A one-person game with two possible outcomes is played as follows: After each play, the player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer $x \geq 1986$ can be obtained as the total score whereas $1985$ and $663$ cannot. Determine $a$ and $b.$

Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered $1$ through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a $50\%$ probability of victory. The tournament unfolds in $n+1$ rounds. In each rounjd, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players $1$ and $2^n$ face each other in the last round? Express your answer in terms of $n$.

When filled with an cold water using a particular cold water tap, a tank will be full in 14 minutes. In 21 minutes, the full tank could be emptied by opening a hole on the base of the tank. If the cold water tap and the hot water tap are opened simultaneously (allowing hot and cold water fill the tank), and the hole on the base of the tank is opened, the tank will be full in 12.6 minutes. Determine the number of minutes needed to fill the tank with hot water until the tank is full, assuming at first the tank is empty and the hole is closed.

For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \text{one} \qquad \textbf{(C)}\ \text{two} \qquad \textbf{(D)}\ \text{more than two, but finitely many}\\ \textbf{(E)}\ \text{infinitely many}$

The sum \[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{2021}{2022!}\] can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$? $ \textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$

Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

Consider the set $E = \{1,2,\ldots,2n\}$. Prove that an element $c \in E$ can belong to a subset $A \subset E$, having $n$ elements, and such that any two distinct elements in $A$ do not divide one each other, if and only if \[c > n \left( \frac{2}{3}\right )^{k+1},\] where $k$ is the exponent of $2$ in the factoring of $c$.

Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$, consider the function \[f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}\] Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$.

Let $n >1$ be a natural number and $T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0$.\\ Assume that for each nonzero real number $t $ we have $T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n} $.\\ Prove that for each $0\le i \le n-1 $ $gcd (a_i,n) >1$. [i]Proposed by Morteza Saghafian[/i]

Let $ABC$ be a triangle with sides $51, 52, 53$. Let $\Omega$ denote the incircle of $\bigtriangleup ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed $r_1 + r_2 + r_3$.

Show that for any positive integers $k$ and $1 \leq a \leq 9$, there exists $n$ such that satisfies the below statement. When $2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots $ $(0 \leq a_i \leq 9$ and $a_i$ is integer), $a_k$ is equal to $a$.

Let $ABC$ be a triangle with incircle $\omega$ and incenter $I$. The circle $\omega$ is tangent to $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. Point $P$ is the foot of the angle bisector from $A$ to $BC$, and point $Q$ is the foot of the altitude from $D$ to $EF$. Suppose $AI=7$, $IP=5$, and $DQ=4$. Compute the radius of $\omega$.

Let $ABC$ be an acute triangle, and let $AA_1, BB_1$, and $CC_1$ be its altitudes. Segments $AA_1$ and $B_1C_1$ meet at point $K$. The perpendicular bisector of segment $A_1K$ intersects sides $AB$ and $AC$ at $L$ and $M$, respectively. Prove that points $A,A_1, L$, and $M$ lie on a circle.

Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.

There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$. (a) The first machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (a + 13, b + 5)$ If the input pair is $(5,2)$, is it possible to obtain the pair $(20,22)$ after a series of transformations? (b) The second machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (2a, 2b)$ If the input pair is $(15,10)$, is it possible to obtain the pair $(27,23)$ after a series of transformations? (c) The third machine can perform two transformations: - $(a, b) \to (a - 2, b + 2)$ - $(a, b) \to (2a - b + 1, 2b - 1 - a)$ If the input pair is $(5,8)$, is it possible to obtain the pair $(13,17)$ after a series of transformations?

In how many rearrangements of the numbers $1, \ 2, \ 3, \ 4, \ 5,\ 6, \ 7, \ 8,\ 9$ do the numbers form a $\textit{hill}$, that is, the numbers form an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in $129876543$ or $258976431$?

Let $H$ be an $n\times n$ matrix all of whose entries are $\pm1$ and whose rows are mutually orthogonal. Suppose $H$ has an $a\times b$ submatrix whose entries are all $1.$ Show that $ab\le n.$

Suppose we have $2016$ points in a $2$-dimensional plane such that no three lie on a line. Two quadrilaterals are not disjoint if they share an edge or vertex, or if their edges intersect. Show that there are at least $504$ quadrilaterals with vertices among these points such that any two of the quadrilaterals are disjoint.

Prove that there does not exist a polynomial $f(x)$ with integer coefficients for which $f(2008) = 0$ and $f(2010) = 1867$.

For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers [i] good [/i] if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good. ([i]In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.[/i]) [i] Lajos Hajdu and Andras Sarkozy, Hungary [/i]

Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1, 2, . . . , ab$, putting the numbers $1, 2, . . . , b$ in the first row, $b + 1, b + 2, . . . , 2b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $ij$ in the cell in row $i$ and column $j$. (Examples are shown for a $3 \times 4$ table below.) [img]https://cdn.artofproblemsolving.com/attachments/6/8/a0855d790069ecd2cd709fbc5e70f21f1fa423.png[/img] Isabella sums up the numbers in her grid, and Vidur sums up the numbers in his grid; the difference between these two quantities is $1200$. Compute $a + b$.

$n$ players participated in a competition. Any two players have played exactly one game, and there was no tie game. For a set of $k(\le n)$ players, if it is able to line the players up so that each player won every player at the back, we call the set [i]ranked[/i]. For each player who participated in the competition, the set of players who lost to the player is ranked. Prove that the whole set of players can be split into three or less ranked sets.

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Rajiv Movva[/i]

Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.