Found problems: 85335
There are $n$ light bulbs in a circle and one of them is marked.
Let operation $A$:
Take a positive divisor $d$ of the number $n,$ starting with the light bulb marked and clockwise, we count around the circumference from $1$ to $dn$, changing the state (on or off) to those light bulbs that correspond to the multiples of $d$.
Let operation $B$ be:
Apply operation$ A$ to all positive divisors of $n$ (to the first divider that is applied is with all the light bulbs off and the remaining divisors is with the state resulting from the previous divisor).
Determine all the positive integers $n$, such that when applying the operation on $B$, all the light bulbs are on.
Show that $(\sum_{i=1}^{n}x_iy_iz_i)^2 \le (\sum_{i=1}^{n}x_i^3) (\sum_{i=1}^{n}y_i^3) (\sum_{i=1}^{n}z_i^3)$ for any positive reals $x_i, y_i, z_i$.
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.
Given a list of the positive integers $1,2,3,4,\dots,$ take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16.$ Continue in this way, crossing off the three smallest remaining numbers and their sum and consider the sequence of sums produced: $6,16,27, 36, \dots.$ Prove or disprove that there is some number in this sequence whose base 10 representation ends with $2015.$
Let $ABCD$ be a convex quadrilateral such that $\vartriangle ABC$ is equilateral. Let $P$ be a point inside the quadrilateral such that $\vartriangle AP D$ is equilateral and $\angle P CD = 30^o$ . Given that $CP = 2$ and $CD = 3$, compute the area of the triangle formed by $P$, the midpoint of segment $\overline{BC}$, and the midpoint of segment $\overline{AB}$.
Let $ K$ be the figure bounded by the curve $ y\equal{}e^x$ and 3 lines $ x\equal{}0,\ x\equal{}1,\ y\equal{}0$ in the $ xy$ plane.
(1) Find the volume of the solid formed by revolving $ K$ about the $ x$ axis.
(2) Find the volume of the solid formed by revolving $ K$ about the $ y$ axis.
Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?
Let $ABC$ be a triangle. Points $D$ and $E$ lie on sides $BC$ and $CA$, respectively, such that $BD=AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of angle $BCA$ meet segments $AD$ and $BE$ at $Q$ and $R$, respectively. Prove that $\frac{PQ}{AD}=\frac{PR}{BE}$.
Let $R$ be the region in the Cartesian plane of points $(x,y)$ satisfying $x\geq 0$, $y\geq 0$, and $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$. Determine the area of $R$.
Show that for all integer $k>1$, there are infinitely many natural numbers $n$ such that $k \cdot 2^{2^n} + 1$ is composite.
The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.
Find the second smallest prime factor of $18!+1.$
[i]Proposed by Kaylee Ji[/i]
The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.
Given an acute triangle $ABC$, let $M$ be the midpoint of $BC$ and $H$ the orthocentre. Let $\Gamma$ be the circle with diameter $HM$, and let $X,Y$ be distinct points on $\Gamma$ such that $AX,AY$ are tangent to $\Gamma$. Prove that $BXYC$ is cyclic.
Let $ G$ be a group such that $ G'$ is abelian and each normal and abelian subgroup of $ G$ is finite. Prove that $ G$ is finite.
Consider the $4\times4$ array of $16$ dots, shown below.
[asy]
size(2cm);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((0,3));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]
Counting the number of squares whose vertices are among the $16$ dots and whose sides are parallel to the sides of the grid, we find that there are nine $1\times1$ squares, four $2\times2$ squares, and one $3\times3$ square, for a total of $14$ squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the $14$ squares is missing at least one vertex?
Let $P$ be a point in the interior of triangle $ABC$. Lines $AP$, $BP$, $CP$ intersect sides $BC$, $CA$, $AB$ at $L$, $M$, $N$, respectively. Prove that $$AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.$$
$\ ABCD$ is a quadrilateral. the feet of the perpendicular from $\ D$ to $\ AB, BC$ are $\ P,Q$ respectively, and the feet of the perpendicular from $\ B$ to $\ AD,CD$ are $\ R,S$ respectively. Show that if $\angle PSR= \angle SPQ$, then $\ PR=QS$.
Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$
Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define
\[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\]
Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$
(Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)
Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite.
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$
Let $A_1A_2 \cdots A_{10}$ be a regular decagon such that $[A_1A_4]=b$ and the length of the circumradius is $R$. What is the length of a side of the decagon?
$
\textbf{a)}\ b-R
\qquad\textbf{b)}\ b^2-R^2
\qquad\textbf{c)}\ R+\dfrac b2
\qquad\textbf{d)}\ b-2R
\qquad\textbf{e)}\ 2b-3R
$
Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$. Prove that $\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|$ is divergent.
In how many ways can $1 + 2 + \cdots + 2007$ be expressed as a sum of consecutive positive integers?