Let $n$ be a positive integer. Is it possible to express $n^2+3n+3$ into the form $ab$ with $a$ and $b$ being positive integers, and such that the difference between $a$ and $b$ is smaller than $2\sqrt{n+1}$?

Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every$ i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No, $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series to one row on his shelf in such a way that there does not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition

Given triangle $ ABC$. Points $ M$, $ N$ are the projections of $ B$ and $ C$ to the bisectors of angles $ C$ and $ B$ respectively. Prove that line $ MN$ intersects sides $ AC$ and $ AB$ in their points of contact with the incircle of $ ABC$.

Consider the sequence of Fibonacci numbers $F_0, F_1, F_2, ... $, given by $F_0 = F_1= 1$ and $F_{n+1} =F_n + F_{n-1}$ for $n\ge 1$. Define the sequence $x_0, x_1, x_2, ....$ by $x_0 = 1$ and $x_{k+1} = x^2_k + F^2_{2^k}$ for $k \ge 0$. Define the sequence $y_0, y_1, y_2, ...$ by $y_0 = 1$ and $y_{k+1} = 2x_ky_k - y^2_k$ for $k \ge 0$. If $$\sum^{\infty}_{k=0} \frac{1}{y_k}= \frac{a -\sqrt{b}}{c}$$ for positive integers a$, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.

a) Prove that if $x,y>0$ then \[ \frac x{y^2} + \frac y{x^2} \geq \frac 1x + \frac 1y. \] b) Prove that if $a,b,c$ are positive real numbers, then \[ \frac {a+b}{c^2} + \frac {b+c}{a^2} + \frac {c+a}{b^2} \geq 2 \left( \frac 1a + \frac 1b + \frac 1c \right). \]

Let $(n+3)a_{n+2}=(6n+9)a_{n+1}-na_n$ and $a_0=1$ and $a_1=2$ prove that all the terms of the sequence are integers

$\odot O_1$ and $\odot O_2$ are escribed circles of $\triangle ABC$ ($\odot O_1$ is in $\angle ACB$, $\odot O_2$ is in $\angle ABC$). $\odot O_1$ touches $CB,CA$ at $E,G$; $\odot O_1$ touches $BC,BA$ at $F,H$. $EG\cap FG=P$, prove that $AP\perp BC$.

Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.

An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$?

Find all real numbers $a$ such that the inequality $3x^2 + y^2 \ge -ax(x + y)$ holds for all real numbers $x$ and $y$.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.

Compute the number of sequences of five positive integers $a_1,..., a_5$ where all $a_i \le 5$ and the greatest common divisor of all five integers is $1$.

In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$

A number $n$ is $\it{bad}$ if there exists some integer $c$ for which $x^x \equiv c \pmod n$ has no integer solutions for $x$. Find the number of bad integers between 2 and 42 inclusive.

How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two? $ \textbf{(A)}\ 96 \qquad \textbf{(B)}\ 104 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 256$

For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

Let $a_{ij}$, $1 \leq i,j \leq 3$, $b_1, b_2, b_3$, and $c_1, c_2, c_3$ be positive real numbers. Let $S$ be the set of triples of positive real numbers $(x, y, z)$ such that: \[ a_{11}x + a_{12}y + a_{13}z \leq b_1, \quad a_{21}x + a_{22}y + a_{23}z \leq b_2, \quad a_{31}x + a_{32}y + a_{33}z \leq b_3. \] Let $M$ be the largest possible value of $f(x, y, z) = c_1x + c_2y + c_3z$ for $(x, y, z) \in S$. Let $T$ be the set of triples $(x_0, y_0, z_0)$ from $S$ such that $f(x_0, y_0, z_0) = M$. Prove that if $T$ contains at least two distinct triples, then $T$ is an infinite set.

In $\triangle ABC$, $AD$ is the bisector of $\angle A$, and $E$, $F$ are the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. $H$ is the intersection of $BE$ and $CF$, and $G$, $I$ are the feet of the perpendiculars from $D$ to $BE$ and $CF$, respectively. Prove that both $AFEH$ and $AEIH$ are cyclic quadrilaterals.

[b]p1.[/b] One chilly morning, $10$ penguins ate a total of $50$ fish. No fish was shared by two or more penguins. Assuming that each penguin ate at least one fish, prove that at least two penguins ate the same number of fish. [b]p2.[/b] A triangle of area $1$ has sides of lengths $a > b > c$. Prove that $b > 2^{1/2}$. [b]p3.[/b] Imagine ducks as points in a plane. Three ducks are said to be in a row if a straight line passes through all three ducks. Three ducks, Huey, Dewey, and Louie, each waddle along a different straight line in the plane, each at his own constant speed. Although their paths may cross, the ducks never bump into each other. Prove: If at three separate times the ducks are in a row, then they are always in a row. [b]p4.[/b] Two computers and a number of humans participated in a large round-robin chess tournament (i.e., every participant played every other participant exactly once). In every game, the winner of the game received one point, the loser zero. If a game ended in a draw, each player received half a point. At the end of the tournament, the sum of the two computers' scores was $38$ points, and all of the human participants finished with the same total score. Describe (with proof) ALL POSSIBLE numbers of humans that could have participated in such a tournament. [b]p5.[/b] One thousand cows labeled $000$, $001$,$...$, $998$, $999$ are requested to enter $100$ empty barns labeled $00$, $01$,$...$,$98$, $99$. One hundred Dalmatians - one at the door of each barn - enforce the following rule: In order for a cow to enter a barn, the label of the barn must be obtainable from the label of the cow by deleting one of the digits. For example, the cow labeled $357$ would be admitted into any of the barns labeled $35$, $37$ or $57$, but would not admitted into any other barns. a) Demonstrate that there is a way for all $1000$ cows to enter the barns so that at least $50$ of the barns remain empty. b) Prove that no matter how they distribute themselves, after all $1000$ cows enter the barns, at most $50$ of the barns will remain empty. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1, a_2, \ldots a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_{1}=a_2 =\cdots=a_{2n+1}.$

In an acute angled triangle $ABC$ with $AB < AC$, let $I$ denote the incenter and $M$ the midpoint of side $BC$. The line through $A$ perpendicular to $AI$ intersects the tangent from $M$ to the incircle (different from line $BC$) at a point $P$> Show that $AI$ is tangent to the circumcircle of triangle $MIP$. [i]Proposed by Tejaswi Navilarekallu[/i]

In $ \triangle ABC$, we have $ \angle C \equal{} 3 \angle A$, $ a \equal{} 27$, and $ c \equal{} 48$. What is $ b$? [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(14,0), C=(10,6); draw(A--B--C--cycle); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$a$", B--C, dir(B--C)*dir(-90)); label("$b$", A--C, dir(C--A)*dir(-90)); label("$c$", A--B, dir(A--B)*dir(-90)); [/asy] $ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ \text{not uniquely determined}$

Let $n$ be a positive integer, and let $S_n$ be the set of all permutations of $1,2,...,n$. let $k$ be a non-negative integer, let $a_{n,k}$ be the number of even permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ and $b_{n,k}$ be the number of odd permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$. Evaluate $a_{n,k}-b_{n,k}$. [i]* * *[/i]