Found problems: 85335
Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.
Let $P$ be an arbitrary point inside $\triangle ABC$ with sides $3,7,8$. What is the probability that the distance of $P$ to at least one vertices of the triangle is less than $1$?
$ \textbf{(A)}\ \frac{\pi}{36}\sqrt 2
\qquad\textbf{(B)}\ \frac{\pi}{36}\sqrt 3
\qquad\textbf{(C)}\ \frac{\pi}{36}
\qquad\textbf{(D)}\ \frac12
\qquad\textbf{(E)}\ \frac 34
$
$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?
For what value of $ m $ is the polynomial $ x^3 + y^3 + z^3 + mxyz $ divisible by $ x + y + z $?
Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: \[\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\] are equal if and only if $n+1$ is a prime number.
[i]Laurentiu Panaitopol[/i]
Prove that the equations $$x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0$$ imply the equation $xy-12x+15y=0$.
Robot "Mag-o-matic" manipulates $101$ glasses, displaced in a row whose positions are numbered from $1$ to $101$. In each glass you can find a ball or not. Mag-o-matic only accepts elementary instructions of the form $(a;b,c)$, which it interprets as
"consider the glass in position $a$: if it contains a ball, then switch the glasses in positions $b$ and $c$ (together with their own content), otherwise move on to the following instruction"
(it means that $a,\,b,\,c$ are integers between $1$ and $101$, with $b$ and $c$ different from each other but not necessarily different from $a$). A $\emph{programme}$ is a finite sequence of elementary instructions, assigned at the beginning, that Mag-o-matic does one by one.
A subset $S\subseteq \{0,\,1,\,2,\dots,\,101\}$ is said to be $\emph{identifiable}$ if there exists a programme which, starting from any initial configuration, produces a final configuration in which the glass in position $1$ contains a ball if and only if the number of glasses containing a ball is an element of $S$.
(a) Prove that the subset $S$ of the odd numbers is identifiable.
(b) Determine all subsets $S$ that are identifiable.
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] $\int (\ln x)^2 dx$
Find the number of pairs of positive integers $a$ and $b$ such that $a\leq 100\,000$, $b\leq 100\,000$, and
$$
\frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}.
$$
In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.
Each number in the second, third, and further rows of the following triangle:
[img]https://cdn.artofproblemsolving.com/attachments/1/5/589d9266749477b0f56f0f503d4f18a6e5d695.png[/img]
is equal to the difference of two neighbouring numbers standing above it. Find the last number (at the bottom of the triangle).
(GW Leibnitz,)
Let $f:\mathbb{R}\to \mathbb{R}$ be a function. Suppose that for every $\varepsilon >0$ , there exists a function $g:\mathbb{R}\to (0,\infty)$ such that for every pair $(x,y)$ of real numbers,
if $|x-y|<\text{min}\{g(x),g(y)\}$, then $|f(x)-f(y)|<\varepsilon$
Prove that $f$ is pointwise limit of a squence of continuous $\mathbb{R}\to \mathbb{R}$ functions i.e., there is a squence $h_1,h_2,...,$ of continuous $\mathbb{R}\to \mathbb{R}$ such that $\lim_{n\to \infty}h_n(x)=f(x)$ for every $x\in \mathbb{R}$
The natural number $n$ can be replaced by $ab$ if $a + b = n$, where $a$ and $b$ are natural numbers. Can the number $2001$ be obtained from $22$ after a sequence of such replacements?
There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?
Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$
Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$.
[img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]
A non-isosceles triangle $\triangle ABC$ has incenter $I$ and the incircle hits $BC, CA, AB$ at $D, E, F$.
Let $EF$ hit the circumcircle of $CEI$ at $P \not= E$. Prove that $\triangle ABC = 2 \triangle ABP$.
Find all $n\in \mathbb Z^+$, so that \[ z_n = \underbrace{ 101\dots101}_{2n+1 \text{ digits} } \] is prime.
Two non-zero-complex numbers $x,y$, satisfy that $x^2+xy+y^2=0$. Then the value of $(\frac{x}{x+y})^{1990}+(\frac{y}{x+y})^{1990}$ is
$\text{(A)}2^{-1989}\qquad\text{(B)}-1\qquad\text{(C)}1\qquad\text{(D)}$none above
Let $ \vartriangle ABC $ be a triangle with $ \angle ACB = 60º $. Let $ E $ be a point inside $ \overline {AC} $ such that $ CE <BC $. Let $ D $ over $ \overline {BC} $ such that $$ \frac {AE} {BD} = \frac {BC} {CE} -1 .$$ Let us call $ P $ the intersection of $ \overline {AD} $ with $ \overline {BE} $ and $ Q $ the other point of intersection of the circumcircles of the triangles $ AEP $ and $ BDP $. Prove that $QE \parallel BC $.
Show that if the series $$\sum_{n=1}^{\infty} \frac{1}{p_n}$$ is convergent, where $p_1,p_2,p_3,\dots, p_n, \dots$ are positive real numbers, then the series $$\sum_{n=1}^{\infty} \frac{n^2}{(p_1+p_2+\dots +p_n)^2}p_n$$ is also convergent.
Let $c$ be fixed natural number. Sequence $(a_n)$ is defined by:
$a_1=1$, $a_{n+1}=d(a_n)+c$ for $n=1,2,...$.
where $d(m)$ is number of divisors of $m$. Prove that there exist $k$ natural such that sequence $a_k,a_{k+1},...$ is periodic.
It is given that the roots of the polynomial $P(z) = z^{2019} - 1$ can be written in the form $z_k = x_k + iy_k$ for $1\leq k\leq 2019$. Let $Q$ denote the monic polynomial with roots equal to $2x_k + iy_k$ for $1\leq k\leq 2019$. Compute $Q(-2)$.
Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \]
[i] Nikolai Nikolov[/i]
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.