Found problems: 85335
Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$:
(a) $f (n)$ is a positive integer,
(b) $f (n + 1) > f (n)$,
(c) $f ( f (n)) = 3n$.
Find $f (2001)$.
Let $ABC$ be a triangle with $AB \neq BC$ and let $BD$ the interior bisectrix of $ \angle ABC$ with $D \in AC$ . Let $M$ be the midpoint of the arc $AC$ that contains the point $B$ in the circumcircle of the triangle $ABC$ .The circumcircle of the triangle $BDM$ intersects the segment $AB$ in $K \neq B$ . Denote by $J$ the symmetric of $A$ with respect to $K$ .If $DJ$ intersects $AM$ in $O$ then prove that $J,B,M,O$ are concyclic.
Let be $ a,b,c $ positive integers.Prove that $ |a-b\sqrt{c}|<\frac{1}{2b} $ is true if and only if $ |a^{2}-b^{2}c|<\sqrt{c} $.
For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that:
\[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \]
then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.
In how many ways can $n$ rooks be placed on a $2n$ x $2n$ chessboard, so that they cover all the white fields?
Let $m = 30030$ and let $M$ be the set of its positive divisors which have exactly $2$ prime factors. Determine the smallest positive integer $n$ with the following property: for any choice of $n$ numbers from $M$, there exist 3 numbers $a$, $b$, $c$ among them satisfying $abc=m$.
There are $N$ houses in a city. Every Christmas, Santa visits these $N$ houses in some order. Show that if $N$ is large enough, then it holds that for three consecutive years there are always are $13$ houses such that they have been visited in the same order for two years (out of the three consecutive years). Determine the smallest $N$ for which this holds.
Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at $P \neq A$. Let $PE$ intersect $\Gamma$ again at $Q \neq P$. Prove that $BQ$ bisects segment $\overline{EF}$.
[i]Proposed by Luke Robitaille[/i]
A little boy wrote the numbers $1,2,\cdots,2011$ on a blackboard. He picks any two numbers $x,y$, erases them with a sponge and writes the number $|x-y|$. This process continues until only one number is left. Prove that the number left is even.
for $k\in\mathbb{N}$ , define $A_n$ for $n=1,2,...$ by
$A_{n+1} = \frac{ nA_n+2(n+1)^{2k} }{n+2} , A_1=1$
Prove $A_n$ is integer for all $n\geq 1$, and $A_n$ is odd if and only if $n\equiv$1 or 2(mod 4)
Find all sequences $\left(a_0,a_1,a_2,a_3\right)$, where for each $k$, $0\leq k\leq3$, $a_k$ is the number of times that the number $k$ appears in the sequence $\left(a_0,a_1,a_2,a_3\right)$.
The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.
Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon
On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions:
$\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$.
$\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$.
$\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally.
$\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$.
Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
There are $3$ unit squares in a row as shown in the figure below. Each side of this figure is painted by one of the three colors: Blue, Green or Red. It is known that for any square, all the three colors are used and no two adjacent sides have the same color. Find the number of possible colorings.
[img]https://cdn.artofproblemsolving.com/attachments/e/c/8963e6716b7d9b23479dd7e106b4bd9a3267c1.png[/img]
A. $48$ B. $96$ C. $108$ D. $192$ E. $216$
Determine the smallest positive integer that has all its digits equal to $4$, and is a multiple of $169$.
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$
[i]Proposed by Igor Voronovich, Belarus[/i]
Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that:
$1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively.
$2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively.
Prove that $KL$ and $ST$ intersect on the line $BC$.
Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.
Let $f(x)=1+\dfrac{x}{2}+\dfrac{x^2}{4}+\dfrac{x^3}{8}+\cdots,$ for $-1\leq x \leq 1$. Find $\sqrt{e^{\int\limits_0^1 f(x)dx}}$.
Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the perpendicular line from $A$ to $BC$ meet $\omega$ at $D$. Let $P$ be a point on $\omega$, and let $Q$ be the foot of the perpendicular line from $P$ to the line $AB$. Prove that if $Q$ is on the outside of $\omega$ and $2\angle QPB = \angle PBC$, then $D,P,Q$ are collinear.
Let be a triangle $ ABC $, the midpoint of the $ AC $ and $ N $ side, and the midpoint of the $ AB $ side. Let $ r $ and $ s $ reflect the straight lines $ BM $ and $ CN $ on the straight $ BC $, respectively. Also define $ D $ and $ E $ as the intersection of the lines $ r $ and $ s $ and the line $ MN $, respectively. Let $ X $ and $ Y $ be the intersection points between the circumcircles of the triangles $ BDM $ and $ CEN $, $ Z $ the intersection of the lines $ BE $ and $ CD $ and $ W $ the intersection between the lines $ r $ and $ s $. Prove that $ XY, WZ $, and $ BC $ are concurrents.
A positive integer $N$ is called [i]googolicious[/i] if there are exactly $10^{100}$ positive integers $x$ that satisfy \[\left\lfloor \frac{N}{\left\lfloor \frac{N}{x} \right\rfloor } \right\rfloor = x,\] where $z$ denotes the greatest integer less than $z.$ Find, with proof, all googolicious integers $N.$