Found problems: 348
1962 Vietnam National Olympiad, 2
Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.
2005 ISI B.Stat Entrance Exam, 6
Let $f$ be a function defined on $(0, \infty )$ as follows:
\[f(x)=x+\frac1x\]
Let $h$ be a function defined for all $x \in (0,1)$ as
\[h(x)=\frac{x^4}{(1-x)^6}\]
Suppose that $g(x)=f(h(x))$ for all $x \in (0,1)$.
(a) Show that $h$ is a strictly increasing function.
(b) Show that there exists a real number $x_0 \in (0,1)$ such that $g$ is strictly decreasing in the interval $(0,x_0]$ and strictly increasing in the interval $[x_0,1)$.
Today's calculation of integrals, 764
Find $\lim_{n\to\infty} \int_0^{\pi} e^{x}|\sin nx|dx.$
2005 Harvard-MIT Mathematics Tournament, 2
How many real numbers $x$ are solutions to the following equation? \[ 2003^x + 2004^x = 2005^x \]
1999 Putnam, 4
Sum the series \[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]
2003 Greece National Olympiad, 1
If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]
1982 IMO Longlists, 47
Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$).
2006 Stanford Mathematics Tournament, 2
Find the minimum value of $ 2x^2\plus{}2y^2\plus{}5z^2\minus{}2xy\minus{}4yz\minus{}4x\minus{}2z\plus{}15$ for real numbers $ x$, $ y$, $ z$.
2010 Romania National Olympiad, 1
Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by
\[F(x)=\int_0^xf(t)\ \text{d}t.\]
Prove that if $F$ has a finite derivative, then $f$ is continuous.
[i]Dorin Andrica & Mihai Piticari[/i]
2013 Stanford Mathematics Tournament, 7
The function $f(x)$ has the property that, for some real positive constant $C$, the expression \[\frac{f^{(n)}(x)}{n+x+C}\] is independent of $n$ for all nonnegative integers $n$, provided that $n+x+C\neq 0$. Given that $f'(0)=1$ and $\int_{0}^{1}f(x) \, dx = C+(e-2)$, determine the value of $C$.
Note: $f^{(n)}(x)$ is the $n$-th derivative of $f(x)$, and $f^{(0)}(x)$ is defined to be $f(x)$.
2009 Today's Calculation Of Integral, 413
Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.
2008 AIME Problems, 14
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
2003 Moldova National Olympiad, 12.2
For every natural number $n\geq{2}$ consider the following affirmation $P_n$:
"Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots."
Are $P_4$ and $P_5$ both true? Justify your answer.
PEN G Problems, 27
Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.
1962 Miklós Schweitzer, 5
Let $ f$ be a finite real function of one variable. Let $ \overline{D}f$ and $ \underline{D}f$ be its upper and lower derivatives, respectively, that is, \[ \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}\] ,
\[ \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}.\] Show that $ \overline{D}f$ and $ \underline{D}f$ are Borel-measurable functions. [A. Csaszar]
1999 Putnam, 1
Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]
2013 ELMO Shortlist, 5
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]
2014 NIMO Problems, 8
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Proposed by Akshaj[/i]
2009 AIME Problems, 4
A group of children held a grape-eating contest. When the contest was over, the winner had eaten $ n$ grapes, and the child in $ k$th place had eaten $ n\plus{}2\minus{}2k$ grapes. The total number of grapes eaten in the contest was $ 2009$. Find the smallest possible value of $ n$.
2009 Putnam, A6
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
1957 AMC 12/AHSME, 36
If $ x \plus{} y \equal{} 1$, then the largest value of $ xy$ is:
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 0.5\qquad
\textbf{(C)}\ \text{an irrational number about }{0.4}\qquad
\textbf{(D)}\ 0.25\qquad
\textbf{(E)}\ 0$
2005 Morocco TST, 1
Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying :
$(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$
2007 ITAMO, 6
a) For each $n \ge 2$, find the maximum constant $c_{n}$ such that
$\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
b) For each $n \ge 2$, find the maximum constant $d_{n}$ such that
$\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
2018 Moscow Mathematical Olympiad, 1
The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this
square trinomial has?
1989 APMO, 5
Determine all functions $f$ from the reals to the reals for which
(1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$,
where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)