Found problems: 1687
2012 Romania National Olympiad, 1
[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]
1962 AMC 12/AHSME, 9
When $ x^9\minus{}x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:
$ \textbf{(A)}\ \text{more than 5} \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 2$
2014 Contests, 901
Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$
2022 CMIMC Integration Bee, 11
\[\int_0^{\pi/2} \frac{\sin(x)}{2-\sin(x)\cos(x)}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
1998 USAMTS Problems, 5
The figure on the right shows the ellipse $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$.
Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$.
[asy]
defaultpen(linewidth(0.7));
pair c=(19,98);
real dist = 30;
real a = sqrt(1998*19),b=sqrt(1998*98);
xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow);
yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow);
draw(ellipse(c,a,b));
label("$R_1$",(100,200));
label("$R_2$",(-80,200));
label("$R_3$",(-60,-150));
label("$R_4$",(70,-150));[/asy]
2009 Today's Calculation Of Integral, 520
Let $ a,\ b,\ c$ be postive constants.
Evaluate $ \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx$.
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$.
PEN S Problems, 32
Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him the value of $a_{1}x_{1}+\cdots+a_{n}x_{n}$. Then Bob chooses another list of positive integers $b_{1}$, $\cdots$, $b_{n}$ and asks Alice for $b_{1}x_{1}+\cdots+b_{n}x_{n}$. Play continues in this way until Bob is able to determine Alice's numbers. How many rounds will Bob need in order to determine Alice's numbers?
2010 Tuymaada Olympiad, 1
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?
2010 Contests, 2
Compute the sum of the series
$\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$
2011 Today's Calculation Of Integral, 695
For a positive integer $n$, let
\[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\]
Answer the following questions:
(1) Show the following inequality.
\[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\]
(2) Express $T_n-2S_n$ in terms of $n$.
(3) Find the limit $\lim_{n\to\infty} T_n.$
2013 Romania National Olympiad, 4
a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$.
b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that
$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$
2010 District Olympiad, 4
Let $ f: [0,1]\rightarrow \mathbb{R}$ a derivable function such that $ f(0)\equal{}f(1)$, $ \int_0^1f(x)dx\equal{}0$ and $ f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]$.
i)Prove that the function $ g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x$ is strictly decreasing.
ii)Prove that for each integer number $ n\ge 1$, we have:
$ \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}$
2016 Danube Mathematical Olympiad, 3
3. Let n > 1 be an integer and $a_1, a_2, . . . , a_n$ be positive reals with sum 1.
a) Show that there exists a constant c ≥ 1/2 so that
$\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$,
where $a_0 = 0$.
b) Show that ’the best’ value of c is at least $\frac{\pi}{4}$.
PEN S Problems, 38
The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.
2011 Today's Calculation Of Integral, 676
Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$.
Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$.
[i]2011 Tokyo University of Science entrance exam/Management[/i]
2007 IMC, 6
Let $ f \ne 0$ be a polynomial with real coefficients. Define the sequence $ f_{0}, f_{1}, f_{2}, \ldots$ of polynomials by $ f_{0}= f$ and $ f_{n+1}= f_{n}+f_{n}'$ for every $ n \ge 0$. Prove that there exists a number $ N$ such that for every $ n \ge N$, all roots of $ f_{n}$ are real.
2019 Korea USCM, 3
Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]
2005 Romania National Olympiad, 3
Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that
\[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \]
[i]Radu Miculescu[/i]
1998 Harvard-MIT Mathematics Tournament, 3
Find the area of the region bounded by the graphs $y=x^2$, $y=x$, and $x=2$.
2011 Today's Calculation Of Integral, 735
Evaluate the following definite integrals:
(a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$
(b) $\int_0^{\frac 13} xe^{3x}\ dx$
(c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$
(d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$
2009 Jozsef Wildt International Math Competition, W. 7
If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$
1997 China Team Selection Test, 3
Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies:
[b]I.[/b] $a_0 = 1, a_1 = 337$;
[b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$;
[b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.
2013 Today's Calculation Of Integral, 881
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.
2009 Today's Calculation Of Integral, 459
Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.