Found problems: 85335
At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is [i]persistent[/i] if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number.
When two stars are very far apart their gravitational potential energy is zero; when they are separated by a distance $d$ the gravitational potential energy of the system is $U$. If the stars are separated by a distance $2d$ the gravitational potential energy of the system is
$ \textbf{(A)}\ U/4\qquad\textbf{(B)}\ U/2 \qquad\textbf{(C)}\ U \qquad\textbf{(D)}\ 2U\qquad\textbf{(E)}\ 4U $
From a deck of playing cards, four [i]threes[/i], four [i]fours[/i] and four [i]fives[/i] are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is
(a) greater than 34;
(b) greater than 37;
loses the game. Which player has a winning strategy?
Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.
Let $ AB$ be the diameter of a circle with a center $ O$ and radius $ 1$. Let $ C$ and $ D$ be two points on the circle such that $ AC$ and $ BD$ intersect at a point $ Q$ situated inside of the circle, and $ \angle AQB\equal{} 2 \angle COD$. Let $ P$ be a point that intersects the tangents to the circle that pass through the points $ C$ and $ D$.
Determine the length of segment $ OP$.
Let $ABCD$ be an tangential trapezoid, $E$ is a point of its diagonals intersection, $r_1,r_2,r_3,r_4$ -- the radiuses of the circles inscribed in the triangles $ABE$, $BCE$, $CDE$, $DAE$ respectively. Prove that $$1/(r_1)+1/(r_3) = 1/(r_2)+1/(r_4).$$
The leader of the Gnome country wants to print banknotes in $12$ different denominations (each with an integer number) in such a way that it is possible to pay an arbitrary amount from $1$ to $6543$ with these banknotes without a balance, using a maximum of $8$ banknotes. (Several bills with the same denomination can be used during payment.)
Can the leader of the land of Gnomes do it?
Consider a parallelogram $ABCD$ where $AB>AD$. Let $X$ and $Y$, distinct from $B$, be points on the ray $BD^\rightarrow$ such that $CX=CB$ and $AY=AB$. Prove that $DX=DY$. Note: The notation $BD^\rightarrow$ denotes the ray originating from point $B$ passing through point $D$.
The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$.
(L.Steingarts)
In triangle $ABC$, angle $C$ is three times the angle $A$, and side $AB$ is twice the side $BC$. What can be the angle $ABC$?
Two circles, $\omega_1$ and $\omega_2$, are internally tangent at $A.$ Let $B$ be the point on $\omega_2$ opposite of $A.$ The radius of $\omega_1$ is $4$ times the radius of $\omega_2.$ Point $P$ is chosen on the circumference of $\omega_1$ such that the ratio $\tfrac{AP}{BP}=\tfrac{2\sqrt{3}}{\sqrt{7}}.$ Let $O$ denote the center of $\omega_2.$ Determine $\tfrac{OP}{AO}.$
Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. Each coefficient $a, b, c, d, e$, and$ f$ is equal to either $ 1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?
Let $1<t<2$ be a real number. Prove that for all sufficiently large positive integers like $d$, there is a monic polynomial $P(x)$ of degree $d$, such that all of its coefficients are either $+1$ or $-1$ and
$$\left|P(t)-2019\right| <1.$$
[i]Proposed by Navid Safaei[/i]
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i]
Let $ABC$ be a triangle. Let $\omega_1$ be circle tangent to $BC$ at $B$ and passes through $A$. Let $\omega_2$ be circle tangent to $BC$ at $C$ and passes through $A$. Let $\omega_1$ and $\omega_2$ intersect again at $P \neq A$. Let $\omega_1$ intersect $AC$ again at $E\neq A$, and let $\omega_2$ intersect $AB$ again at $F\neq A$. Let $R$ be the reflection of $A$ about $BC$, Prove that lines $BE, CF, PR$ are concurrent.
Let $(a_n )$ be a sequence of real numbers satisfying the inequalities
$$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$
and such that the series
$$\sum_{n=0}^{\infty} a_n $$
converges. Prove that
$$\lim_{n\to \infty} n a_n = 0.$$
Rectangle $ABCD$ has $AB=3$ and $BC=4.$ Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}.$ What is the area of $\triangle ADE?$
$\textbf{(A)} \text{ 1} \qquad \textbf{(B)} \text{ }\frac{42}{25} \qquad \textbf{(C)} \text{ }\frac{28}{15} \qquad \textbf{(D)} \text{ 2} \qquad \textbf{(E)} \text{ }\frac{54}{25}$
On hypotenuse $ AB$ of a right triangle $ ABC$ a second right triangle $ ABD$ is constructed with hypotenuse $ AB$. If $ \overline{BC} \equal{} 1, \overline{AC} \equal{} b$, and $ \overline{AD} \equal{} 2$, then $ \overline{BD}$ equals:
$ \textbf{(A)}\ \sqrt {b^2 \plus{} 1} \qquad\textbf{(B)}\ \sqrt {b^2 \minus{} 3} \qquad\textbf{(C)}\ \sqrt {b^2 \plus{} 1} \plus{} 2$
$ \textbf{(D)}\ b^2 \plus{} 5 \qquad\textbf{(E)}\ \sqrt {b^2 \plus{} 3}$
A convex quadrilateral with area $1$ is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area $< \frac38$.
Let $ a $ be a positive real number and $ \left( x_n\right)_{n\ge 1} $ be a sequence of pairwise distinct real numbers satisfying the properties:
$ \text{(i) } x_n\in (0,a) , $ for any natural numbers $ n $
$ \text{(ii) } \left| x_n-x_m \right|\geqslant\frac{m+n}{amn} , $ for all pairs $ (m,n) $ of distinct natural numbers
Show that $ a\geqslant 2. $
Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.
Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.
In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.
Find all functions $f: R \to R$ such that, for any $x, y \in R$:
$f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$
There are $6$ points on the plane such that no three of them are collinear. It's known that between every $4$ points of them, there exists a point that it's power with respect to the circle passing through the other three points is a constant value $k$.(Power of a point in the interior of a circle has a negative value.)
Prove that $k=0$ and all $6$ points lie on a circle.
[i]Proposed by Morteza Saghafian[/I]