying in the plane. Can the quotient of the perimeter and area of U be arbitrarily large?
@below: i think "single" means "connected".
2009 Today's Calculation Of Integral, 462
Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.
2004 Putnam, B5
Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
2008 ITest, 76
During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic, \[x^2-sx+p,\] with roots $r_1$ and $r_2$. He notices that \[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\cdots=r_1^{2007}+r_2^{2007}.\] He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of \[\dfrac1{r_1^{2008}}+\dfrac1{r_2^{2008}}.\] Help Michael by computing this maximum.
2022 Romania National Olympiad, P3
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]
Today's calculation of integrals, 865
Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.
2005 Today's Calculation Of Integral, 38
Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure
which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis.
If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.
1979 IMO Longlists, 60
Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$
2011 Today's Calculation Of Integral, 712
Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left\{\frac{1}{\tan x\ (\ln \sin x)}+\frac{\tan x}{\ln \cos x}\right\}\ dx.$
2010 Today's Calculation Of Integral, 656
Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$
2010 Gheorghe Vranceanu, 2
Let be a natural number $ n, $ a nonzero number $ \alpha, \quad n $ numbers $ a_1,a_2,\ldots ,a_n $ and $ n+1 $ functions $ f_0,f_1,f_2,\ldots ,f_n $ such that $ f_0=\alpha $ and the rest are defined recursively as
$$ f_k (x)=a_k+\int_0^x f_{k-1} (x)dx . $$
Prove that if all these functions are everywhere nonnegative, then the sum of all these functions is everywhere nonnegative.
2005 AMC 12/AHSME, 11
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
$ \textbf{(A)}\ 41\qquad
\textbf{(B)}\ 42\qquad
\textbf{(C)}\ 43\qquad
\textbf{(D)}\ 44\qquad
\textbf{(E)}\ 45$
2010 CIIM, Problem 4
Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$, and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\displaystyle \lim_{n\to+\infty} d(A_n) = 0$, where $d(A)$ is the diameter of the set $A$.
Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$
2010 Today's Calculation Of Integral, 649
Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$,
$I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$
Find $\lim_{n\to\infty} I_n.$
[i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]
2019 Jozsef Wildt International Math Competition, W. 48
Let $f : (0,+\infty) \to \mathbb{R}$ a convex function and $\alpha, \beta, \gamma > 0$. Then $$\frac{1}{6\alpha}\int \limits_0^{6\alpha}f(x)dx\ +\ \frac{1}{6\beta}\int \limits_0^{6\beta}f(x)dx\ +\ \frac{1}{6\gamma}\int \limits_0^{6\gamma}f(x)dx$$ $$\geq \frac{1}{3\alpha +2\beta +\gamma}\int \limits_0^{3\alpha +2\beta +\gamma}f(x)dx\ +\ \frac{1}{\alpha +3\beta +2\gamma}\int \limits_0^{\alpha +3\beta +2\gamma}f(x)dx\ $$ $$+\ \frac{1}{2\alpha +\beta +3\gamma}\int \limits_0^{2\alpha +\beta +3\gamma}f(x)dx$$
2005 Brazil Undergrad MO, 6
Prove that for any natural numbers $0 \leq i_1 < i_2 < \cdots < i_k$ and $0 \leq j_1 < j_2 < \cdots < j_k$, the matrix $A = (a_{rs})_{1\leq r,s\leq k}$, $a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}$ ($1\leq r,s\leq k$) is nonsingular.
1991 Arnold's Trivium, 1
Sketch the graph of the derivative and the graph of the integral of a function given by a free-hand graph.
2005 Today's Calculation Of Integral, 13
Calculate the following integarls.
[1] $\int x\cos ^ 2 x dx$
[2] $\int \frac{x-1}{(3x-1)^2}dx$
[3] $\int \frac{x^3}{(2-x^2)^4}dx$
[4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$
[5] $\int (\ln x)^2 dx$
2007 Today's Calculation Of Integral, 181
For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$
2011 Today's Calculation Of Integral, 760
Prove that there exists a positive integer $n$ such that $\int_0^1 x\sin\ (x^2-x+1)dx\geq \frac {n}{n+1}\sin \frac{n+2}{n+3}.$
2021 Nigerian Senior MO Round 3, 5
Let $f(x)=\frac{P(x)}{Q(x)}$. Where $P(x), Q(x)$ are two non constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$ for all infinitely many values of $x$.
a. Show that $deg(P) <deg(Q).$
b. Show that $P'(1)=2Q'(1)- deg(Q). Q(1)$
Here $P'(x)$ denotes the derivatives of $P(x)$ as usual
2005 Today's Calculation Of Integral, 25
Let $|a|<\frac{\pi}{2}$.
Evaluate
\[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]