An equilateral triangle with side length 2023 has area $A$ and a regular hexagon with side length 289 has area $B$. If $\frac{A}{B}$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime, find $m+n$. [i]Proposed by Andy Xu[/i]

What is the remainder when $201820182018... $ [$2018$ times] is divided by $15$?

For every positive integer $m$ and for all non-negative real numbers $x,y,z$ denote \begin{align*} K_m =x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m \end{align*} [list=a] [*] Prove that $K_m \geq 0$ for every odd positive integer $m$ [*] Let $M$ $= \prod_{cyc} (x-y)^2$. Prove, $K_7+M^2 K_1 \geq M K_4$

Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

Find all real numbers $a,b$ satisfying the following system of equations \begin{align*} 2a^2 -2ab+b^2 &=a\\ 4a^2 -5ab +2b^2 & =b. \end{align*}

Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?

Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.

Show that if a circumscribable quadrilateral of sides $a,b,c,d$ has area $A= \sqrt{abcd},$ then it is also inscribable.

A triangle with integral sides has perimeter $8$. The area of the triangle is $\textbf{(A) }2\sqrt{2}\qquad \textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad \textbf{(C) }2\sqrt{3}\qquad \textbf{(D) }4\qquad \textbf{(E) }4\sqrt{2}$

In trapezoid $ABCD$, $AB < CD$, $AB\perp BC$, $AB\parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD\perp DC$. If \[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then $\frac{BP} {AP}-\frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i]

A polynomial $p$ can be written as \begin{align*} p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d. \end{align*} Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$.

Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold: $$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$ $$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$ for all $x,y \in \mathbb{Q}$.

Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

Let $ ABC $ be an isosceles triangle with $ AC=BC $ . Take points $ D $ on side $AC$ and $E$ on side $BC$ and $ F $ the intersection of bisectors of angles $ DEB $ and $ADE$ such that $ F$ lies on side $AB$. Prove that $F$ is the midpoint of $AB$.

Find $n$ such that $\dfrac1{2!9!}+\dfrac1{3!8!}+\dfrac1{4!7!}+\dfrac1{5!6!}=\dfrac n{10!}$.

Let $ a @ b$ represent the operation on two numbers, $ a$ and $ b$, which selects the larger of the two numbers, with $ a@a \equal{} a.$ Let $ a ! b$ represent the operator which selects the smaller of the two numbers, with $ a ! a \equal{} a.$ Which of the following three rules is (are) correct? $ \textbf{(1)}\ a@b \equal{} b@a \qquad \textbf{(2)}\ a@(b@c) \equal{} (a@b)@c \qquad \textbf{(3)}\ a ! (b@c) \equal{} (a ! b) @ (a ! c)$ $ \textbf{(A)}\ (1)\text{ only} \qquad \textbf{(B)}\ (2) \text{ only} \qquad \textbf{(C)}\ \text{(1) and (2) only} \qquad \textbf{(D)}\ \text{(1) and (3) only} \qquad \textbf{(E)}\ \text{all three}$

Find all positive integer solutions of the equation $n^5+n^4=7^{m}-1$

Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]

In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.

In a triangle $ABC$ with $AB\neq AC$, the internal and external bisectors of angle $A$ meet the line $BC$ at $D$ and $E$ respectively. If the feet of the perpendiculars from a point $F$ on the circle with diameter $DE$ to $BC,CA,AB$ are $K,L,M$, respectively, show that $KL=KM$.

Squares $ BCA_{1}A_{2}$ , $ CAB_{1}B_{2}$ , $ ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $ \triangle ABC$. If $ AB_{1}A'C_{2}$ , $ BC_{1}B'A_{2}$ , $ CA_{1}C'B_{2}$ are parallelograms then prove that: (i) Lines $ BC$ and $ AA'$ are orthogonal. (ii)Triangles $ \triangle ABC$ and $ \triangle A'B'C'$ have common centroid

Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors. A move consists of an operation of one of the following three forms: [list] [*] If a duck picking rock sits behind a duck picking scissors, they switch places. [*] If a duck picking paper sits behind a duck picking rock, they switch places. [*] If a duck picking scissors sits behind a duck picking paper, they switch places. [/list] Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take place, over all possible initial configurations.

Let $n$'s positive divisors sum is $T(n)$. For all $n \geq 3$'s prove that, $$(T(n))^3<n^4$$