Let $ABC$ be an acute triangle inscribed in a circle centered at $O$. Let $M$ be a point on the small arc $AB$ of the triangle's circumcircle. The perpendicular dropped from $M$ on the ray $OA$ intersects the sides $AB$ and $AC$ at the points $K$ and $L$, respectively. Similarly, the perpendicular dropped from $M$ on the ray $OB$ intersects the sides $AB$ and $BC$ at $N$ and $P$, respectively. Assume that $KL=MN$. Find the size of the angle $\angle{MLP}$ in terms of the angles of the triangle $ABC$.

Let $\pi$ be a permutation of the numbers from $ 2$ through $2012$. Find the largest possible value of $$\log_2 \pi(2) \cdot \log_3 \pi(3) ...\log_{2012} \pi(2012).$$

Nikees has a set $S$ of $n$ points on a plane and decides to colour them. All $\dbinom{n}{2}$ line segments are drawn and they have distinct lengths. Find the maximum number of colours that are used at least once, given that: (a) For each point $P$, the two endpoints of the longest line segment connecting $P$ must be of the same colour. (b) For each point $P$, the two endpoints of the shortest line segment connecting $P$ must be of the same colour.

Let $ABC$ be a triangle. $P$, $Q$ and $R$ are the points of contact of the incircle with sides $AB$, $BC$ and $CA$, respectively. Let $L$, $M$ and $N$ be the feet of the altitudes of the triangle $PQR$ from $R$, $P$ and $Q$, respectively. a) Show that the lines $AN$, $BL$ and $CM$ meet at a point. b) Prove that this points belongs to the line joining the orthocenter and the circumcenter of triangle $PQR$. [i]Aarón Ramírez, El Salvador[/i]

(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$ (2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$

Determine, with proof, whether or not there exists a [i]non-isosceles[/i] trapezoid $ABCD$ such that the lengths $AC$ and $BD$ both lie in the set $\{ DA+AB, AB+BC, BC+CD, CD+DA, AB+CD, BC+DA \}$.

Prove that for $n>1$ , $n$ does not divide $2^{n-1}+1$

How many integers with four different digits are there between $1,000$ and $9,999$ such that the absolute value of the difference between the first digit and the last digit is $2$? $\textbf {(A) } 672 \qquad \textbf {(B) } 784 \qquad \textbf {(C) } 840 \qquad \textbf {(D) } 896 \qquad \textbf {(E) } 1008$

Prove that for any positive number $a,b,c$ and any nonnegative integer $p$ $$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$

Let $(f_n) _{n\ge 0}$ be the sequence defined by$ f_0 = 0, f_1 = 1, f_{n + 2 }= f_{n + 1} + f_n$ for $n> 0$ (Fibonacci string) and let $t_n =$ ${n+1}\choose{2}$ for $n \ge 1$ . Prove that: a) $f_1^2+f_2^2+...+f_n^2 = f_n \cdot f_{n+1}$ for $n \ge 1$ b) $\frac{1}{n^2} \cdot \Sigma_{k=1}^{n}\left( \frac{t_k}{f_k}\right)^2 \ge \frac{t_{n+1}^2}{9 f_n \cdot f_{n+1}}$

Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$. [i]Proposed by Evan Chen[/i]

Let $n$ be a positive integer. Mariano divides a rectangle into $n^2$ smaller rectangles by drawing $n-1$ vertical lines and $n-1$ horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer $k$ for which it is possible that Emilio can do $k$ conveniently thought steps in such a way that with the received information, he can determine the area of each one of the $n^2$ smaller rectangles.

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 $ABC$.

Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

The quadrilateral $ABCD$ is inscribed in the circle and the lengths of the sides $BC$ and $DC$ are equal, and the length of the side $AB$ is equal to the length of the diagonal $AC$. Let the point $P$ be the midpoint of the arc $CD$, which does not contain point $A$, and $Q$ is the point of intersection of diagonals $AC$ and $BD$. Prove that the lines $PQ$ and $AB$ are perpendicular.

The number of terms in the expansion of $ [(a\plus{}3b)^2(a\minus{}3b)^2]^2$ when simplified is: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form $x^2 +px+q$, whose coeficients $p$ and $q$ run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values? [i]I. Rubanov[/i]

A square is inscribed into an acute-angled triangle: two vertices of this square lie on the same side of the triangle and two remaining vertices lies on two remaining sides. Two similar squares are constructed for the remaining sides. Prove that three segments congruent to the sides of these squares can be the sides of an acute-angled triangle.

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$. [i]Proposed by Anant Mudgal, India[/i]

The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$

$9$ chess players participate in a chess tournament. According to the regulation, each participant plays a single game with each of the others. At a certain moment of the competition it was found that exactly two participants played the same number of party. To prove that in this case, not a single chess player played any the game, or just one chess player played with everyone else.

[b]p1.[/b] Find $$2^{\left(0^{\left(2^3\right)}\right)}$$ [b]p2.[/b] Amy likes to spin pencils. She has an $n\%$ probability of dropping the $n$th pencil. If she makes $100$ attempts, the expected number of pencils Amy will drop is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p3.[/b] Determine the units digit of $3 + 3^2 + 3^3 + 3^4 +....+ 3^{2022} + 3^{2023}$. [b]p4.[/b] Cyclic quadrilateral $ABCD$ is inscribed in circle $\omega$ with center $O$ and radius $20$. Let the intersection of $AC$ and $BD$ be $E$, and let the inradius of $\vartriangle AEB$ and $\vartriangle CED$ both be equal to $7$. Find $AE^2 - BE^2$. [b]p5.[/b] An isosceles right triangle is inscribed in a circle which is inscribed in an isosceles right triangle that is inscribed in another circle. This larger circle is inscribed in another isosceles right triangle. If the ratio of the area of the largest triangle to the area of the smallest triangle can be expressed as $a+b\sqrt{c}$, such that $a, b$ and $c$ are positive integers and no square divides $c$ except $1$, find $a + b + c$. [b]p6.[/b] Jonny has three days to solve as many ISL problems as he can. If the amount of problems he solves is equal to the maximum possible value of $gcd \left(f(x), f(x+1) \right)$ for $f(x) = x^3 +2$ over all positive integer values of $x$, then find the amount of problems Jonny solves. [b]p7.[/b] Three points $X$, $Y$, and $Z$ are randomly placed on the sides of a square such that $X$ and $Y$ are always on the same side of the square. The probability that non-degenerate triangle $\vartriangle XYZ$ contains the center of the square can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p8.[/b] Compute the largest integer less than $(\sqrt7 +\sqrt3)^6$. [b]p9.[/b] Find the minimum value of the expression $\frac{(x+y)^2}{x-y}$ given $x > y > 0$ are real numbers and $xy = 2209$. [b]p10.[/b] Find the number of nonnegative integers $n \le 6561$ such that the sum of the digits of $n$ in base $9$ is exactly $4$ greater than the sum of the digits of $n$ in base $3$. [b]p11.[/b] Estimation (Tiebreaker) Estimate the product of the number of people who took the December contest, the sum of all scores in the November contest, and the number of incorrect responses for Problem $1$ and Problem $2$ on the October Contest. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If the sum of the real roots $x$ to each of the equations \[2^{2x}-2^{x+1}+1-\frac{1}{k^2}=0\] for $k=2,3,\dots,2023$ is $N$, what is $2^N$?