In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the $n$ -th switch the $n$ -th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from the first switch until he has to turn on a bulb; as soon as he turns a bulb on, he leaves the room. For example the first person goes in, turns the first switch on and leaves. Then the second man goes in, seeing that the first switch on turns it off and then lights the second bulb. Then the third person goes in, finds the first switch off and turns it on and leaves the room. Then the fourth person enters and switches off the first and second bulbs and switches on the third. The process continues in this way. Finally we find out that first 10 bulbs are off and the 11 -th bulb is on. Then how many people were involved in the entire process?

The triangle $ ABC$ has semiperimeter $ p$. Find the side length $ BC$ and the area $ S$ in terms of $ \angle A$, $ \angle B$ and $ p$. In particular, find $ S$ if $ p \approx 23.6$, $ \angle A \approx 52^{\circ}42'$, $ \angle B \approx 46^{\circ}16'$.

Mary is $ 20\%$ older than Sally, and Sally is $ 40\%$ younger than Danielle. The sum of their ages is 23.2 years. How old will Mary be on her next birthday? $ \textbf{(A) } 7\qquad \textbf{(B) } 8\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 11$

We say that a set of points $M$ in the plane is convex if for any two points $A, B \in M$, all the points from the segment $(AB)$ also belong to $M$. Let $n \ge 2$ be an integer and let $F$ be a family of $n$ disjoint convex sets in the plane. Prove that there exists a line $\ell$ in the plane, a set $M \in F$ and a subset $S \subset F$ with at least $\lceil \frac{n}{12} \rceil $ elements such that $M$ is contained in one closed half-plane determined by $\ell$, and all the sets $N \in S$ are contained in the complementary closed half-plane determined by $\ell$.

Each of the squares in a $50 \times 50$ square board is filled with a number from $1$ to $50$ so that each of the numbers $1,2, ..., 50$ appears exactly $50$ times. Prove that there is a row or a column containing at least $8$ distinct numbers.

Let $ABCD$ is a tetrahedron. Show that a)If $AB=CD,AC=DB,AD=BC$ then triangles $ABC,ABD,ACD,BCD$ are acute. b)If the triangles $ABC,ABD,ACD,BCD$ have the same area , then $AB=CD,AC=DB,AD=BC$.

You have an infinite stack of T-shaped tetrominoes (composed of four squares of side length 1), and an n × n board. You are allowed to place some tetrominoes on the board, possibly rotated, as long as no two tetrominoes overlap and no tetrominoes extend off the board. For which values of n can you cover the entire board?

There are $2017$ mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn. Find all possible numbers of tangent segments when Luciano stops drawing.

A table has $3 000 000$ rows and $100$ columns, divided into unit squares. Each row contains the numbers from $1$ to $100$, each exactly once, and no two rows are the same. Above each column, the number of distinct entries in that column is written in red. Find the smallest possible sum of the red numbers.

Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.

A convex pentagon $ABCDE$ satisfies that the sidelengths and $AC,AD\leq \sqrt 3.$ Let us choose $2011$ distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least $403$ points out of the $2011$ given points. {Edited} {I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}

Positive real numbers $x, y, z$ satisfy $\sqrt x + \sqrt y + \sqrt z = 1$. Prove that $$\frac{x^4+y^2z^2}{x^{\frac 52}(y+z)} + \frac{y^4+z^2x^2}{y^{\frac 52}(z+x)} + \frac{z^4+y^2x^2}{z^{\frac 52}(y+x)} \geq 1.$$ [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 15)[/i]

