An equation $P(x)=Q(y)$ is called [b]Interesting[/b] if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$. An interesting equation $P(x)=Q(y)$ [b]yields in[/b] interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$. (a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$.$S$ [i]is an answer[/i] of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation. (b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$. An interesting equation is called [b]primary[/b] if there's no other interesting equation with lower degree that yields in it. Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$. Time allowed for this question was 2 hours.

The quiz scores of a class with $k>12$ students have a mean of $8.$ The mean of a collection of $12$ of these quiz scores is $14.$ What is the mean of the remaining quiz scores in terms of $k$? $\textbf{(A) } \frac{14-8}{k-12} \qquad \textbf{(B) } \frac{8k-168}{k-12} \qquad \textbf{(C) } \frac{14}{12} - \frac{k}{8} \qquad \textbf{(D) } \frac{14(k-12)}{k^2} \qquad \textbf{(E) } \frac{14(k-12)}{8k}$

Let $x_1,x_2,\ldots x_n$ be positive integers such that no one of them is an initial fragment of any other (for example, $12$ is an initial fragment of $\underline{12},\underline{12}5$ and $\underline{12}405$). Prove that \[\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}<3. \]

Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.

In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon. [img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]

The plane is divided by three infinite sets of parallel lines into equilateral triangles of equal area. Let $M$ be the set of their vertices, and $A$ and $B$ be two vertices of such an equilateral triangle. One may rotate the plane through $120^o$ around any vertex of the set $M$. Is it possible to move the point $A$ to the point $B$ by a number of such rotations (N Vasiliev, Moscow)

Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?

For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$. $\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$

A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\] are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),\\T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of $|S-T|$.

Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$

The number of roots satisfying the equation $ \sqrt{5 \minus{} x} \equal{} x\sqrt{5 \minus{} x}$ is: $ \textbf{(A)}\ \text{unlimited}\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 0$

If $a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 = 0$ where $a,b,c,d$ are real numbers, what is $a$? $ \textbf{(A)}\ \frac 23 \qquad\textbf{(B)}\ \frac {\sqrt 2} 3 \qquad\textbf{(C)}\ \frac {\sqrt 3} 2 \qquad\textbf{(D)}\ \frac 15 \qquad\textbf{(E)}\ \text{None of the above} $

Bob has a $12$ foot by $20$ foot garden. He wants to put fencing around it to keep out the neighbor’s dog. Normal fence posts cost $\$2$ each while strong ones cost $\$3$ each. If he needs one fence post for every $2$ feet and has $\$70$ to spend on the fence posts, what is the largest number of strong fence posts he can buy?

In triangle $ABC$, the incircle, with center $I$, touches the sides $BC$ at point $D$. Line $DI$ meets $AC$ at $X$. The tangent line from $X$ to the incircle (different from $AC$) intersects $AB$ at $Y$. If $YI$ and $BC$ intersect at point $Z$, prove that $AB=BZ$. [i]Proposed by Hooman Fattahimoghaddam[/i]

Given a number, we calculate its square and add $1$ to the sum of the digits in this square, obtaining a new number. If we start with the number $7$ we will obtain, in the first step, the number $1+(4+9)=14$, since $7^2 = 49$. What number will we obtain in the $1999$th step?

Consider two mirrors placed at a right angle to each other and two points A at $ (x,y)$ and B at $ (a,b)$. Suppose a person standing at point A shines a laser pointer so that it hits both mirrors and then hits a person standing at point B (as shown in the picture). What is the total distance that the light ray travels, in terms of $ a$, $ b$, $ x$, and $ y$? Assume that $ x$, $ y$, $ a$, and $ b$ are positive. [asy]draw((0,4)--(0,0)--(4,0),linewidth(1)); draw((1,3)--(0,2),MidArrow); draw((0,2)--(2,0),MidArrow); draw((2,0)--(3,1),MidArrow); dot((1,3)); dot((3,1)); label("$A (x,y)$", (1,3),NE); label("$B (a,b)$", (3,1),NE);[/asy]

Let $ABC$ be a triangle such that $AB = 13$, $BC = 14$, $CA = 15$ and let $E$, $F$ be the feet of the altitudes from $B$ and $C$, respectively. Let the circumcircle of triangle $AEF$ be $\omega$. We draw three lines, tangent to the circumcircle of triangle $AEF$ at $A$, $E$, and $F$. Compute the area of the triangle these three lines determine.

Source: 1976 Euclid Part B Problem 3 ----- $I$ is the centre of the inscribed circle of $\triangle{ABC}$. $AI$ meets the circumcircle of $\triangle{ABC}$ at $D$. Prove that $D$ is equidistant from $I$, $B$, and $C$.

Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have?

What maximal number of the men in checkers game can be put on the chess-board $8\times 8$ so, that every man can be taken by at least one other man ?

Proove that in every five positive numbers there is a pair, say $a,b$, for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$

Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions with $g(x)=2f(x)+f(x^2),$ for all $x \in \mathbb{R}.$ a) Prove that, if $f$ is bounded in a neighbourhood of the origin and $g$ is continuous in the origin, then $f$ is continuous in the origin. b) Provide an example of function $f$, discontinuous in the origin, for which the function $g$ is continuous in the origin.

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

Determine the smallest positive integer $n$ for which the following statement holds true: From any $n$ consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by $2019$. [i]Proposed by Kartal Nagy, Hungary[/i]