For any function $f(x)$, define $f^1(x) = f(x)$ and $f^n (x) = f(f^{n-1}(x))$ for any integer $n \ge 2$. Does there exist a quadratic polynomial $f(x)$ such that the equation $f^n(x) = 0$ has exactly $2^n$ distinct real roots for every positive integer $n$? [i](6 points)[/i]

Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 75^o$ and $AB = 2$. The points $P$ and $Q$ on the sides $AC$ and $BC$ respectively are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$ . Calculate the measurement of the segment $QA $.

Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that $$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$

Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

Find all functions $\displaystyle f : \mathbb N^\ast \to M$ such that \[ \displaystyle 1 + f(n) f(n+1) = 2 n^2 \left( f(n+1) - f(n) \right), \, \forall n \in \mathbb N^\ast , \] in each of the following situations: (a) $\displaystyle M = \mathbb N$; (b) $\displaystyle M = \mathbb Q$. [i]Dinu Şerbănescu[/i]

Elza draws $2013$ cities on the map and connects some of them with $N$ roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest $N$ for which Elza has a winning strategy.

Given right hexagon. The lines parallel to all the sides are drawn from all the vertices and midpoints of the sides (consider only the interior, with respect to the hexagon, parts of those lines). Thus the hexagon is divided onto $24$ triangles, and the figure has $19$ nodes. $19$ different numbers are written in those nodes. Prove that at least $7$ of $24$ triangles have the property: the numbers in its vertices increase (from the least to the greatest) counterclockwise.

We call a tetrahedron divisor of a parallelepiped if the parallelepiped can be divided into $6$ copies of that tetrahedron. Does there exist a parallelepiped that it has at least two different divisor tetrahedrons?

Find all triples $(m,p,q)$ such that \begin{align*} 2^mp^2 +1=q^7, \end{align*} where $p$ and $q$ are ptimes and $m$ is a positive integer.

Let $M$ be an empty set of real numbers. For any $x \in M$ the functions $f: M\to M$ and $g: M\to M$ satisfy the relations $f (g (x)) = g (f (x)) = x$ and $f (x) + g (x) = x$. Show that $- x \in M$ ¸ and $f (-x) = -f (x)$ whatever $x \in M$.

Let $ A_1,A_2,...,A_n$ be the vertices of a closed convex $ n$-gon $ K$ numbered consecutively. Show that at least $ n\minus{}3$ vertices $ A_i$ have the property that the reflection of $ A_i$ with respect to the midpoint of $ A_{i\minus{}1}A_{i\plus{}1}$ is contained in $ K$. (Indices are meant $ \textrm{mod} \;n\ .$)

In Abaliba country there are twenty cities and two airline companies, Blue Planes and Red Planes. The flights are planned as follows: $\bullet$ Given any two cities, one and only one of the two companies operates direct flights (in both directions and without stops) between the two cities. Furthermore: $\bullet$There are two cities A and B between which it is not possible to fly (with possible stops) using only Red Planes. Prove that, given any two cities, a passenger can travel from one to the other using only Blue Planes, making at most one stop in a third city.

The set of all finite ordered sets of $0$ and $ 1$ is somehow partitioned into two disjoint classes. Prove that any infinite sequence of $0$ and $1$ can be cut into non-intersecting finite parts such that all of these parts (except perhaps the first) belong to the same class.

Prove the following inequalities: (1) For $0\leq x\leq 1$, \[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\] (2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$

Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides. [i]A. Berzin’sh, O. Musin[/i]

Solve the equation $\big(2cos(x-\frac{\pi}{4})+tgx\big)^3=54 sin^2x$, $x\in \big[0,\frac{\pi}{2}\big)$

Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define: \[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \] where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.

Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if [list] [*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and [*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and [*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list] Determine the number of good $p$-tuples.

Points $E$ and $F$ lie inside a square $ABCD$ such that the two triangles $ABF$ and $BCE$ are equilateral. Show that $DEF$ is an equilateral triangle.

An integer $n\ge2$ is called [i]resistant[/i], if it is coprime to the sum of all its divisors (including $1$ and $n$). Determine the maximum number of consecutive resistant numbers. For instance: * $n=5$ has sum of divisors $S=6$ and hence is resistant. * $n=6$ has sum of divisors $S=12$ and hence is not resistant. * $n=8$ has sum of divisors $S=15$ and hence is resistant. * $n=18$ has sum of divisors $S=39$ and hence is not resistant.

Let $P$ be a convex pentagon with all sides equal. Prove that if two of the angles of $P$ add to $180^o$, then it is possible to cover the plane with $P$, without overlaps.

Consider the shape formed from taking equilateral triangle $ABC$ with side length $6$ and tracing out the arc $BC$ with center $A$. Set the shape down on line $l$ so that segment $AB$ is perpendicular to $l$, and $B$ touches $l$. Beginning from arc $BC$ touching $l$, we roll $ABC$ along $l$ until both points $A$ and $C$ are on the line. The area traced out by the roll can be written in the form $n\pi$, where $n$ is an integer. Find $n$.

