A positive integer n is called [i]primary divisor [/i] if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.

Let $ a,b,c$ be positive real numbers which satisfy $ 5(a^2\plus{}b^2\plus{}c^2)<6(ab\plus{}bc\plus{}ca)$. Prove that these three inequalities hold: $ a\plus{}b>c$, $ b\plus{}c>a$, $ c\plus{}a>b$.

[u]Round 9[/u] [b]p25.[/b] What is the largest integer that cannot be expressed as the sum of nonnegative multiples of $7$, $11$, and $13$? [b]p26.[/b] Evaluate $12{3 \choose3}+ 11{4\choose 3}+ 10{5\choose 3}+ ...+ 2{13\choose 3}+{14 \choose 3}$. [b]p27.[/b] Worker Bob drives to work at $30$ mph half the time and $60$ mph half the time. He returns home along the same route at $30$ mph half the distance and $60$ mph half the distance. What is his average speed along the entire trip, in mph? [u]Round 10[/u] [b]p28.[/b] In quadrilateral $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$ with $BP = 4$, $P D = 6$, $AP = 8$, $P C = 3$, and $AB = 6$. What is the length of $AD$? [b]p29.[/b] Find all positive integers $x$ such that$ x^2 + 17x + 17$ is a square number. [b]p30.[/b] Zach has ten weighted coins that turn up heads with probabilities $\frac{2}{11^2}$ ,$\frac{2}{10^2}$ ,$\frac{2}{9^2}$ $, . . $.,$\frac{2}{2^2}$ . If he flips all ten coins simultaneously, then what is the probability that he will get an even number of heads? [u]Round 11[/u] [b]p31.[/b] Given a sequence $a_1, a_2, . . .$ such that $a_1 = 3$ and $a_{n+1} = a^2_n - 2a_n + 2$ for $n \ge 1$, find the remainder when the product a1a2 · · · a2012 is divided by 100. [b]p32.[/b] Let $ABC$ be an equilateral triangle and let $O$ be its circumcircle. Let $D$ be a point on $\overline{BC}$, and extend $\overline{AD}$ to intersect $O$ at $P$. If $BP = 5$ and $CP = 4$, then what is the value of $DP$? [b]p33.[/b] Surya and Hao take turns playing a game on a calendar. They start with the date January $1$ and they can either increase the month to a later month or increase the day to a later day in that month but not both. The first person to adjust the date to December $31$ is the winner. If Hao goes first, then what is the first date that he must choose to ensure that he does not lose? [u]Round 12[/u] [b]p34.[/b] On May $5$, $1868$, exactly $144$ years before today, Memorial Day in the United States was officially proclaimed. The first Memorial Day took place that year on May $30$ at Waterloo, New York. On May $5$, $2012$, at $12:00$ PM, how many results did the search “memorial day” on Google return? The search phrase is in quotes, so Google will only return sites that have the words memorial and day next to each other in that order. Let $N = max-\{0, \rfloor 15.5 \times \frac{ Your\,\,\, Answer}{Actual \,\,\,Answer} \rfloor \}$. You will earn the number of points equal to $min\{N, max\{0, 30 - N\}\}$. [b]p35.[/b] Estimate the side length of a regular pentagon whose area is $2012$. You will earn the number of points equal to $max\{0, 15 - \lfloor 5 \times |Your \,\,\,Answer - Actual \,\,\,Answer| \rfloor \}$. [b]p36.[/b] Write down one integer between $1$ and $15$, inclusive. (If you do not, then you will receive $0$ points.) Let the number that you submit be $x$. Let $\overline{x}$ be the arithmetic mean of all of the valid numbers submitted by all of the teams. If $x > \overline{x}$, then you will receive $0$ points; otherwise, you will receive $x$ points. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 6-8 [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Inside a square of side $16$ are placed $1000$ points. Show that it is possible to put a equilateral triangle of side $2\sqrt3$ in the plane so that it covers at least $16$ of these points.

On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.

Let $I$ be the center of a circle inscribed in triangle $ABC$, in which $\angle BAC = 60 ^o$ and $AB \ne AC$. The points $D$ and $E$ were marked on the rays $BA$ and $CA$ so that $BD = CE = BC$. Prove that the line $DE$ passes through the point $I$.

