Define $a \star b$ to be $2ab + a + b$. What is $((3 \star 4) \star 5) - (4 \star (5 \star 3))$ ?

On a plane several straight lines are drawn in such a way that each of them intersects exactly $15$ other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer. Poland

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$, $N$ and $R$ be the midpoints of $AB$, $BC$ an $AH$, respectively. If $A\hat{B}C=70^\large\circ$, compute $M\hat{N}R$.

Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators. [i](Proposed by Bogdan Rublov)[/i]

A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies [list] [*]$f(0)=0$ [*]$f(6)=1$ [*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list] Find $f(2)^2+f(10)^2$.

Let $A_1A_2A_3A_4$, $B_1B_2B_3B_4$, and $C_1C_2C_3C_4$ be three regular tetrahedra in $3$-dimensional space, no two of which are congruent. Suppose that, for each $i\in \{1,2,3,4\}$, $C_i$ is the midpoint of the line segment $A_iB_i$. Determine whether the four lines $A_1B_1$, $A_2B_2$, $A_3B_3$, and $A_4B_4$ must concur.

Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$. [i]Laurențiu Panaitopol[/i]

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

Let $x, y$ be numbers in the interval (0,1) such that for some $a>0, a \neq 1$ $$\log _{x} a+\log _{y} a=4 \log _{x y} a$$Prove that $x=y$

Source: 1976 Euclid Part A Problem 5 ----- If $\log_8 m+\log_8 \frac{1}{6}=\frac{2}{3}$, then $m$ equals $\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{2}{3} \qquad \textbf{(C) } \frac{23}{6} \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 24$

Let $A$ be a ring and let $D$ be the set of its non-invertible elements. If $a^2=0$ for any $a \in D,$ prove that: [b]a)[/b] $axa=0$ for all $a \in D$ and $x \in A$; [b]b)[/b] if $D$ is a finite set with at least two elements, then there is $a \in D,$ $a \neq 0,$ such that $ab=ba=0,$ for every $b \in D.$ [i]Ioan Băetu[/i]

Two circles in the plane intersect at $C$ and $D$. A chord $AB$ of the first circle and a chord $EF$ of the second circle pass through a point on the common chord $CD$. Show that the points $A,B,E,F$ lie on a circle.

Show that there are infinitely many noncongruent triangles which satisfy the following conditions: i) the side lengths are relatively prime integers; ii)the area is an integer number; iii)the altitudes' lengths are not integer numbers.

Suppose that $x$ and $y$ are different real numbers such that $\frac{x^n-y^n}{x-y}$ is an integer for some four consecutive positive integers $n$. Prove that $\frac{x^n-y^n}{x-y}$ is an integer for all positive integers n.

Find a sequence of $ 2012 $ distinct integers bigger than $ 0 $ such that their sum is a perfect square and their product is a perfect cube.

Given $T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1$ for $n>0.$ Prove: (i) If $m \ne n,$ $T_m$ and $T_n$ have no common factor greater than $1.$ (ii) $\sum_{i=1}^{\infty} \frac{1}{T_i }=1.$

For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ?

Find all continuously differentiable functions $f:[0,1]\to(0,\infty)$ such that $\frac{f(1)}{f(0)}=e$ and $$\int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2.$$

Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered [u]clockwise[/u]. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$. Prove that $\Delta A_3MN$ is an equilateral triangle.

We consider three distinct half-lines $Ox, Oy, Oz$ in a plane. Prove the existence and uniqueness of three points $A \in Ox, B \in Oy, C \in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p > 0.$

A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.

$ABC$ is an isosceles right triangle with right angle $B$ and $AB = 1$. $ABC$ has an incenter at $E$. The excircle to $ABC$ at side $AC$ is drawn and has center $P$. Let this excircle be tangent to $AB$ at $R$. Draw $T$ on the excircle so that $RT$ is the diameter. Extend line $BC$ and draw point $D$ on $BC$ so that $DT$ is perpendicular to $RT$. Extend $AC$ and let it intersect with $DT$ at $G$. Let $F$ be the incenter of $CDG$. Find the area of $\vartriangle EFP$.

$30$ segments of lengths$$1,\quad \sqrt{3},\quad \sqrt{5},\quad \sqrt{7},\quad \sqrt{9},\quad \ldots ,\quad \sqrt{59} $$ have been drawn on a blackboard. In each step, two of the segments are deleted and a new segment of length equal to the hypotenuse of the right triangle with legs equal to the two deleted segments is drawn. After $29$ steps only one segment remains. Find the possible values of its length.

In a village $X_0$ there are $80$ tourists who are about to visit $5$ nearby villages $X_1,X_2,X_3,X_4,X_5$.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among $X_1,X_2,X_3,X_4,X_5$.Each tourist visits only the village he has chosen and the villages he is forced to.If $X_1,X_2,X_3,X_4,X_5$ are totally visited by $40,60,65,70,75$ tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs $(X_i,X_j):i,j\in \{1,2,3,4,5\}$ which are such that,the visit in $X_i$ forces the visitor to visit $X_j$ as well.