Find the number of ways to completely cover a $2 \times 10$ rectangular grid of unit squares with $2 \times 1$ rectangles $R$ and $\sqrt{2}$ - $\sqrt{2}$ - $2$ triangles $T$ such that the following all hold: [list] [*] a placement of $R$ must have all of its sides parallel to the grid lines, [*] a placement of $T$ must have its longest side parallel to a grid line, [*] the tiles are non-overlapping, and [*] no tile extends outside the boundary of the grid. [/list] (The figure below shows an example of such a tiling; consider tilings that differ by reflections to be distinct.) [asy] unitsize(1cm); fill((0,0)--(10,0)--(10,2)--(0,2)--cycle, grey); draw((0,0)--(10,0)--(10,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((2,1)--(4,1)); draw((5,0)--(5,2)); draw((7,0)--(7,2)); draw((5,1)--(7,1)); draw((8,0)--(8,2)); draw((8,0)--(10,2)); draw((8,2)--(10,0)); [/asy]

[u]Round 6[/u] [b]p16.[/b] Given that $x$ and $y$ are positive real numbers such that $x^3+y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a\sqrt{b}}{c}$ +d where $a$, $b$, $c$, and $d$ are positive integers such that $gcd(a,c) = 1$ and $b$ is square-free. Find $a +b +c +d$. [b]p17.[/b] In $\vartriangle DRK$ , $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the intersection of the altitudes of $\vartriangle DRK$. Find the value of $\lfloor DE +RE +KE \rfloor$. [b]p18.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find the expected number of jumps it takes Subaru to reach pad $7$. [u]Round 7[/u] This set has problems whose answers depend on one another. [b]p19.[/b] Let $B$ be the answer to Problem $20$ and let $C$ be the answer to Problem $21$. Given that $$f (x) = x^3-Bx-C = (x-r )(x-s)(x-t )$$ where $r$, $s$, and $t$ are complex numbers, find the value of $r^2+s^2+t^2$. [b]p20.[/b] Let $A$ be the answer to Problem $19$ and let $C$ be the answer to Problem $21$. Circles $\omega_1$ and $\omega_2$ meet at points $X$ and $Y$ . Let point $P \ne Y$ be the point on $\omega_1$ such that $PY$ is tangent to $\omega_2$, and let point $Q \ne Y$ be the point on $\omega_2$ such that $QY$ is tangent to $\omega_1$. Given that $PX = A$ and $QX =C$, find $XY$ . [b]p21.[/b] Let $A$ be the answer to Problem $19$ and let $B$ be the answer to Problem $20$. Given that the positive difference between the number of positive integer factors of $A^B$ and the number of positive integer factors of $B^A$ is $D$, and given that the answer to this problem is an odd prime, find $\frac{D}{B}-40$. [u]Round 8[/u] [b]p22.[/b] Let $v_p (n)$ for a prime $p$ and positive integer $n$ output the greatest nonnegative integer $x$ such that $p^x$ divides $n$. Find $$\sum^{50}_{i=1}\sum^{i}_{p=1} { v_p (i )+1 \choose 2},$$ where the inner summation only sums over primes $p$ between $1$ and $i$ . [b]p23.[/b] Let $a$, $b$, and $c$ be positive real solutions to the following equations. $$\frac{2b^2 +2c^2 -a^2}{4}= 25$$ $$\frac{2c^2 +2a^2 -b^2}{4}= 49$$ $$\frac{2a^2 +2b^2 -c^2}{4}= 64$$ The area of a triangle with side lengths $a$, $b$, and $c$ can be written as $\frac{x\sqrt{y}}{z}$ where $x$ and $z$ are relatively prime positive integers and $y$ is square-free. Find $x + y +z$. [b]p24.[/b] Alan, Jiji, Ina, Ryan, and Gavin want to meet up. However, none of them know when to go, so they each pick a random $1$ hour period from $5$ AM to $11$ AM to meet up at Alan’s house. Find the probability that there exists a time when all of them are at the house at one time. [b]Round 9 [/b] [b]p25.[/b] Let $n$ be the number of registered participantsin this $LMT$. Estimate the number of digits of $\left[ {n \choose 2} \right]$ in base $10$. If your answer is $A$ and the correct answer is $C$, then your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ [b]p26.[/b] Let $\gamma$ be theminimum value of $x^x$ over all real numbers $x$. Estimate $\lfloor 10000\gamma \rfloor$. If your answer is $A$ and the correct answer is $C$, then your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ [b]p27.[/b] Let $$E = \log_{13} 1+log_{13}2+log_{13}3+...+log_{13}513513.$$ Estimate $\lfloor E \rfloor$. If your answer is $A$ and the correct answer is $C$, your score will be $$\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.$$ PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167127p28823220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.