Let the set consisting of the squares of the positive integers be called $u$; thus $u$ is the set $1, 4, 9, 16 . . .$. If a certain operation on one or more members of the set always yields a member of the set, we say that the set is closed under that operation. Then $u$ is closed under: ${{ \textbf{(A)}\ \text{Addition}\qquad\textbf{(B)}\ \text{Multiplication} \qquad\textbf{(C)}\ \text{Division} \qquad\textbf{(D)}\ \text{Extraction of a positive integral root} }\qquad\textbf{(E)} \text{None of these} } $

Sally's salary in 2006 was $\$37,500$. For 2007 she got a salary increase of $x$ percent. For 2008 she got another salary increase of $x$ percent. For 2009 she got a salary decrease of $2x$ percent. Her 2009 salary is $\$34,825$. Suppose instead, Sally had gotten a $2x$ percent salary decrease for 2007, an $x$ percent salary increase for 2008, and an $x$ percent salary increase for 2009. What would her 2009 salary be then?

Let $A$ be a subset with seven elements of the set $\{1,2,3, ...,26\}$. Show that there are two distinct elements of $A$, having the same sum of their elements.

Robin has obtained a circular pizza with radius $2$. However, being rebellious, instead of slicing the pizza radially, he decides to slice the pizza into $4$ strips of equal width both vertically and horizontally. What is the area of the smallest piece of pizza?

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

The sides of a triangle are consecutive natural numbers, and the radius of the inscribed circle is $4$. Find the radius of the circumscribed circle.

Let $n$ be a positive integer and let $S \subseteq \{0, 1\}^n$ be a set of binary strings of length $n$. Given an odd number $x_1, \dots, x_{2k + 1} \in S$ of binary strings (not necessarily distinct), their [i]majority[/i] is defined as the binary string $y \in \{0, 1\}^n$ for which the $i^{\text{th}}$ bit of $y$ is the most common bit among the $i^{\text{th}}$ bits of $x_1, \dots,x_{2k + 1}$. (For example, if $n = 4$ the majority of 0000, 0000, 1101, 1100, 0101 is 0100.) Suppose that for some positive integer $k$, $S$ has the property $P_k$ that the majority of any $2k + 1$ binary strings in $S$ (possibly with repetition) is also in $S$. Prove that $S$ has the same property $P_k$ for all positive integers $k$. [i]Proposed by Joshua Brakensiek[/i]

Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.

The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

Given four straight lines, $m_1, m_2, m_3, m_4$, intersecting at $O$ and numbered clockwise with $O$ as the center of the clock, we draw a line through an arbitrary point $A_1$ on $m_1$ parallel to $m_4$ until the line meets $m_2$ at $A_2$. We draw a line through $A_2$ parallel to $m_1$ until it meets $m_3$ at $A_3$. We also draw a line through $A_3$ parallel to $m_2$ until it meets $m_4$ at $A_4$. Now, we draw a line through$ A_4$ parallel to $m_3$ until it meets $m_1$ at $B$. Prove that a) $OB< \frac{OA_1}{2}$ . b) $OB \le \frac{OA_1}{4}$ . [img]https://cdn.artofproblemsolving.com/attachments/5/f/5ea08453605e02e7e1253fd7c74065a9ffbd8e.png[/img]

Let $AXBY$ be a cyclic quadrilateral, and let line $AB$ and line $XY$ intersect at $C.$ Suppose $AX \cdot AY = 6, BX \cdot BY=5,$ and $CX \cdot CY=4.$ Compute $AB^2.$

Solve the following equation in the set of positive integers $x + y^2 + (GCD(x, y))^2 = xy \cdot GCD(x, y)$.

Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$. [b]a)[/b] Prove that $D,I,J$ collinear. [b]b)[/b] $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.

Let $n$ be an odd integer. Consider a square lattice of size $n \times n$, consisting of $n^2$ unit squares and $2n(n +1)$ edges. All edges are painted in red or blue so that the number of red edges does not exceed $n^2$. Prove that there is a cell that has at least three blue edges.

Let $ABCD$ be a convex quadrilateral. Let $n \geq 2$ be a whole number. Prove that there are $n$ triangles with the same area that satisfy all of the following properties: a) Their interiors are disjoint, that is, the triangles do not overlap. b) Each triangle lies either in $ABCD$ or inside of it. c) The sum of the areas of all of these triangles is at least $\frac{4n}{4n+1}$ the area of $ABCD$.

Find the largest integer $k$ ($k \ge 2$), for which there exists an integer $n$ ($n \ge k$) such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions: 1. each chosen number is not divisible by $6$, by $7$, nor by $8$; 2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the numbers $6$, $7$, and $8$.

Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.

Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that a) $\triangle OMN \sim \triangle OBA$; b) $OE \perp AB$. [i]Claudiu-Stefan Popa[/i]

[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that $\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$ [b](N.17)[/b]

Find minimal natural $n$ for which there exist integers $a_1, a_2,\ldots, a_n$ such that quadratic trinom $$x^2-2(a_1+a_2+\cdots+a_n)^2x+(a_1^4+a_2^4+\cdots+a_n^4+1)$$ has at least one integral root.

What is $3(2 \log_4 (2(2 \log_3 9)))$ ?

In a circle, consider two chords $[AB], [CD]$ that intersect at $E$, lines $AC$ and $BD$ meet at $F$. Let $G$ be the projection of $E$ onto $AC$. We denote by $M,N,K$ the midpoints of the segment lines $[EF] ,[EA]$ and $[AD]$, respectively. Prove that the points $M, N,K,G$ are concyclic.

If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)\minus{}2p^{2002}(x)\plus{}p^{2001}(x)$ is divisible by $ p(x)\minus{}x$

Does there exist an infinite set $S$ consisted of positive integers,so that for any $x,y,z,w\in S,x<y,z<w$,if $(x,y)\ne (z,w)$,then $\gcd(xy+2022,zw+2022)=1$?