Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$

There are 100 merchants who are selling salmon for Durer dollars around the circular shore of the island of Durerland. Since the beginning of times good and bad years have been alternating on the island. (So after a good year, the next year is bad; and after a bad year, the next year is good.) In every good year all merchants set their price as the maximum value between their own selling price from the year before and the selling price of their left-hand neighbour from the year before. In turn, in every bad year they sell it for the minimum between their own price from the year before and their left-hand neighbour’s price from the year before. Paul and Pauline are two merchants on the island. This year Paul is selling salmon for 17 Durer dollars a kilogram. Prove that there will come a year when Pauline will sell salmon for 17 Durer dollars a kilogram. [i]Note: The merchants are immortal, they have been selling salmon on the island for thousands of years and will continue to do so until the end of time.[/i]

Show that the number $ 3^{{4}^{5}} \plus{} 4^{{5}^{6}}$ can be expresed as the product of two integers greater than $ 10^{2009}$

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

Consider the randomly generated base 10 real number $r = 0.\overline{p_0p_1p_2\ldots}$, where each $p_i$ is a digit from $0$ to $9$, inclusive, generated as follows: $p_0$ is generated uniformly at random from $0$ to $9$, inclusive, and for all $i \geq 0$, $p_{i + 1}$ is generated uniformly at random from $p_i$ to $9$, inclusive. Compute the expected value of $r$.

Let $\mathcal P$ be the parabola $y^2=2px$, and let $T_0$ be a point on it. Point $T_0'$ is such that the midpoint of the segment $T_0T_0'$ lies on the axis of the parabola. For a variable point $T$ on $\mathcal P$, the perpendicular from $T_0'$ to the line $T_0T$ intersects the line through $T$ parallel to the axis of $\mathcal P$ at a point $T'$. Find the locus of $T'$.

Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

A parallelepiped has surface area 216 and volume 216. Show that it is a cube.

Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related. Note that $1$ and $n$ are included as divisors.

Three pairwairs distinct positive integers $a,b,c,$ with $gcd(a,b,c)=1$, satisfy \[a|(b-c)^2 ,b|(a-c)^2 , c|(a-b)^2\] Prove that there doesnt exist a non-degenerate triangle with side lengths $a,b,c.$

Let $ a$ and $ b$ be integers such that the equation $ x^3\minus{}ax^2\minus{}b\equal{}0$ has three integer roots. Prove that $ b\equal{}dk^2$, where $ d$ and $ k$ are integers and $ d$ divides $ a$.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

For integral $m$, let $p(m)$ be the greatest prime divisor of $m.$ By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty.$ Find all polynomials $f$ with integer coefficients such that the sequence \[ \{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0} \] is bounded above. (In particular, this requires $f \left (n^2 \right ) \neq 0$ for $n \geq 0.$)

$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken? [img]https://s8.uupload.ir/files/photo14908575660_2r3g.jpg[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [b]C1.[/b] A circle has circumference $6\pi$. Find the area of this circle. [b]C2 / G2.[/b] Points $A$, $B$, and $C$ are on a line such that $AB = 6$ and $BC = 11$. Find all possible values of $AC$. [b]C3.[/b] A trapezoid has area $84$ and one base of length $5$. If the height is $12$, what is the length of the other base? [b]C4 / G1.[/b] $27$ cubes of side length 1 are arranged to form a $3 \times 3 \times 3$ cube. If the corner $1 \times 1 \times 1$ cubes are removed, what fraction of the volume of the big cube is left? [b]C5.[/b] There is a $50$-foot tall wall and a $300$-foot tall guard tower $50$ feet from the wall. What is the minimum $a$ such that a flat “$X$” drawn on the ground $a$ feet from the side of the wall opposite the guard tower is visible from the top of the guard tower? [b]C6.[/b] Steven’s pizzeria makes pizzas in the shape of equilateral triangles. If a pizza with side length 8 inches will feed 2 people, how many people will a pizza of side length of 16 inches feed? [b]C7 / G3.[/b] Consider rectangle $ABCD$, with $1 = AB < BC$. The angle bisector of $\angle DAB$ intersects $\overline{BC}$ at $E$ and $\overline{DC}$ at $F$. If $FE = FD$, find $BC$. [b]C8 / G6.[/b] $\vartriangle ABC$. is a right triangle with $\angle A = 90^o$. Square $ADEF$ is drawn, with $D$ on $\overline{AB}$, $F$ on $\overline{AC}$, and $E$ inside $\vartriangle ABC$. Point $G$ is chosen on $\overline{BC}$ such that $EG$ is perpendicular to $BC$. Additionally, $DE = EG$. Given that $\angle C = 20^o$, find the measure of $\angle BEG$. [b]G4.[/b] Consider a lamp in the shape of a hollow cylinder with the circular faces removed with height $48$ cm and radius $7$ cm. A point source of light is situated at the center of the lamp. The lamp is held so that the bottom of the lamp is at a height $48$ cm above an infinite flat sheet of paper. What is the area of the illuminated region on the flat sheet of paper, in $cm^2$? [img]https://cdn.artofproblemsolving.com/attachments/c/6/6e5497a67ae5ff5a7bff7007834a4271ce3ca7.png[/img] [b]G5.[/b] There exist two triangles $ABC$ such that $AB = 13$, $BC = 12\sqrt2$, and $\angle C = 45^o$. Find the positive difference between their areas. [b]G7.[/b] Let $ABC$ be an equilateral triangle with side length $2$. Let the circle with diameter $AB$ be $\Gamma$. Consider the two tangents from $C$ to $\Gamma$, and let the tangency point closer to $A$ be $D$. Find the area of $\angle CAD$. [b]G8.[/b] Let $ABC$ be a triangle with $\angle A = 60^o$, $AB = 37$, $AC = 41$. Let $H$ and $O$ be the orthocenter and circumcenter of $ABC$, respectively. Find $OH$. [i]The orthocenter of a triangle is the intersection point of the three altitudes. The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.[/i] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A container the shape of a pyramid has a 12 × 12 square base, and the other four edges each have length 11. The container is partially filled with liquid so that when one of its triangular faces is lying on a flat surface, the level of the liquid is half the distance from the surface to the top edge of the container. Find the volume of the liquid in the container. [center][img]https://snag.gy/CdvpUq.jpg[/img][/center]

The sequence $(a_n)$ is defined as: $$a_1=1007$$ $$a_{i+1}\geq a_i+1$$ Prove the inequality: $$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?

A square $ABCD$ and point $E$ are drawn in a plane such that lengths $DE < BE$ and $\vartriangle ACE$ is equilateral. Compute $\angle BAE$.

If integers $m,n,k$ satisfy $m^2+n^2+1=kmn$, what values can $k$ have?

On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.

Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.