An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.

The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. Find the fourth term. ${{ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt[7]{2} \qquad\textbf{(C)}\ \sqrt[8]{2} \qquad\textbf{(D)}\ \sqrt[9]{2} }\qquad\textbf{(E)}\ \sqrt[10]{2} } $

Let $f : [0, 1] \rightarrow \mathbb{R}$ be a monotonically increasing function such that $$f\left(\frac{x}{3}\right) = \frac{f(x)}{2}$$ $$f(1 0 x) = 2018 - f(x).$$ If $f(1) = 2018$, find $f\left(\dfrac{12}{13}\right)$.

A particle moves in the Cartesian Plane according to the following rules: 1. From any lattice point $ (a,b)$, the particle may only move to $ (a \plus{} 1,b)$, $ (a,b \plus{} 1)$, or $ (a \plus{} 1,b \plus{} 1)$. 2. There are no right angle turns in the particle's path. How many different paths can the particle take from $ (0,0)$ to $ (5,5)$?

Find the average of the quantity \[(a_1 - a_2)^2 + (a_2 - a_3)^2 +\cdots + (a_{n-1} -a_n)^2\] taken over all permutations $(a_1, a_2, \dots , a_n)$ of $(1, 2, \dots , n).$

The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: $\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 100$

Given is an acute triangle \( \triangle ABC \) with \( AB < AC \). Let \( M \) be the midpoint of side \( BC \), and let \( X \) and \( Y \) be points on segments \( BM \) and \( CM \), respectively, such that \( BX = CY \). Let \( \omega_1 \) be the circumcircle of \( \triangle ABX \), and \( \omega_2 \) the circumcircle of \( \triangle ACY \). The common tangent \( t \) to \( \omega_1 \) and \( \omega_2 \), which lies closer to point \( A \), touches \( \omega_1 \) and \( \omega_2 \) at points \( P \) and \( Q \), respectively. Let the line \( MP \) intersect \( \omega_1 \) again at \( U \), and the line \( MQ \) intersect \( \omega_2 \) again at \( V \). Prove that the circumcircle of triangle \( \triangle MUV \) is tangent to both \( \omega_1 \) and \( \omega_2 \).

A real function $ f$ satisfies $ 4f(f(x))\minus{}2f(x)\minus{}3x\equal{}0$ for all real numbers $ x$. Prove that $ f(0)\equal{}0$.

Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$. Prove that $\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|$ is divergent.

Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Alex Li[/i]

There are 4 coins, 3 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights are such that if equal weights are placed on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is sure to pull. How to definitely identify a counterfeit coin in three weighings and easily establish what is it or is it heavier than the others?

Let $p>5$ be a prime number. Prove that the sum \[\left(\frac{(p-1)!}{1}\right)^p+\left(\frac{(p-1)!}{2}\right)^p+\cdots+\left(\frac{(p-1)!}{p-1}\right)^p\]is divisible by $p^3$.

Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.

A sequence $a_n$ is defined by $$a_0=0,\qquad a_1=3;$$$$a_n=8a_{n-1}+9a_{n-2}+16\text{ for }n\ge2.$$Find the least positive integer $h$ such that $a_{n+h}-a_n$ is divisible by $1999$ for all $n\ge0$.

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{R} \right) $ that don’t commute. [b]a)[/b] If $ A^3=B^3, $ then $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n. $ [b]b)[/b] If $ A^n\neq B^n $ and $ \text{tr} \left( A^n \right) =\text{tr} \left( B^n \right) , $ for all natural numbers $ n, $ then find some of the matrices $ A,B. $

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: $$\log_2\left({x \over yz}\right) = {1 \over 2}$$ $$\log_2\left({y \over xz}\right) = {1 \over 3}$$ $$\log_2\left({z \over xy}\right) = {1 \over 4}$$ Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$

The positive integer $n$ is the sum of two positive integers that divide $n+6$. Find all possible values of $n$

In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.

Laura has $2010$ lamps connected with $2010$ buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons simultaneously, so the lamps are lit simultaneously, too. a) If Richard chooses the buttons to be pressed, what is the maximum number of different combinations of buttons he can press until Laura can assign the buttons to the lamps correctly? b) Supposing that Laura will choose the combinations of buttons to be pressed, what is the minimum number of attempts she has to do until she is able to associate the buttons with the lamps in a correct way?

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

The points $A, B, C, D$ lie in that order on a circle. The segments $AC$ and $BD$ intersect at the point $P$. The point $B'$ lies on the line $AB$ such that $A$ is between $B$ and $B'$ and $|AB'| = |DP |$. The point $C'$ lies on the line $CD$ such that $D$ is between $C$ and $C'$ lies and $|DC' | = |AP|$. Prove that $\angle B'PC' = \angle ABD'$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/7ec65246ff5ecfebc25ca13f3709d1791ceb6c.png[/img] =

A regular hexagon is cut up into $N$ parallelograms of equal area. Prove that $N$ is divisible by three. (V. Prasolov, I. Sharygin, Moscow)

Let $ P(x,y)$ be a polynomial in two variables $ x,y$ such that $ P(x,y)\equal{}P(y,x)$ for every $ x,y$ (for example, the polynomial $ x^2\minus{}2xy\plus{}y^2$ satisfies this condition). Given that $ (x\minus{}y)$ is a factor of $ P(x,y)$, show that $ (x\minus{}y)^2$ is a factor of $ P(x,y)$.