$O$ is a circumcircle of non-isosceles triangle $ABC$ and the angle bisector of $A$ meets $BC$ at $D$. If the line perpendicular to $BC$ passing through $D$ meets $AO$ at $E$, show that $ADE$ is an isosceles triangle.

Let $A$ and $B$ be positive self-adjoint operators on a complex Hilbert space $H$. Prove that \[\limsup_{n \to \infty} \|A^n x\|^{1/n} \le \limsup_{n \to \infty} \|B^n x\|^{1/n}\] holds for every $x \in H$ if and only if $A^n \le B^n$ for each positive integer $n$.

Prove that for every natural $n$ the following inequality is held: $$(2n + 1)^n \ge (2n)^n + (2n - 1)^n$$

Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that $f(x,y)+f(y,z)+f(z,x)=0$. Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.

Consider the sequence given by $a_0 = 3$ and such that for $i \ge 1$, we have $ai = 2^{a_{i-1}} + 1$. Let $m$ be the smallest integer such that $a^3_3$ divides $a_m$. Let $m'$ the smallest integer such that $a^3_m$ divides $a_{m'}$ . Find the value of $m'$.

Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.

Evaluate $$\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx.$$

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$ [b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests? [b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side. [b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of? [b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes? [b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes? [b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$? [img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img] [b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $. [b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$? [b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes? [b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$, the bisector $CL$ was drawn. The incircles of triangles $CAL$ and $CBL$ touch $AB$ at points $M$ and $N$ respectively. Points $M$ and $N$ are marked on the picture, and then the whole picture except the points $A, L, M$, and $N$ is erased. Restore the triangle using a compass and a ruler. (V.Protasov)

For what minimal natural $a$ the polynomial $ax^2 + bx + c$ with the integer $c$ and $b$ has two different positive roots both less than one.

Let $ABC$ be a triangle with $AB=13$, $AC=14$, and $BC=15$. Let $G$ be the point on $AC$ such that the reflection of $BG$ over the angle bisector of $\angle B$ passes through the midpoint of $AC$. Let $Y$ be the midpoint of $GC$ and $X$ be a point on segment $AG$ such that $\frac{AX}{XG}=3$. Construct $F$ and $H$ on $AB$ and $BC$, respectively, such that $FX \parallel BG \parallel HY$. If $AH$ and $CF$ concur at $Z$ and $W$ is on $AC$ such that $WZ \parallel BG$, find $WZ$.

A set \( A \) of integers is said to be \textit{admissible} if it satisfies the property: \[ \text{If } x, y \in A, \text{ then } x^2 + kxy + y^2 \in A \text{ for all } k \in \mathbb{Z}. \] Determine all pairs \( (m, n) \) of nonzero integers such that the only admissible set containing both \( m \) and \( n \) is the set of all integers.

Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$. .

Find all positive integers $n$ with the folowing property: for all triples ($a$,$b$,$c$) of positive real there is a triple of non negative integers ($l$,$j$,$k$) such that $an^k$, $bn^j$ and $cn^l$ are sides of a non degenate triangle

Let $ f(x)$ be a polynomial with the following properties: (i) $ f(0)\equal{}0$; (ii) $ \frac{f(a)\minus{}f(b)}{a\minus{}b}$ is an integer for any two different integers $ a$ and $ b$. Is there a polynomial which has these properties, although not all of its coefficients are integers?

Find all pairs of natural numbers $(m,n)$, $m\leq n$, such that there exists a table with $m$ rows and $n$ columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.

For the sequence \[S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},\]find the limit \[\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).\]

Nine journalists from different countries attend a press conference. None of these speaks more than three languages, and each pair of the journalists share a common language. Show that there are at least five journalists sharing a common language.

How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$?

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$ $\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Consider a point $P$ inside the triangle having $PA=1$, $PB=2$ and $PC=3$. Find the maximum possible area of the triangle $ABC$.

Let $n\ge 2$ be a positive integer. $S$ is a set of $2n$ airports. For two arbitrary airports $A,B$, if there is an airway from $A$ to $B$, then there is an airway from $B$ to $A$. Suppose that $S$ has only one independent set of $n$ airports. Let the independent set $X$. Prove that there exists an airport $P\in S\setminus X$ which satisfies following condition. [b]Condition[/b] : For two arbitrary distinct airports $A,B\in S\setminus \{P\}$, if there exists a path connecting $A$ and $B$, then there exists a path connecting $A$ and $B$ which does not pass $P$.

The altitude $CH$ of a right triangle $ABC$, with $\angle{C}=90$, cut the angles bisectors $AM$ and $BN$ at $P$ and $Q$, and let $R$ and $S$ be the midpoints of $PM$ and $QN$. Prove that $RS$ is parallel to the hypotenuse of $ABC$

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. The tangent line of the circumcircle of triangle $BHC$ at $H$ meets $AB$ and $AC$ at $E$ and $F$ respectively. If $O$ is the circumcenter of triangle $AEF$, prove that the circumcircle of triangle $EOF$ is tangent to $\Omega$.