Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

Let $ABC$ be an acute triangle with $BC = 48$. Let $M$ be the midpoint of $BC$, and let $D$ and $E$ be the feet of the altitudes drawn from $B$ and $C$ to $AC$ and $AB$ respectively. Let $P$ be the intersection between the line through $A$ parallel to $BC$ and line $DE$. If $AP = 10$, compute the length of $PM$,

Let $p$ be a prime number. For each $k$, $1\le k\le p-1$, there exists a unique integer denoted by $k^{-1}$ such that $1\le k^{-1}\le p-1$ and $k^{-1}\cdot k=1\pmod{p}$. Prove that the sequence \[1^{-1},\quad 1^{-1}+2^{-1},\quad 1^{-1}+2^{-1}+3^{-1},\quad \ldots ,\quad 1^{-1}+2^{-1}+\ldots +(p-1)^{-1} \] (addition modulo $p$) contains at most $\frac{p+1}{2}$ distinct elements.

A game consists in pushing a flat stone along a sequence of squares $S_0, S_1, S_2, . . .$ that are arranged in linear order. The stone is initially placed on square $S_0$. When the stone stops on a square $S_k$ it is pushed again in the same direction and so on until it reaches $S_{1987}$ or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly $n$ squares is $\frac{1}{2^n}$. Determine the probability that the stone will stop exactly on square $S_{1987}.$

One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day? $\textbf{(A) }2.5\qquad\textbf{(B) }3.0\qquad\textbf{(C) }3.5\qquad\textbf{(D) }4.0\qquad \textbf{(E) }4.5$

In a convex trapezoid $ABCD$, side $AD$ is twice the length of the other sides. Let $E$ and $F$ be points on segments $AC$ and $BD$, respectively, such that $\angle BEC = 70^\circ$ and $\angle BFC = 80^\circ$. Determine the ratio of the areas of quadrilaterals $BEFC$ and $ABCD$. [i]Proposed by Zaza Meliqidze, Georgia [/i]

Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]

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]