Found problems: 85335
A sequence $a_n$ is defined by
$$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.
Juan rolls a fair regular octahedral die marked with the numbers $ 1$ through $ 8$. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of $ 3$?
$ \textbf{(A)}\ \frac{1}{12} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{7}{12} \qquad
\textbf{(E)}\ \frac{2}{3}$
Let $p(n)\geq 0$ for all positive integers $n$. Furthermore, $x(0)=0, v(0)=1$, and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\]
Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$. Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$.
Does there exist any continuous function $ f$ such that $ f(f(x))=-x^{2019}\ \forall\ x \in \mathbb{R}$
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, \[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\]
[i]Proposed by Aritra Mondal[/i]
Simplify $\sqrt[4]{17 + 12\sqrt2} - \sqrt[4]{17 - 12\sqrt2}$.
Let $C$ be the part of the graph $y=\frac{1}{x}\ (x>0)$. Take a point $P\left(t,\ \frac{1}{t}\right)\ (t>0)$ on $C$.
(i) Find the equation of the tangent $l$ at the point $A(1,\ 1)$ on the curve $C$.
(ii) Let $m$ be the line passing through the point $P$ and parallel to $l$. Denote $Q$ be the intersection point of the line $m$ and the curve $C$ other than $P$. Find the coordinate of $Q$.
(iii) Express the area $S$ of the part bounded by two line segments $OP,\ OQ$ and the curve $C$ for the origin $O$ in terms of $t$.
(iv) Express the volume $V$ of the solid generated by a rotation of the part enclosed by two lines passing through the point $P$ and pararell to the $y$-axis and passing through the point $Q$ and pararell to $y$-axis, the curve $C$ and the $x$-axis in terms of $t$.
(v) $\lim_{t\rightarrow 1-0} \frac{S}{V}.$
Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.
Alice lives on a continent with $6$ countries labeled $1$ through $6$. Each country randomly chooses one other country to allow entry from. Alice can travel to any country that allows entry from the country she is currently in, and can travel along a path through multiple countries in this manner. If Alice starts in county $1$, what is the expected number of countries that she can reach (including country $1$)?
A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure:
Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet.
[img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]
Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
f(x+f(y)) &= g(x) + h(y) \\
g(x+g(y)) &= h(x) + f(y) \\
h(x+h(y)) &= f(x) + g(y)
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)
[i]Proposed by Evan Chen[/i]
Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs?
A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of 5 units due east. Each night the bug rotates $60 ^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point $P$. Then $OP^2 = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $ ABC$ be an acute-angled triangle. Let $ E$ be a point such $ E$ and $ B$ are on distinct sides of the line $ AC,$ and $ D$ is an interior point of segment $ AE.$ We have $ \angle ADB \equal{} \angle CDE,$ $ \angle BAD \equal{} \angle ECD,$ and $ \angle ACB \equal{} \angle EBA.$ Prove that $ B, C$ and $ E$ lie on the same line.
Let \(n\) be a non-negative integer and define \(a_n = 2^n - n\). Determine all non-negative integers \(m\) such that \(s_m = a_0 + a_1 + \dots + a_m\) is a power of 2.
Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers.
[i]Proposed by Michael Ren[/i]
Find all pairs $(x,y)$ of real numbers such that
\[16^{x^{2}+y} + 16^{x+y^{2}} = 1\]
Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$?
[img]https://cdn.artofproblemsolving.com/attachments/1/3/03abbd4df2932f4a1d16a34c2b9e15b683dedb.png[/img]
$\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3$
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ , and let $O$ , $H$ be the triangle's circumcenter and orthocenter respectively . Let also $A^{'}$ be the point where the angle bisector of the angle $BAC$ meets $\omega$ . If $A^{'}H=AH$ , then find the measure of the angle $BAC$.
Find all value of parameter $a$ such that equations $x^2-ax+1=0$ and $x^2-x+a=0$ have at least one same solution.
For this value $a$ find same solution of this equations(real or imaginary).
Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f(f^{(n)}(x))$.
Let $p(n)$ be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation $f^{(n)}(n)=p(n)$ has exactly one solution $f$ that maps the positive integers into the positive integers?
[i]Submitted by Dávid Matolcsi and Kristóf Szabó, Budapest[/i]
For each $P$ inside the triangle $ABC$, let $A(P), B(P)$, and $C(P)$ be the points of intersection of the lines $AP, BP$, and $CP$ with the sides opposite to $A, B$, and $C$, respectively. Determine $P$ in such a way that the area of the triangle $A(P)B(P)C(P)$ is as large as possible.
An exterior common tangent to two non-intersecting circles with centers and $O_2$ touches them at the points $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects the circles at the points $B_1$ and $B_2$ respectively. $C$ is the point where the straight lines $A_1B_1$ and $A_2B_2$ meet. $D$ is the point on the line $A_1A_2$ such that $CD$ is perpendicular to $B_1B_2$. Prove that $A_1D = DA_2$.
Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.