Found problems: 85335
Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$.
Let $n$ be a positive integer. Suppose that the diophantine equation
\[z^n = 8 x^{2009} + 23 y^{2009} \]
uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.
Let $a, b$, and $c$ be real numbers. Let $u = a^2 + b^2 + c^2$ and $v = 2ab + 2bc + 2ca$. Suppose $2018u = 1001v + 1024$. Find the maximum possible value of $35a - 28b - 3c$.
let $ABC$ be a right angled triangle with inradius $1$
find the minimum area of triangle $ABC$
Let $\alpha$ be a root of the equation $x^3-5x+3=0$ and let $f(x)$ be a polynomial with rational coefficients. Prove that if $f(\alpha)$ be the root of equation above, then $f(f(\alpha))$ is a root, too.
a) Find :
$A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$
b) Prove that for any $(a,b,c) \in A$ next inequality hold :
\begin{align*}
\frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8}
\end{align*}
How many ordered pairs $(a, b)$ of positive integers satisfy the equation
$$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$
where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$
Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides.
[i]A. Berzin’sh, O. Musin[/i]
Let $ABCD$ be a fixed convex quadrilateral. Say a point $K$ is [i]pastanaga[/i] if there's a rectangle $PQRS$ centered at $K$ such that $A\in PQ, B\in QR, C\in RS, D\in SP$. Prove there exists a circle $\omega$ depending only on $ABCD$ that contains all pastanaga points.
Let $N$ and $d$ be two positive integers with $N\geq d+2$. There are $N$ countries connected via two-way direct flights, where each country is connected to exactly $d$ other countries. It is known that for any two different countries, it is possible to go from one to another via several flights. A country is \emph{important} if after removing it and all the $d$ countries it is connected to, there exist two other countries that are no longer connected via several flights.
Show that if every country is important, then one can choose two countries so that more than $2d/3$ countries are connected to both of them via direct flights.
[i]Proposed by usjl[/i]
Let $ABCDE$ be a convex pentagon and area of $\vartriangle ABC =$ area of $\vartriangle BCD =$ area of $\vartriangle CDE=$ area of $\vartriangle DEA =$ area of $\vartriangle EAB$. Given that area of $\vartriangle ABCDE = 2$. Evaluate the area of area of $\vartriangle ABC$.
Is it possible to represent number $2011... 2011$, where number $2011$ is written $20112011$ times, as a product of some number and sum of its digits?
Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$.
Find all $n$ such that the cardinality of $n$ is finite.
The following part problems may attract partial credit.
$\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$.
$\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$.
$\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality
of its trajectory is $k$.
$\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.
How many lattice points $P$ in or on the circle $x^2+y^2=25$ have the property that there exists a unique line with rational slope through $P$ that divides the circle into two parts with equal areas?
[i]Proposed by Nathan Ramesh
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.
Define $\eta(f)$ to be the number of roots that are repeated of the complex-valued polynomial $f$ (e.g. $\eta((x-1)^3\cdot(x+1)^4)=5$). Prove that for nonconstant, relatively prime $f,g\in\mathbb C[x]$,
$$\eta(f)+\eta(g)+\eta(f+g)<\deg f+\deg g$$
Let $n$ be a fixed positive integer. There are $n \geq 1$ lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer $i$ ($1 \leq i \leq n$) and switch the state of the first $i$ lamps from the left. Determine the smallest number $k$ with the property that we can make all of the lamps be switched on using at most $k$ moves, no matter what the initial configuration was.
[i]Proposed by Viktor Simjanoski and Nikola Velov[/i]
Let $ABC$ be an acute triangle with side $AB$ of length $1$. Say we reflect the points $A$ and $B$
across the midpoints of $BC$ and $AC$, respectively to obtain the points $A’$ and $B’$ . Assume that the orthocenters of triangles $ ABC$, $A’BC$ and $B’AC$ form an equilateral triangle.
a) Prove that triangle $ABC$ is isosceles.
b) What is the length of the altitude of $ABC$ through $C$?
The largest integer not exceeding $[(n+1)a]-[na]$ where $n$ is a natural number, $a=\frac{\sqrt{2013}}{\sqrt{2014}}$ is:
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
Prove that there exists a topological space $ T$ containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over $ T$. Show that the real line cannot be an everywhere-dense subset of such a space $ T$.
[i]A. Csaszar[/i]
Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$
In the convex pentagon $ABCDE$, the following equalities hold: $\angle CDE=90^\circ$, $AC=AD$, and $BD=BE$. Prove that triangle $ABD$ and quadrilateral $ABCE$ have the same area.
The distance between towns $A$ and $B$ is $999$ km.
At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows:
$\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$
How many signs are there, with both distances written with the help of only two distinct digits?
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]
Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.