$a_1=-5$, $a_2=-6$ and for all $n \geq 2$ the ${(a_n)^\infty}_{n=1}$ sequence defined as, \[a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).\] If a prime $p$ divides $na_n+1$ for a natural number n, prove that there is a integer $m$ such that $m^2\equiv5(modp)$

If $a$ and $b$ are complex numbers such that $a^2 + b^2 = 5$ and $a^3 + b^3 = 7$, then their sum, $a + b$, is real. The greatest possible value for the sum $a + b$ is $\tfrac{m+\sqrt{n}}{2}$ where $m$ and $n$ are integers. Find $n.$

[u]Round 6 [/u] [b]p16.[/b] Let $a$ be a solution to $x^3 -x +1 = 0$. Find $a^6 -a^2 +2a$. [b]p17.[/b] For a positive integer $n$, $\phi (n)$ is the number of positive integers less than $n$ that are relatively prime to $n$. Compute the sum of all $n$ for which $\phi (n) = 24$. [b]p18.[/b] Let $x$ be a positive integer such that $x^2 \equiv 57$ (mod $59$). Find the least possible value of $x$. [u]Round 7[/u] [b]p19.[/b] In the diagram below, find the number of ways to color each vertex red, green, yellow or blue such that no two vertices of a triangle have the same color. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01418af242c7e2c095a53dd23e997b8d1f3686.png[/img] [b]p20.[/b] In a set with $n$ elements, the sum of the number of ways to choose $3$ or $4$ elements is a multiple of the sumof the number of ways to choose $1$ or $2$ elements. Find the number of possible values of $n$ between $4$ and $120$ inclusive. [b]p21.[/b] In unit square $ABCD$, let $\Gamma$ be the locus of points $P$ in the interior of $ABCD$ such that $2AP < BP$. The area of $\Gamma$ can be written as $\frac{a\pi +b\sqrt{c}}{d}$ for integers $a,b,c,d$ with $c$ squarefree and $gcd(a,b,d) = 1$. Find $1000000a +10000b +100c +d$. [u]Round 8 [/u] [b]p22.[/b] Ephram, GammaZero, and Orz walk into a bar. Each write some permutation of the letters “LMT” once, then concatenate their permutations one after the other (i.e. LTMTLMTLM would be a possible string, but not LLLMMMTTT). Suppose that the probability that the string “LMT” appears in that order among the new $9$-character string can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] In $\vartriangle ABC$ with side lengths $AB = 27$, $BC = 35$, and $C A = 32$, let $D$ be the point at which the incircle is tangent to $BC$. The value of $\frac{\sin \angle C AD }{\sin\angle B AD}$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Let $A$ be the greatest possible area of a square contained in a regular hexagon with side length $1$. Let B be the least possible area of a square that contains a regular hexagon with side length $1$. The value of $B-A$ can be expressed as $a\sqrt{b}-c$ for positive integers $a$, $b$, and $c$ with $b$ squarefree. Find $10000a +100b +c$. [u]Round 9[/u] [b]p25.[/b] Estimate how many days before today this problem was written. If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{2} \right| \right \rfloor , 0 \right)$ points. [b]p26.[/b] Circle $\omega_1$ is inscribed in unit square $ABCD$. For every integer $1 < n \le 10,000$, $\omega_n$ is defined as the largest circle which can be drawn inside $ABCD$ that does not overlap the interior of any of $\omega_1$,$\omega_2$, $...$,$\omega_{n-1}$ (If there are multiple such $\omega_n$ that can be drawn, one is chosen at random). Let r be the radius of ω10,000. Estimate $\frac{1}{r}$ . If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{200} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Answer with a positive integer less than or equal to $20$. We will compare your response with the response of every other team that answered this problem. When two equal responses are compared, neither team wins. When two unequal responses $A > B$ are compared, $A$ wins if $B | A$, and $B$ wins otherwise. If your team wins n times, you will receive $\left \lfloor \frac{n}{2} \right \rfloor$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167135p28823324]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ \{x_n\}_{n\geq 1}$ be a sequences, given by $ x_1 \equal{} 1$, $ x_2 \equal{} 2$ and \[ x_{n \plus{} 2} \equal{} \frac { x_{n \plus{} 1}^2 \plus{} 3 }{x_n} . \] Prove that $ x_{2008}$ is the sum of two perfect squares.

Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$. As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?

The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one? [i]Author: Ray Li[/i]

Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$. $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of arc $BC$ of $\Gamma$ (the one that doesn't contain $A$). $X$ and $Y$ are points on $\Gamma$ such that $BX\parallel CY\parallel AM$. Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$. Let $\omega$ be the circumcircle of triangle $ZMN$. Line $AM$ meets $\omega$ for the second time at $P$. Let $K$ be a point on $\omega$ such that $KN\parallel AM$, $\omega_b$ be a circle that passes through $B$, $X$ and tangents to $BC$ and $\omega_c$ be a circle that passes through $C$, $Y$ and tangents to $BC$. Prove that circle with center $K$ and radius $KP$ is tangent to 3 circles $\omega_b$, $\omega_c$ and $\Gamma$. [i]Proposed by Tran Quan - Vietnam[/i]

Let $ABC$ be a triangle and $H$ be its orthocenter. If it is given that $B$ is $(0,0)$, $C$ is $(1,2)$ and $H$ is $(5,0)$, find $A$.

