A set of $n$ lights, numbered $1$ to $n$, are initially off. At every moment, it is possible to perform one of the following operations: $\bullet$ change the state of lamp $1$, $\bullet$ change the state of lamp $2$, as long as lamp $1$ is on, $\bullet$ change the state of lamp $k > 2$, as long as lamp $k - 1$ is on and all lamps $1, . . . , k - 2$ are off. It shows that it is possible, after a certain number of operations, to have only the lamp left on.

The cost of $3$ hamburgers, $5$ milk shakes, and $1$ order of fries at a certain fast food restaurant is $\$23.50$. At the same restaurant, the cost of $5$ hamburgers, $9$ milk shakes, and $1$ order of fries is $\$39.50$. What is the cost of $2$ hamburgers, $2$ milk shakes and $2$ orders of fries at this restaurant?

Prove that among the elements of the sequence $\left( \left\lfloor n \sqrt 2 \right\rfloor + \left\lfloor n \sqrt 3 \right\rfloor \right)_{n \geq 0}$ are an infinity of even numbers and an infinity of odd numbers.

Let $ABC$ be a triangle with $AB < AC$, let $L$ be midpoint of arc $BC$(the point $A$ is not in this arc) of the circumcircle $w$($ABC$). Let $E$ be a point in $AC$ where $AE = \frac{AB + AC}{2}$, the line $EL$ intersects $w$ in $P$. If $M$ and $N$ are the midpoints of $AB$ and $BC$, respectively, prove that $AL, BP$ and $MN$ are concurrents

There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?

Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$. [i]S. Berlov[/i]

Find, with proof, a polynomial $f(x,y,z)$ in three variables, with integer coefficients, such that for all $a,b,c$ the sign of $f(a,b,c)$ (that is, positive, negative, or zero) is the same as the sign of $a+b\sqrt[3]{2}+c\sqrt[3]{4}$.

How many perfect squares are greater than $0$ but less than or equal to $100$? $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied: a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$. b) For any $x \in M$, one has that $-x \in M$. c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$. For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.

Find all prime numbers $p$ such that $p^3$ divides the determinant \[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]

In the equation below, $x$ is a nonzero real number such that \[\frac1{729}\left(3^t\right)=3^x.\] Which of the following is equal to $t$? $\textbf{(A) } \dfrac16x\qquad\textbf{(B) } \dfrac13x\qquad\textbf{(C) } 6x\qquad\textbf{(D) } x-6\qquad\textbf{(E) } x+6$

[b]p1.[/b] If Suzie has $6$ coins worth $23$ cents, how many nickels does she have? [b]p2.[/b] Let $a * b = (a - b)/(a + b)$. If $8 * (2 * x) = 4/3$, what is $x$? [b]p3.[/b] How many digits does $x = 100000025^2 - 99999975^2$ have when written in decimal form? [b]p4.[/b] A paperboy’s route covers $8$ consecutive houses along a road. He does not necessarily deliver to all the houses every day, but he always traverses the road in the same direction, and he takes care never to skip over $2$ consecutive houses. How many possible routes can he take? [b]p5.[/b] A regular $12$-gon is inscribed in a circle of radius $5$. What is the sum of the squares of the distances from any one fixed vertex to all the others? [b]p6.[/b] In triangle $ABC$, let $D, E$ be points on $AB$, $AC$, respectively, and let $BE$ and $CD$ meet at point $P$. If the areas of triangles $ADE$, $BPD$, and $CEP$ are $5$, $8$, and $3$, respectively, find the area of triangle ABC. [b]p7.[/b] Bob has $11$ socks in his drawer: $3$ different matched pairs, and $5$ socks that don’t match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly $9$ socks? [b]p8.[/b] Consider the unit cube $ABCDEFGH$. The triangle bound to $A$ is the triangle formed by the $3$ vertices of the cube adjacent to $A$ (and similarly for the other vertices of the cube). Suppose we slice a knife through each of the $8$ triangles bound to vertices of the cube. What is the volume of the remaining solid that contains the former center of the cube? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

In triangle $\triangle ABC$ with median from $B$ not perpendicular to $AB$ nor $BC$, we call $X$ and $Y$ points on $AB$ and $BC$, which lie on the axis of the median from $B$. Find all such triangles, for which $A,C,X,Y$ lie on one circumrefference.

A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter as shown. It is then tipped over by some angle $\gamma$ with respect to the table. At what minimum angle $\gamma$ will the disk lose balance and tumble over? Express your answer in degrees, rounded to the nearest integer. [asy] draw(arc((2,0), 1, 0,180)); draw((0,0)--(4,0)); draw((0,-2.5)--(4,-2.5)); draw(arc((3-sqrt(2)/2, -4+sqrt(2)/2+1.5), 1, -45, 135)); draw((3-sqrt(2), -4+sqrt(2)+1.5)--(3, -4+1.5)); draw(anglemark((3-sqrt(2), -4+sqrt(2)+1.5), (3, -4+1.5), (0, -4+1.5))); label("$\gamma$", (2.8, -3.9+1.5), WNW, fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

Prove that $n!\cdot(n+1)!\cdot(n+2)!$ divides $(3n)!$ for every integer $n \geq 3$.

Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$ [i]Proposed by Krit Boonsiriseth[/i]

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

The square was cut into $n^2$ rectangles with sides $a_i \times b_j, i , j= 1,..., n$. For what is the smallest $n$ in the set $\{a_1, b_1, ..., a_n, b_n\}$ all the numbers can be different?

The parabola $P$ has focus a distance $m$ from the directrix. The chord $AB$ is normal to $P$ at $A.$ What is the minimum length for $AB?$

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Semicircle $O$ has diameter $AB = 12$. Arc $AC = 135^o$ . Let $D$ be the midpoint of arc $AC$. Compute the region bounded by the lines $CD$ and $DB$ and the arc $CB$.

Pythagoras drew some points in the plane and and connected some of these with segments. Now Tortillagoras wants to write a positive integer next to every point, such that one number divides another number if and only if these numbers are written next to points that Pythagoras has connected.Can Tortillagoras do this for the following drawings? [i]In part b), vertices in the same row or column but not adjacent are not connected.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/e/7356e39e44e45e3263275292af3719595e2dd2.png[/img]

(a) Let $x$ and $y$ be integers. Prove that $x = y$ if $x^n \equiv y^n$ mod $n$ for all positive integers $n$. (b) For which pairs of integers $(x, y)$ are there infinitely many positive integers $n$ such that $x^n \equiv y^n$ mod $n$?

Let $ p, r $ be prime numbers, and $ q $ natural. Solve the equation $ (p+q+r)^2=2p^2+2q^2+r^2 $.