Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

Erick stands in the square in the 2nd row and 2nd column of a 5 by 5 chessboard. There are \$1 bills in the top left and bottom right squares, and there are \$5 bills in the top right and bottom left squares, as shown below. \[\begin{tabular}{|p{1em}|p{1em}|p{1em}|p{1em}|p{1em}|} \hline \$1 & & & & \$5 \\ \hline & E & & &\\ \hline & & & &\\ \hline & & & &\\ \hline \$5 & & & & \$1 \\ \hline \end{tabular}\] Every second, Erick randomly chooses a square adjacent to the one he currently stands in (that is, a square sharing an edge with the one he currently stands in) and moves to that square. When Erick reaches a square with money on it, he takes it and quits. The expected value of Erick's winnings in dollars is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle [i]gergonnians[/i]. (a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent. (b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.

What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle $(a)$ of size $6\times 3$? $(b)$ of size $5\times 3$?

Let $K$ be a finite field, $n \ge 2$ an integer, $f \in K[X]$ an irreducible polynomial of degree $n,$ and $g$ the product of all the nonconstant polynomials in $K[X]$ of degree at most $n-1.$ Prove that $f$ divides $g-1.$

Real number $x,y$ satisfy that $4x^2-5xy+4y^2=5,S=x^2+y^2$, then $\frac{1}{S_\text{max}}+\frac{1}{S_\text{min}}=$________.

In the triangle $ABC$ the relationship $AB+AC = 2BC$ holds. Let $I$ and $M$ be the incenter and intersection point of the medians of triangle $ABC$ respectively, $AL$ its angle bisector, and point $P$ the orthocenter of triangle $BIC$. Prove that the points $L, M, P$ lie on a straight line. (Matvii Kurskyi)

If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$

On Monday Taye has \$2. Everyday he either gains \$3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later? $\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

Suppose that $n$ is a positive integer and that $a$ is the integer equal to $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$ If the sum of the digits of $a$ is 567, what is the value of $n$?

For every positive real pair $(x,y)$ satisfying the equation $x^3+y^4 = x^2y$, if the greatest value of $x$ is $A$, and the greatest value of $y$ is $B$, then $A/B = ?$ $ \textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{512}{729} \qquad \textbf{(C)}\ \frac{729}{1024} \qquad \textbf{(D)}\ \frac{3}{4} \qquad \textbf{(E)}\ \frac{243}{256}$

A sequence infinity $a_{1}, a_{2},...,$ is $generadora$ if: $a_{1}=1,2$ and $a_{n+1}$ is obtained by placing a digit 1 on the left or a digit 2 on the right for all natural n. Prove that there is an infinite $generadora$ sequence such that it does not contain any multiples of 7.

Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?

Determine all positive integers that are expressible in the form \[a^{2}+b^{2}+c^{2}+c,\] where $a$, $b$, $c$ are integers.

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[\frac{2^{n}+2}{n}\] is also an integer.

Let $n$ be a positive integer. Navinim writes down all positive square numbers that divide $n$ on a blackboard. For each number $k$ on the blackboard, Navagem replaces it with $d(k)$. Show that the sum of all numbers on the blackboard now is a perfect square. (Note: $d(k)$ denotes the number of divisors of $k$.) [i](Proposed by Ivan Chan Guan Yu)[/i]

Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.

A triple of numbers $(a_1,a_2,a_3)=(3,4,12)$ is given. The following operation is performed a finite number of times: choose two numbers $a,b$ from the triple and replace them by $0.6x-0.8y$ and $0.8x+0.6y$. Is it possible to obtain the (unordered) triple $(2,8,10)$?

Consider the fourteen numbers, $1^4,2^4,...,14^4$. The smallest natural numebr $n$ such that they leave distinct remainders when divided by $n$ is:

We consider $n$ real variables $x_i(1 \le i \le n)$, where $n$ is an integer and $n \ge 2$. The product of these variables will be denoted by $p$, their sum by $s$, and the sum of their squares by $S$. Furthermore, let $\alpha$ be a positive constant. We now study the inequality $ps \le S\alpha$. Prove that it holds for every $n$-tuple $(x_i)$ if and only if $\alpha=\frac{n+1}{2}$

No matter how $n$ real numbers on the interval $[1,2013]$ are selected, if it is possible to find a scalene polygon such that its sides are equal to some of the numbers selected, what is the least possible value of $n$? $ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10 $

A function $f$ is defined recursively by $f(1)=f(2)=1$ and $$f(n)=f(n-1)-f(n-2)+n$$ for all integers $n \geq 3$. What is $f(2018)$? $\textbf{(A)} \text{ 2016} \qquad \textbf{(B)} \text{ 2017} \qquad \textbf{(C)} \text{ 2018} \qquad \textbf{(D)} \text{ 2019} \qquad \textbf{(E)} \text{ 2020}$

A point object of mass $M$ hangs from the ceiling of a car from a massless string of length $L$. It is observed to make an angle $\theta$ from the vertical as the car accelerates uniformly from rest. Find the acceleration of the car in terms of $\theta$, $M$, $L$, and $g$. [asy] size(250); import graph; // Left draw((-3,0)--(-23,0),linewidth(1.5)); draw((-13,0)--(-13,-14)); filldraw(circle((-13,-15),2),gray); draw((-13,-15)--(-21,-15),dashed); draw((-21,-14)--(-21,-1),EndArrow(size=5)); draw((-21,-1)--(-21,-14),EndArrow(size=5)); label(scale(1.5)*"$L$",(-21,-7.5),2*E); // Right draw((3,0)--(23,0),linewidth(1.5)); draw((13,0)--(13,-19),dashed); draw((13,0)--(5,-12)); filldraw(circle((3.89,-13.66),2),gray); label(scale(1.5)*"$\theta$",(12,-9),1.5*W); real f(real x){ return 5x^2/12-95x/12+25; } draw(graph(f,12,7),Arrows); [/asy] (A) $Mg \sin \theta$ (B) $MgL \tan \theta$ (C) $g \tan \theta$ (D) $g \cot \theta$ (E) $Mg \tan \theta$

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be positive real numbers ($ n \ge 2$) whose sum is $ S$. Show that \[ \sum_{i\equal{}1}^n\frac{a_i^{2^{k}}}{\left(S\minus{}a_i\right)^{2^t\minus{}1}}\ge\frac{S^{1\plus{}2^k\minus{}2^t}}{\left(n\minus{}1\right)^{2^t\minus{}1}n^{2^k\minus{}2^t}}\] for any nonnegative integers $ k$, $ t$ with $ k \ge t$. When does equality occur?