Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.

We call a $5$-tuple of integers [i]arrangeable[/i] if its elements can be labeled $a, b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all $2017$-tuples of integers $n_1, n_2, . . . , n_{2017}$ such that if we place them in a circle in clockwise order, then any $5$-tuple of numbers in consecutive positions on the circle is arrangeable. [i]Warut Suksompong, Thailand[/i]

Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers.

Consider the numbers $\{24,27,55,64,x\}$. Given that the mean of these five numbers is prime and the median is a multiple of $3$, compute the sum of all possible positive integral values of $x$.

Let $n$ be a positive integer. Each cell of an $n \times n$ table is coloured in one of $k$ colours where every colour is used at least once. Two different colours $A$ and $B$ are said to touch each other, if there exists a cell coloured in $A$ sharing a side with a cell coloured in $B$. The table is coloured in such a way that each colour touches at most $2$ other colours. What is the maximal value of $k$ in terms of $n$?

Let $a, b$ and $c$ be positive integers satisfying the equation $(a, b) + [a, b]=2021^c$. If $|a-b|$ is a prime number, prove that the number $(a+b)^2+4$ is composite. [i]Proposed by Serbia[/i]

Determine all the natural numbers $n$ such that $21$ divides $2^{2^{n}}+2^n+1.$

In $\Delta ABC$ $(AB>BC)$ $BM$ and $BL$ $(M,L\in AC)$ are a median and an angle bisector respectively. Let the line through $M$, parallel to $AB$, intersect $BL$ in point $D$ and the line through $L$, parallel to $BC$, intersect $BM$ in point $E$. Prove that $DE\perp BL$.

A polygon is called [i]convex[/i] if all its internal angles are smaller than 180$^{\circ}$. Given a convex polygon, prove that one can find three distinct vertices $A$, $P$, and $Q$, where $PQ$ is a side of the polygon, such that the perpendicular from $A$ to the line $PQ$ meets the segment $PQ$ (possible at $P$ of $Q$).

A convex quadrilateral $ABCD$ is inscribed in a semicircle with diameter $AB$. The diagonals $AC,BD$ intersect at $S$, and $T$ is the projection of $S$ on $AB$. Show that $ST$ bisects angle $CTD$.

Two different functions $f, g$ of $x$ are selected from the set of real-valued functions $$\left \{sin x, e^{-x}, x \ln x, \arctan x, \sqrt{x^2 + x} -\sqrt{x^2 + x} -x, \frac{1}{x} \right \}$$ to create a product function $f(x)g(x)$. For how many such products is $\lim_{x\to infty} f(x)g(x)$ finite?

Let $a_1,a_2, \dots a_n$ be positive real numbers. Define $b_1,b_2, \dots b_n$ as follows. \begin{align*} b_1&=a_1 \\ b_2&=max(a_1,a_2)\\ b_i&=max(b_{i-1},b_{i-2}+a_i) \text{ for } i=3,4 \dots n \end{align*} Also define $c_1,c_2 \dots c_n$ as follows. \begin{align*} c_n&=a_n \\ c_{n-1}&=max(a_n,a_{n-1})\\ c_i&=max(c_{i+1},c_{i+2}+a_i) \text{ for } i=n-2,n-3 \dots 1 \end{align*} Prove that $b_n=c_1$.\\

Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

Set $A=\{(x,y)||x|+|y|=a,a>0\},B=\{(x,y)||xy|+1=|x|+|y|\}$. If $A\cap B$ is a set of eight vertices of a regular octagon, then $a=$________.

Consider a convex $100$-gon $A_1A_2...A_{100}$. Draw the diagonal $A_{43}A_{81}$ which divides it into two convex polygons $P_1,P_2$. How many vertices and how diagonals, has each of the polygons $P_1,P_2$?

Jane's mother bakes cookies for Jane to share with her $6$ friends. When the cookies are evenly divided among the $7$ children (Jane and her $6$ friends), there is one cookie left over. Given that each child receives at least $1$ cookie, and Jane's mother baked less than $100$ cookies, how many different numbers of cookies could Jane's mother have baked? For example, she could have baked $15$ cookies, because each child receives $2$ cookies, with $1$ left over. $\textbf{(A) }9\qquad\textbf{(B) }11\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }17$

a) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+2d]$? b) Do there exist positive integers $a$ and $d$ such that $[a, a+d] = [a, a+4d]$? Here $[a, b]$ denotes the least common multiple of integers $a, b$.

[b]p1.[/b] Find all prime factors of $8051$. [b]p2.[/b] Simplify $$[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],$$ where $x = 628$, $y = 233$, $z = 340$. [b]p3.[/b] In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G. Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s? [b]p4.[/b] What is the smallest positive $n$ so that $17^n + n$ is divisible by $29$? [b]p5.[/b] The legs of the right triangle shown below have length $a = 255$ and $b = 32$. Find the area of the smaller rectangle (the one labeled $R$). [img]https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png[/img] [b]p6.[/b] A $3$ dimensional cube contains ”cubes” of smaller dimensions, ie: faces ($2$-cubes),edges ($1$-cubes), and vertices ($0$-cubes). How many 3-cubes are in a $5$-cube? PS. You had better use hide for answers.

$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$. Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.

Quadratics $g(x) = ax^2 + bx + c$ and $h(x) = dx^2 + ex + f$ are such that the six roots of $g,h$, and $g - h$ are distinct real numbers (in particular, they are not double roots) forming an arithmetic progression in some order. Determine all possible values of $a/d$.

Find all functions $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that: $f(x^3+xf(xy))=f(xy)+x^2f(x+y) \forall x,y \in \mathbb{R}^{\ge 0}$

Given a non-right triangle $ABC$ with $BC>AC>AB$. Two points $P_1 \neq P_2$ on the plane satisfy that, for $i=1,2$, if $AP_i, BP_i$ and $CP_i$ intersect the circumcircle of the triangle $ABC$ at $D_i, E_i$, and $F_i$, respectively, then $D_iE_i \perp D_iF_i$ and $D_iE_i = D_iF_i \neq 0$. Let the line $P_1P_2$ intersects the circumcircle of $ABC$ at $Q_1$ and $Q_2$. The Simson lines of $Q_1$, $Q_2$ with respect to $ABC$ intersect at $W$. Prove that $W$ lies on the nine-point circle of $ABC$.

In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\omega_C$ have a common point $X$ other than $I$, and that $\angle AXO = \angle OXA'$. [i]Proposed by Sammy Luo[/i]

The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$