Parallelograms $ABGF$, $CDGB$ and $EFGD$ are drawn so that $ABCDEF$ is a convex hexagon, as shown. If $\angle ABG = 53^o$ and $\angle CDG = 56^o$, what is the measure of $\angle EFG$, in degrees? [img]https://cdn.artofproblemsolving.com/attachments/9/f/79d163662e02bc40d2636a76b73f632e59d584.png[/img]

Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.

An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.

Find all the values of $n \in N$ such that $n^2 = 2^n$.

Let $a$, $b$, and $c$ be the roots of the polynomial $$x^3+4x^2-7x-1.$$ Which of the following has roots $ab$, $bc$, and $ac$? $\text{(A) }x^3-4x^2+7x-1\qquad\text{(B) }x^3-7x^2+4x-1\qquad\text{(C) }x^3+7x^2-4x-1\qquad\text{(D) }x^3-4x^2+7x+1\qquad\text{(E) }x^3+7x^2-4x+1$

In triangle \( ABC \), \( D \), \( E \), and \( F \) are the feet of the perpendiculars from the vertices \( A \), \( B \), and \( C \), respectively. The parallel to \( EF \) through \( D \) intersects \( AB \) at \( P_B \) and \( AC \) at \( P_C \). Let \( X \) be the intersection of \( EF \) and \( BC \). Prove that the circumcircle of triangle \( P_B P_C X \) passes through the midpoint of side \( BC \).

Harry wants to label the points of a regular octagon with numbers $1,2,\ldots ,8$ and label the edges with $1,2,\ldots, 8$. There are special rules he must follow: If an edge is numbered even, then the sum of the numbers of its endpoints must also be even. If an edge is numbered odd, then the sum of the numbers of its endpoints must also be odd. Two octagon labelings are equivalent if they can be made equal up to rotation, but not up to reflection. If $N$ is the number of possible octagon labelings, find the remainder when $N$ is divided by $100$. [i]Proposed by Harry Kim[/i]

Let the $JHIZ$ be a rectangle and let $A$ and $C$ be points on the sides $ZI$ and $ZJ$, respectively. The perpendicular from $A$ on $CH$ intersects line $HI$ at point $X$ and perpendicular from $C$ on $AH$ intersects line $HJ$ at point $Y$. Show that points $X, Y$, and $Z$ are collinear.

In a convex quadrilateral $ABCD$ the angles $BAD$ and $BCD$ are equal. Points $M$ and $N$ lie on the sides $(AB)$ and $(BC)$ such that the lines $MN$ and $AD$ are parallel and $MN=2AD$. The point $H$ is the orthocenter of the triangle $ABC$ and the point $K$ is the midpoint of $MN$. Prove that the lines $KH$ and $CD$ are perpendicular.

For a positive integer $k$, define the $k$-[i]pop[/i] of a positive integer $n$ as the infinite sequence of integers $a_1, a_2, ...$ such that $a_1 = n$ and $$a_{i+1}= \left\lfloor \frac{a_i}{k} \right\rfloor , i = 1, 2, ..$$ where $ \lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$. Furthermore, define a positive integer $m$ to be $k$-[i]pop avoiding[/i] if $k$ does not divide any nonzero term in the $k$-pop of $m$. For example, $14$ is 3-pop avoiding because $3$ does not divide any nonzero term in the $3$-pop of $14$, which is $14, 4, 1, 0, 0, ....$ Suppose that the number of positive integers less than $13^{2018}$ which are $13$-pop avoiding is equal to N. What is the remainder when $N$ is divided by $1000$?

Compute the smallest positive integer that can be expressed as the product of four distinct integers. [i]Proposed by Yannick Yao[/i]

For real numbers, if $ |x| \plus{} |y| \equal{} 13$, then $ x^2 \plus{} 7x \minus{} 3y \plus{} y^2$ cannot be $\textbf{(A)}\ 208 \qquad\textbf{(B)}\ 15\sqrt {2} \qquad\textbf{(C)}\ \frac {35}{2} \qquad\textbf{(D)}\ 37 \qquad\textbf{(E)}\ \text{None}$

Prove that the diagonals of a quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides equals that of the other.

Let $P:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $P(X)=X$ has no real solution. Prove that $P(P(X))=X$ has no real solution.

Does there exist points $A,B$ on the curve $y=x^3$ and on $y=x^3+|x|+1$ respectively such that distance between $A,B$ is less than $\frac{1}{100}$ ?

Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left.

Let a real number $k$ and points $S,A,SA=1$ in plane be given. Denote $A'$ the image of $A$ under rotation by an oriented angle $\varphi$ with respect to center $S$. Similarly, let $A''$ be the image of $A'$ under homothety with the factor $\frac{1}{\cos\varphi-k\sin\varphi}$ with respect to center $S.$ Denote the locus \[\ell=\bigl\{A''\mid\varphi\in(-\pi,\pi],\cos\varphi-k\sin\varphi\neq0\bigr\}.\] Show that $\ell$ is a line containing $A.$

We say distinct positive integers  $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. [i]Proposed by Morteza Saghafian[/i]

Let $ABC$ be a triangle for which the tangent from $A$ to the circumcircle intersects line $BC$ at $D$, and let $O$ be the circumcenter. Construct the line $l$ that passes through $A$ and is perpendicular to $OD$. $l$ intersects $OD$ at $E$ and $BC$ at $F$. Let the circle passing through $ADO$ intersect $BC$ again at $H$. It is given that $AD=AO=1$. a) Find $OE$ b) Suppose for this part only that $FH=\frac{1}{\sqrt{12}}$: determine the area of triangle $OEF$. c) Suppose for this part only that $BC=\sqrt3$: determine the area of triangle $OEF$. d) Suppose that $B$ lies on the angle bisector of $DEF$. Find the area of the triangle $OEF$.

Assume we are given a set of weights, $x_1$ of which have mass $d_1$, $x_2$ have mass $d_2$, etc, $x_k$ have mass $d_k$, where $x_i,d_i$ are positive integers and $1\le d_1<d_2<\ldots<d_k$. Let us denote their total sum by $n=x_1d_1+\ldots+x_kd_k$. We call such a set of weights [i]perfect[/i] if each mass $0,1,\ldots,n$ can be uniquely obtained using these weights. (a) Write down all sets of weights of total mass $5$. Which of them are perfect? (b) Show that a perfect set of weights satisfies $$(1+x_1)(1+x_2)\cdots(1+x_k)=n+1.$$ (c) Conversely, if $(1+x_1)(1+x_2)\cdots(1+x_k)=n+1$, prove that one can uniquely choose the corresponding masses $d_1,d_2,\ldots,d_k$ with $1\le d_1<\ldots<d_k$ in order for the obtained set of weights is perfect. (d) Determine all perfect sets of weights of total mass $1993$.

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

Show that for all real numbers $x$ and $y$ with $x > -1$ and $y > -1$ and $x + y = 1$ the inequality $$\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23$$ holds. When does equality apply? [i](Walther Janous)[/i]

Let $a, b, c, d$ be positive reals strictly smaller than $1$, such that $a+b+c+d=2$. Prove that $$\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}. $$

Let $a$ and $b$ be different positive rational numbers such that the there exist an infinity of positive integers $n$ for which $a^n - b^n$ is an integer. Prove that $a$ and $b$ are also integers.

Determine all ordered pairs $(x, y)$ of nonnegative integers that satisfy the equation $$3x^2 + 2 \cdot 9^y = x(4^{y+1}-1).$$