Found problems: 1687
2010 Today's Calculation Of Integral, 589
Evaluate $ \int_0^1 \frac{x}{\{(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\plus{}(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.
Today's calculation of integrals, 858
On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$.
$(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$.
$(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.
2011 Today's Calculation Of Integral, 718
Find $\sum_{n=1}^{\infty} \frac{1}{2^n}\int_{-1}^1 (1-x)^2(1+x)^n dx\ (n\geq 1).$
2010 Today's Calculation Of Integral, 531
(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that
\[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\]
Explain the fact by using graph. Note that you don't need to prove the statement.
(2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$,
Prove that there exists $ \theta$ such that
\[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]
1941 Putnam, B6
Assuming that $f(x)$ is continuous in the interval $(0,1)$, prove that
$$\int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x)f(y)f(z)\;dz dy dx= \frac{1}{6}\left(\int_{0}^{1} f(t)\; dt\right)^{3}.$$
2006 Moldova National Olympiad, 12.2
Let $a, b, n \in \mathbb{N}$, with $a, b \geq 2.$ Also, let $I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx $ and $I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.$ Find $\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.$
2009 Today's Calculation Of Integral, 495
Evaluate the following definite integrals.
(1) $ \int_0^{\frac {1}{2}} \frac {x^2}{\sqrt {1 \minus{} x^2}}\ dx$
(2) $ \int_0^1 \frac {1 \minus{} x}{(1 \plus{} x^2)^2}\ dx$
(3) $ \int_{ \minus{} 1}^7 \frac {dx}{1 \plus{} \sqrt [3]{1 \plus{} x}}$
2022 CMIMC Integration Bee, 9
\[\int_e^{e^2} (\log(x))^{\log(x)}(2+\log(\log(x)))\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2007 Mathematics for Its Sake, 3
Let be three positive real numbers $ a,b,c, $ a natural number $ n, $ and the functions $ f:\mathbb{R}\longrightarrow\mathbb{R} ,g:(0,\infty )\longrightarrow\mathbb{R} $ defined as:
$$ f(x)=\frac{2(n+1)x^n(x^{n+1}-a) +nx^{n+1} +2a^2x+a}{x^{2n+2}-2ax^{n+1} +a^2x^2+a^2} , $$
$$ g(x)=\frac{a+bx^n}{x+cx^{2n+1}} $$
Calculate the antiderivatives of $ f $ and $ g. $
[i]Nicolae Sanda[/i]
2010 Today's Calculation Of Integral, 592
Prove the following inequality.
\[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]
1966 Miklós Schweitzer, 4
Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that
a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and
b) $ I$ contains of a polynomial with constant term $ 1$.
Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$.
[i]Gy. Szekeres[/i]
2012 Today's Calculation Of Integral, 809
For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$.
Find the maximum area of $S(a)$.
2007 Today's Calculation Of Integral, 172
Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$
2009 Today's Calculation Of Integral, 463
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx$.
2010 Putnam, A6
Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
2010 Morocco TST, 2
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$
1964 Miklós Schweitzer, 2
Let $ p$ be a prime and let \[ l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ .\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $ (\xi,\eta)$ of integers, not both divisible by $ p$, the values $ l_k(\xi,\eta), \;1\leq k\leq p^2 $, represent every residue class $ \textrm{mod} \;p$ exactly $ p$ times. Prove that the set of pairs $ \{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $ \textrm{mod} \;p$ with the set $ \{(m,n): 0\leq m,n \leq p\minus{}1 \}.$
PEN E Problems, 27
Prove that for each positive integer $n$, there exist $n$ consecutive positive integers none of which is an integral power of a prime number.
2007 Today's Calculation Of Integral, 229
Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.
2007 Bulgaria National Olympiad, 3
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
2007 Today's Calculation Of Integral, 236
Let $a$ be a positive constant. Evaluate the following definite integrals $A,\ B$.
\[A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx\].
[i]1998 Shinsyu University entrance exam/Textile Science[/i]
2003 Romania National Olympiad, 2
Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities.
$$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$
[i]Titu Andreescu[/i]
1999 Harvard-MIT Mathematics Tournament, 1
Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0$, $f(0)=19$, and $f(1)=99$.
2021 Science ON grade XII, 1
Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that
\begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\
g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*}
and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\
[i] (Nora Gavrea)[/i]
2006 Taiwan National Olympiad, 3
If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.