Find all ordered pairs $(a,b)$ of positive integers that satisfy $a>b$ and the equation $(a-b)^{ab}=a^bb^a$.

Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.

For $0 \leq p \leq 1/2,$ let $X_1, X_2, \ldots$ be independent random variables such that $$X_i=\begin{cases} 1 & \text{with probability } p, \\ -1 & \text{with probability } p, \\ 0 & \text{with probability } 1-2p, \end{cases} $$ for all $i \geq 1.$ Given a positive integer $n$ and integers $b,a_1, \ldots, a_n,$ let $P(b, a_1, \ldots, a_n)$ denote the probability that $a_1X_1+ \ldots + a_nX_n=b.$ For which values of $p$ is it the case that $$P(0, a_1, \ldots, a_n) \geq P(b, a_1, \ldots, a_n)$$ for all positive integers $n$ and all integers $b, a_1,\ldots, a_n?$

A positive integer is called [i]lonely [/i] if the sum of the inverses of its positive divisors (including $1$ and itself) is not equal with the some of the inverses of the positive divisors of any other positive integer. a) Show that any prime number is lonely. b) Prove that there are infinitely many numbers that are not lonely

Prove that $\forall n\in\mathbb{Z}^+_0:(\exists b\in\mathbb{Z}^+_0:(\forall m\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(x+m = b))\lor(\exists s\in\mathbb{Z}^+_0:(\exists p\in\mathbb{Z}^+_0:((\neg(\exists x\in\mathbb{Z}^+_0:(p+x = 1)))\land(\neg(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:(p = (x+2) \cdot (y+2)))))\land(\exists x\in\mathbb{Z}^+_0:(p = m+x+1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + m) + y))))))\land(\forall u\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(u = p+x))\lor(u = 0)\lor(u = n+1)\lor(\neg(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + u) + y)))))))\lor(\exists v\in\mathbb{Z}^+_0:(\exists k\in\mathbb{Z}^+_0:((\neg(v = 0))\land((u = v \cdot (k+2))\lor(u = v \cdot (k+2) + 1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + v) + y)))))))))))))))))$. Proposed by [i]Warren Bei[/i]

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Compare the magnitude of the following three numbers. $$ \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}} $$

A positive integer is [i]detestable[/i] if the sum of its digits is a multiple of $11$. How many positive integers below $10000$ are detestable? [i]Proposed by Giacomo Rizzo[/i]

For three real numbers $ a,b,c>1, $ prove the inequality: $ \log_{a^2b} a +\log_{b^2c} b +\log_{c^2a} c\le 1. $

What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.

Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\ y+\log\left(y+\sqrt{y^2+1}\right)=z \\ z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

A sequence of integers $(a_0,...,a_k)$ is said to be [i]medaled[/i] if, for each $i = 0,...,k$, there are exactly $a_i$ elements of the sequence equal to $i$. For example, $(1,2,1,0)$ is a [i]medaled [/i] seqence. Indicates all [i]medaled [/i] sequences $(a_0,...,a_{2015})$.

Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

We will say that an octagon is integral if its is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer. For example, the figure on the right shows an integral octagon of area $21$. Determine, with proof, the smallest positive integer $K$ so that for every positive integer $k\geq K$, there is an integral octagon of area $k$. [asy] size(200); defaultpen(linewidth(0.8)); draw((-1/2,0)--(17/2,0)^^(0,-1/2)--(0,15/2)); for(int i=1;i<=6;++i){ draw((0,i)--(17/2,i),linetype("4 4")); } for(int i=1;i<=8;++i){ draw((i,0)--(i,15/2),linetype("4 4")); } draw((2,1)--(1,2)--(1,3)--(4,6)--(5,6)--(7,4)--(7,3)--(5,1)--cycle,linewidth(1)); label("$1$",(1,0),S); label("$2$",(2,0),S); label("$x$",(17/2,0),SE); label("$1$",(0,1),W); label("$2$",(0,2),W); label("$y$",(0,15/2),NW); [/asy]