[b]p1.[/b] Martin’s car insurance costed $\$6000$ before he switched to Geico, when he saved $15\%$ on car insurance. When Mayhem switched to Allstate, he, a safe driver, saved $40\%$ on car insurance. If Mayhem and Martin are now paying the same amount for car insurance, how much was Mayhem paying before he switched to Allstate? [b]p2.[/b] The $7$-digit number $N$ can be written as $\underline{A} \,\, \underline{2} \,\,\underline{0} \,\,\underline{B} \,\,\underline{2} \,\, \underline{1} \,\,\underline{5}$. How many values of $N$ are divisible by $9$? [b]p3.[/b] The solutions to the equation $x^2-18x-115 = 0$ can be represented as $a$ and $b$. What is $a^2+2ab+b^2$? [b]p4.[/b] The exterior angles of a regular polygon measure to $4$ degrees. What is a third of the number of sides of this polygon? [b]p5.[/b] Charlie Brown is having a thanksgiving party. $\bullet$ He wants one turkey, with three different sizes to choose from. $\bullet$ He wants to have two or three vegetable dishes, when he can pick from Mashed Potatoes, Saut´eed Brussels Sprouts, Roasted Butternut Squash, Buttery Green Beans, and Sweet Yams; $\bullet$ He wants two desserts out of Pumpkin Pie, Apple Pie, Carrot Cake, and Cheesecake. How many different combinations of menus are there? [b]p6.[/b] In the diagram below, $\overline{AD} \cong \overline{CD}$ and $\vartriangle DAB$ is a right triangle with $\angle DAB = 90^o$. Given that the radius of the circle is $6$ and $m \angle ADC = 30^o$, if the length of minor arc $AB$ is written as $a\pi$, what is $a$? [img]https://cdn.artofproblemsolving.com/attachments/d/9/ea57032a30c16f4402886af086064261d6828b.png[/img] [b]p7.[/b] This Halloween, Owen and his two friends dressed up as guards from Squid Game. They needed to make three masks, which were black circles with a white equilateral triangle, circle, or square inscribed in their upper halves. Resourcefully, they used black paper circles with a radius of $5$ inches and white tape to create these masks. Ignoring the width of the tape, how much tape did they use? If the length can be expressed $a\sqrt{b}+c\sqrt{d}+ \frac{e}{f} \pi$ such that $b$ and $d$ are not divisible by the square of any prime, and $e$ and $f$ are relatively prime, find $a + b + c + d + e + f$. [img]https://cdn.artofproblemsolving.com/attachments/0/c/bafe3f9939bd5767ba5cf77a51031dd32bbbec.png[/img] [b]p8.[/b] Given $LCM (10^8, 8^{10}, n) = 20^{15}$, where $n$ is a positive integer, find the total number of possible values of $n$. [b]p9.[/b] If one can represent the infinite progression $\frac{1}{11} + \frac{2}{13} + \frac{3}{121} + \frac{4}{169} + \frac{5}{1331} + \frac{6}{2197}+ ...$ as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, what is $a$? [b]p10.[/b] Consider a tiled $3\times 3$ square without a center tile. How many ways are there to color the squares such that no two colored squares are adjacent (vertically or horizontally)? Consider rotations of an configuration to be the same, and consider the no-color configuration to be a coloring. [b]p11.[/b] Let $ABC$ be a triangle with $AB = 4$ and $AC = 7$. Let $AD$ be an angle bisector of triangle $ABC$. Point $M$ is on $AC$ such that $AD$ intersects $BM$ at point $P$, and $AP : PD = 3 : 1$. If the ratio $AM : MC$ can be expressed as $\frac{a}{b}$ such that $a$, $b$ are relatively prime positive integers, find $a + b$. [b]p12.[/b] For a positive integer $n$, define $f(n)$ as the number of positive integers less than or equal to $n$ that are coprime with $n$. For example, $f(9) = 6$ because $9$ does not have any common divisors with $1$, $2$, $4$, $5$, $7$, or $8$. Calculate: $$\sum^{100}_{i=2} \left( 29^{f(i)}\,\,\, mod \,\,i \right).$$ [b]p13.[/b] Let $ABC$ be an equilateral triangle. Let $P$ be a randomly selected point in the incircle of $ABC$. Find $a+b+c+d$ if the probability that $\angle BPC$ is acute can be expressed as $\frac{a\sqrt{b} -c\pi}{d\pi }$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c, d) = 1$ and $b$ is not divisible by the square of any prime. [b]p14.[/b] When the following expression is simplified by expanding then combining like terms, how many terms are in the resulting expression? $$(a + b + c + d)^{100} + (a + b - c - d)^{100}$$ [b]p15.[/b] Jerry has a rectangular box with integral side lengths. If $3$ units are added to each side of the box, the volume of the box is tripled. What is the largest possible volume of this box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

a) Prove that we can assign one of the numbers $1$ or $-1$ to the vertices of a cube such that the product of the numbers assigned to the vertices of any face is equal to $-1$. b) Prove that for a hexagonal prism such a mapping is not possible.

Let $\mathcal{C}$ be a right circular cone with height $\sqrt{15}$ and base radius $1$. Let $V$ be the vertex of $\mathcal{C}$, $B$ be a point on the circumference of the base of $\mathcal{C}$, and $A$ be the midpoint of $VB$. An ant travels at constant velocity on the surface of the cone from $A$ to $B$ and makes two complete revolutions around $\mathcal{C}$. Find the distance the ant travelled.

Let $ x,y,z$ be positive real numbers such that $ x\plus{}2y\plus{}3z\equal{}\frac{11}{12}$. Prove the inequality $ 6(3xy\plus{}4xz\plus{}2yz)\plus{}6x\plus{}3y\plus{}4z\plus{}72xyz\le \frac{107}{18}$.

The sum of the following seven numbers is exactly 19: \[a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71,\] \[a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86.\] It is desired to replace each $a_i$ by an integer approximation $A_i$, $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "errors" $|A_i - a_i|$, is as small as possible. For this minimum $M$, what is $100M$?

Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$

Let $ABC$ be an acute triangle with $AC\neq BC$, $M$ the midpoint of side $AB$, $H$ is the orthocenter of $\triangle ABC$, $D$ on $BC$ is the foot of the altitude from $A$ and $E$ on $AC$ is the foot of the perpendicular from $B$. Prove that the lines $AB, DE$ and the perpendicular to $MH$ through $C$ are concurrent.

The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$. [img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]

In a triangle $ABC$, $D$ is the reflection of $A$ about the sideline $BC$. A circle $(K)$ with diameter $AD$ meets $DB,DC$ at $M,N$ which are distinct from $D$. Let $E,F$ be the midpoint of $CA,AB$. The circumcircles of $KEM,KFN$ meet each other again at $L$, distinct from $K$. Let $KL$ meets $EF$ at $X$; points $Y,Z$ are defined in the same manner. Prove that three lines $AX,BY,CZ$ are concurrent. Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.

There are n points marked on the circle. It is known that among all possible distances between two marked points there are no more than $100$ different ones. What is the largest possible value for $n$?

Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.

Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points?

Given triangle $ABC$. Consider lines $l$ with the next property: the reflections of $l$ in the sidelines of the triangle concur. Prove that all these lines have a common point.