Found problems: 348
2010 Romania National Olympiad, 4
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and
\[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\]
Prove that
a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$.
b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$.
[i]Calin Popescu[/i]
2024 Mexican University Math Olympiad, 6
Let \( p \) be a monic polynomial with all distinct real roots. Show that there exists \( K \) such that
\[
(p(x)^2)'' \leq K(p'(x))^2.
\]
1986 Iran MO (2nd round), 2
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\]
Is the function $g$ continuous in the point $x=0 \ ?$
[b](d)[/b] Sketch the diagram of $g.$
Today's calculation of integrals, 893
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
2005 Today's Calculation Of Integral, 41
Evaluate
\[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]
VII Soros Olympiad 2000 - 01, 11.3
The function $F (x)$ is defined on $R$ and has a second derivative for each value of the variable. Prove that there is a point $x_0$ such that the product $ F(x_0) F''(x_0)$ is non-negative.
PS. In my [url=http://www.1543.su/olympiads/soros/20002001/1/1soros00.htm]source[/url], it is not clear if it means $ F(x_0) F''(x_0)$ or $ F(x_0) F'(x_0)$.
2008 USAPhO, 4
Two beads, each of mass $m$, are free to slide on a rigid, vertical hoop of mass $m_h$. The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio $m/m_h$ such that the hoop always remains in contact with the ground? Neglect friction.
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2012 Today's Calculation Of Integral, 785
For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$
1950 AMC 12/AHSME, 17
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline
x&0&1&2&3&4\\\hline
y&100&90&70&40&0\\\hline
\end{tabular}\]
$\textbf{(A)}\ y=100-10x \qquad
\textbf{(B)}\ y=100-5x^2 \qquad
\textbf{(C)}\ y=100-5x-5x^2 \qquad\\
\textbf{(D)}\ y=20-x-x^2 \qquad
\textbf{(E)}\ \text{None of these}$
2010 Today's Calculation Of Integral, 603
Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$.
Please solve the problem without using partial differentiation for those who don't learn it.
1961 Waseda University entrance exam/Science and Technology
2005 Putnam, A5
Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$
1989 China Team Selection Test, 3
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
2005 Georgia Team Selection Test, 3
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
2007 Princeton University Math Competition, 9
Find the value of $x+y$ for which the expression
\[\frac{6x^2}{y^6} + \frac{6y^2}{x^6}+9x^2y^2+\frac{4}{x^6y^6}\]
is minimized.
2007 Romania National Olympiad, 4
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$.
a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$.
b) Give an example of a non-constant function $f$ with property $(P)$.
c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.
2014 Online Math Open Problems, 25
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$, find $p+q$.
[i]Proposed by Michael Kural[/i]
2007 German National Olympiad, 6
For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$
2007 ISI B.Stat Entrance Exam, 2
Use calculus to find the behaviour of the function
\[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\]
and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.
2006 Romania National Olympiad, 1
Find the maximal value of \[ \left( x^3+1 \right) \left( y^3 + 1\right) , \] where $x,y \in \mathbb R$, $x+y=1$.
[i]Dan Schwarz[/i]
2014 Taiwan TST Round 2, 1
Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$,
\[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]
2008 Romania National Olympiad, 1
Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have
\[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.
2005 Georgia Team Selection Test, 3
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
PEN I Problems, 8
Prove that $\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}+\sqrt[3]{n+2}\rfloor =\lfloor \sqrt[3]{27n+26}\rfloor$ for all positive integers $n$.
1996 China National Olympiad, 2
Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that
\[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]
1958 AMC 12/AHSME, 46
For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression
\[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2}
\]
has:
$ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\
\textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\
\textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\
\textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\
\textbf{(E)}\ \text{a maximum value of }{\minus{}1}$