Positive integers $a,b,c$ satisfy that $a+b=b(a-c)$ and c+1 is a square of a prime. Prove that $a+b$ or $ab$ is a square.

Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$$:$ {$1,2,...n$}$\rightarrow${$1,2,...,n$} and a binary function $C:$ {$1,2,...,n$}$\rightarrow${$0,1$} "revengeful" if it satisfies the two following conditions: $1)$For every $i$ $\in$ {$1,2,...,n$}, there exist $j$ $\in$ $S_{i}=${$i, \pi(i),\pi(\pi(i)),...$} such that $C(j)=1$. $2)$ If $C(k)=1$, then $k$ is one of the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$, where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$, for every positive integer $t$. Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$. Determine $\frac{V}{P}$.

Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of you two uniform numbers is today's date. What number does Caitlin wear? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$

There are $12$ monsters in a plane. Each monster is capable of spraying fire in a $30$-degree cone. Prove that monsters can destroy the plane.

Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.

$p$ passengers get on a train with $n$ wagons. Find the probability of being at least one passenger at each wagon.

A triangle $OAB$ with $\angle A=90^{\circ}$ lies inside another triangle with vertex $O$. The altitude of $OAB$ from $A$ until it meets the side of angle $O$ at $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$.

Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$. [i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]

We call a permutation of the numbers $1$, $2$, $3$, $\dots$ , $n$ 'kawaii' if there is exactly one number that is greater than its position. For example: $1$, $4$, $3$, $2$ is a kawaii permutation (when $n=4$) because only the number $4$ is greater than its position $2$. How many kawaii permutations are there if $n=14$?

There are 1000 towns $A_{1},A_{2},\ldots ,A_{1000}$ with airports in a country and some of them are connected via flights. It's known that the $i$-th town is connected with $d_{i}$ other towns where $d_{1}\leq d_{2}\leq \ldots \leq d_{1000}$ and $d_{j}\geq j+1$ for every $j=1,2,\ldots 999-d_{999}$. Prove that if the airport of any town $A_{k}$ is closed, then we'd still be able to get from any town $A_{i}$ to any $A_{j}$ for $i,j\neq k$ (possibly by more than one flight).

Georg is putting his $250$ stamps in a new album. On the first page he places one stamp and then on every page just as many or twice as many stamps as on the preceding page. In this way he ends up precisely having put all $250$ stamps in the album. How few pages are sufficient for him?

Three circles with centers $V_0$, $V_1$, $V_2$ and radii $33$, $30$, $25$ respectively and mutually externally tangent: $P_i$ is the tangency point between circles $V_{i+1}$ and $V_{i+2}$, where indeces are taken modulo $3$. For $i=0,1,2$, line $P_{i+1}P_{i+2}$ intersects circle $V_{i+1}$ at $P_{i+2}$ and $Q_i$, and the same line intersects circle $V_{i+2}$ at $P_{i+1}$ and $R_i$. If $Q_0R_1$ intersects $Q_2R_0$ at $X$, then the distance from $X$ to line $R_1Q_2$ can be expressed as $\tfrac{a\sqrt b}c$, where the integer $b$ is not divisible by the square of any prime, and positive integers $a$ and $c$ are relatively prime. Find the value of $b+c$.

Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use? $\textbf{(A) }48\qquad\textbf{(B) }87\qquad\textbf{(C) }91\qquad\textbf{(D) }96\qquad \textbf{(E) }120$

Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? $\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16$

Find all positive integers $ x$ and $ y$ such that $ 5^{x}-3^{y}= 16$.

A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$? $ \textbf{(A) }\log_2\frac{\sqrt3}7\qquad \textbf{(B) }\log_2\frac3{\sqrt7}\qquad \textbf{(C) }\log_2\frac7{\sqrt3}\qquad \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad $

Each square on an $8\times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total? [img]https://cdn.artofproblemsolving.com/attachments/0/8/e46929e7ec3fff9be4892ef954ae299e0cb8c7.png[/img]

Let $ABC$ be an acute triangle and let $D$ be the foot of the $A$-bisector. Moreover, let $M$ be the midpoint of $AD$. The circle $\omega_1$ with diameter $AC$ meets $BM$ at $E$, while the circle $\omega_2$ with diameter $AB$ meets $CM$ at $F$. Assume that $E$ and $F$ lie inside $ABC$. Prove that $B$, $E$, $F$, $C$ are concyclic.

How many three-digit positive integers have an odd number of even digits? $\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

Let be a natural number $ a\ge 2. $ [b]a)[/b] Show that there is no infinite sequence $ \left( k_n \right)_{n\ge 1} $ of pairwise distinct natural numbers greater than $ 1 $ having the property that the sequence $ \left( a^{1/k_n} \right)_{n\ge 1} $ is a geometric progression. [b]b)[/b] Show that there are finite sequences $ \left( l_i \right)_i, $ of any length, of pairwise distinct natural numbers greater than $ 1 $ with the property that $ \left( a^{1/l_i} \right)_{i} $ is a geometric progression. [i]Bogdan Enescu[/i]

Assume that the number of offspring for every man can be $0,1,\ldots, n$ with with probabilities $p_0,p_1,\ldots,p_n$ independently from each other, where $p_0+p_1+\cdots+p_n=1$ and $p_n\neq 0$. (This is the so-called Galton-Watson process.) Which positive integer $n$ and probabilities $p_0,p_1,\ldots,p_n$ will maximize the probability that the offspring of a given man go extinct in exactly the tenth generation?

Let $S$ be the sum of all distinct real solutions of the equation \[ \sqrt{x + 2015} = x^2 - 2015. \] Compute $\lfloor 1/S \rfloor$. Recall that if $r$ is a real number, then $\lfloor r \rfloor$ (the [i]floor[/i] of $r$) is the greatest integer that is less than or equal to $r$.

An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive $\sqrt{3}$’s in a row of twelve $\sqrt{3}$’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to $12\sqrt{3}$?

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove the inequality $$\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.$$ [i]Proposed by Anastasija Trajanova[/i]

The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30 ^\circ$ larger than the other. What is the degree measure of the largest angle in the triangle? $ \textbf{(A)}\ 69 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 108 $