Assume the $n$ sets $A_1, A_2..., A_n$ are a partition of the set $A=\{1,2,...,29\}$, and the sum of any elements in $A_i$ , $(i=1,2,...,n)$ is not equal to $30$. Find the smallest possible value of $n$.

Let $p=4k+3$ be a prime. Prove that if \[\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}\] (where the fraction $\dfrac {m} {n}$ is in reduced terms), then $p \mid 2m-n$. [i]Proposed by A. Golovanov[/i]

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

For integers $a$, $b$, $c$, and $d$, let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$. Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2)) = g(f(4)) = 0$.

For any positive integer $n$, let $S_n$ denote the set of positive integers which cannot be written in the form $an+2017b$ for nonnegative integers $a$ and $b$. Let $A_n$ denote the average of the elements of $S_n$ if the cardinality of $S_n$ is positive and finite, and $0$ otherwise. Compute \[\left\lfloor\displaystyle\sum_{n=1}^{\infty}\frac{A_n}{2^n}\right\rfloor.\] [i]Proposed by Tristan Shin[/i]

How many positive integers $n$ are there such that there are exactly $20$ positive odd integers that are less than $n$ and relatively prime with $n$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None of above} $

Find the smallest positive integer $n$ with the following property: in every sequence of $n$ positive integers such that the sum of the $n$ numbers is equal to $2013$, there are some consecutive terms whose sum is equal to $31$.

A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website? [i](5 points)[/i]

Three pairwise externally tangent circles $\omega_1,\omega_2$ and $\omega_3$ are given. Let $K_{12}$ be the point of tangency between $\omega_1$ and $\omega_2$ and define $K_{23}$ and $K_{31}$ similarly. Consider the point $A_1$ on $\omega_1$. Let $A_2$ be the second intersection of the line $A_1K_{12}$ with $\omega_2$. The line $A_2K_{23}$ then intersects $\omega_3$ the second time at $A_3$, and then line $A_3K_{31}$ intersects $\omega_1$ again at $A_4$ and so on. [list=a] [*]Prove that after six steps, the process will loop; that is, $A_7=A_1$. [*]Prove that the lines $A_1A_2$ and $A_4A_5$ are perpendicular. [*]Prove that the triples of lines $A_1A_2,A_3A_4$ and $A_5A_6$ and $A_2A_3,A_4A_5$ and $A_6A_1$ intersect at two diametrically opposite points on the circle $(K_{12}K_{23}K_{31})$. [/list] [i]Proposed by E. Morozov[/i]

Let $ABC$ an acute triangle with orthocenter $H$. Let $M$ be a point on segment $BC$. The line through $M$ and perpendicular to $BC$ intersects lines $BH$ and $CH$ in points $P$ and $Q$ respectively. Prove that the orthocenter of triangle $HPQ$ lies on the line $AM$.

Find all real numbers $x$ such that the expression \[\log_2 |1 + \log_2 |2 + \log_2 |x| | |\] does not have a defined value.

Let $ ABC$ be an acute angled triangle whose circumcenter is $ O$. The three diameters of the circumcircle that pass through $ A$, $ B$, and $ C$, meet the opposite sides $ BC$, $ CA$, and $ AB$ at the points $ A_1$, $ B_1$ and $ C_1$, respectively. The circumradius of $ ABC$ is of length $ 2P$, where $ P$ is a prime number. The lengths of $ OA_1$, $ OB_1$, $ OC_1$ are integers. What are the lengths of the sides of the triangle?

Let $XY$ be a chord of a circle $\Omega$, with center $O$, which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$, where $Q$ lies between $P$ and $X$. Let $\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\ell$ and $\Omega$, which is closer to $P$. Prove that $$ MP \cdot XY \ge 2 \cdot QX \cdot PY$$

Methanol can be gently oxidized with hot copper metal. What is(are) the product(s) of this oxidation? $ \textbf{(A) }\text{Acetic acid}\qquad\textbf{(B) }\text{Carbon dioxide + Water}\qquad\textbf{(C) }\text{Ethanol} \qquad\textbf{(D) }\text{Methanal} \qquad$

Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.

Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$. Show that $ PMID$ ist cyclic.

Let $\omega \ne 1$ be a $13$th root of unity. Find the remainder when \[ \prod_{k=0}^{12} \left(2 - 2\omega^k + \omega^{2k} \right) \] is divided by $1000$.