Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

The trapezoid below has bases with lengths 7 and 17 and area 120. Find the difference of the areas of the two triangles. [center] [img]https://i.snag.gy/BlqcSQ.jpg[/img] [/center]

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

Let $m,n$ be positive integers. An $n\times n$ board has rows and columns numbered $1,2,\dots,n$ from left to right and top to bottom, respectively. This board is colored with colors $r_1,r_2,\dots,r_m$ such that the cell at the intersection of $i$th row and $j$th column is colored with $r_{i+j-1}$ where indices are taken modulo $m$. After the board is colored, Ahmet wants to put $n$ stones to the board so that each row and column has exactly one stone, also he wants to put the same amount of stones to each color. Find all pairs $(m,n)$ for which he can accomplish his goal. Proposed by [i]Sena Başaran[/i]

Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of $$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$

A square table of size $7\times 7$ with the four corner squares deleted is given. [list=a] [*] What is the smallest number of squares which need to be colored black so that a $5-$square entirely uncolored Greek cross (Figure 1) cannot be found on the table? [*] Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive. [/list] [asy] size(3.5cm); usepackage("amsmath"); MP("\text{Figure }1.", (1.5, 3.5), N); DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black); [/asy]

Without using Dirichlet's theorem, show that there are infinitely many primes ending in the digit $9$.

Is there a power of $2$ that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of $2$?

Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$.

Let $x_i\ (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$. Find the maximum possible value of $$(x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}})$$

We call the positive integer $n$ a $3-$[i]stratum[/i] number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others. $a)$ Find a $3-$stratum number. $b)$ Prove that there are infinitely many $3-$stratum numbers.

In a triangle $ABC$, let $M$ be the midpoint of $AC$. If $BC = \frac{2}{3} MC$ and $\angle{BMC}=2 \angle{ABM}$, determine $\frac{AM}{AB}$.

Cells of a $n*n$ square are filled with positive integers in the way that in the intersection of the $i-$th column and $j-$th row, the number $i+j$ is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to $n$ and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the $i-$th column and $j-$row is written $2n+2-i-j$.

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

Find all triples $(x,y,z)$ of real numbers that satisfy the system $\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$

What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$? $\textbf{(A) }-1010\qquad\textbf{(B) }-1009\qquad\textbf{(C) }1008\qquad\textbf{(D) }1009\qquad \textbf{(E) }1010$

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

Ice cream costs $2000$ rubles. Petya has $$400^5 - 399^2\cdot (400^3 + 2\cdot 400^2 + 3\cdot 400 + 4)$$ rubles. Does Petya have enough money for ice cream?

What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

Triangle $ABC$ has right angle $A$, with $AB=AC=6$. Point $P$ is located within triangle $ABC$ such that the areas of triangles $ABP$, $BCP$, and $ACP$ are equal. Express $CP$ in simplest radical form.

[b]p1.[/b] Minseo writes all of the divisors of $1,000,000$ on the whiteboard. She then erases all of the numbers which have the digit $0$ in their decimal representation. How many numbers are left? [b]p2.[/b] $n < 100$ is an odd integer and can be expressed as $3k - 2$ and $5m + 1$ for positive integers $k$ and $m$. Find the sum of all possible values of $n$. [b]p3.[/b] Mr. Pascal is a math teacher who has the license plate $SQUARE$. However, at night, a naughty student scrambles Mr. Pascal’s license plate to $UQRSEA$. The math teacher luckily has an unscrambler that is able to move license plate letters. The unscrambler swaps the positions of any two adjacent letters. What is the minimum number of times Mr. Pascal must use the unscrambler to restore his original license plate? [b]p4.[/b] Find the number of distinct real numbers $x$ which satisfy $x^2 + 4 \lfloor x \rfloor + 4 = 0$. [b]p5.[/b] All four faces of tetrahedron $ABCD$ are acute. The distances from point $D$ to $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ are all $7$, and the distance from point $D$ to face $ABC$ is $5$. Given that the volume of tetrahedron $ABCD$ is $60$, find the surface area of tetrahedron $ABCD$. [b]p6.[/b] Forrest has a rectangular piece of paper with a width of $5$ inches and a height of $3$ inches. He wants to cut the paper into five rectangular pieces, each of which has a width of $1$ inch and a distinct integer height between $1$ and $5$ inches, inclusive. How many ways can he do so? (One possible way is shown below.) [img]https://cdn.artofproblemsolving.com/attachments/7/3/205afe28276f9df1c6bcb45fff6313c6c7250f.png[/img] [b]p7.[/b] In convex quadrilateral $ABCD$, $AB = CD = 5$, $BC = 4$ and $AD = 8$. If diagonal $\overline{AC}$ bisects $\angle DAB$, find the area of quadrilateral $ABCD$. [b]p8.[/b] Let $x$ and $y$ be real numbers such that $$x + y = x^3 + y^3 + \frac34 = \frac{1}{8xy}.$$ Find the value of $x + y$. [b]p9.[/b] Four blue squares and four red parallelograms are joined edge-to-edge alternately to form a ring of quadrilateral as shown. The areas of three of the red parallelograms are shown. Find the area of the fourth red parallelogram. [img]https://cdn.artofproblemsolving.com/attachments/9/c/911a8d53604f639e2f9bd72b59c7f50e43e258.png[/img] [b]p10.[/b] Define $f(x, n) =\sum_{d|n}\frac{x^n-1}{x^d-1}$ . For how many integers $n$ between $1$ and $2023$ inclusive is $f(3, n)$ an odd integer? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show how to construct a chord $BPC$ of a given angle $A$ through a given point $P$ such that $\tfrac{1}{BP}\plus{} \tfrac{1}{PC}$ is a maximum. [asy] size(200); defaultpen(linewidth(0.7)); pair A = origin, B = (5,0), C = (4.2,3), P = waypoint(B--C,0.65); pair Bp = 1.3 * B, Cp = 1.2 * C; draw(A--B--C--A^^Bp--A--Cp); dot(P); label("$A$",A,W); label("$B$",B,S); label("$C$",C,dir(B--C)); label("$P$",P,dir(A--P)); [/asy]