Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? $\text{(A)}\ \text{1 hr} \qquad \text{(B)}\ \text{1 hr 10 min} \qquad \text{(C)}\ \text{1 hr 20 min} \qquad \text{(D)}\ \text{1 hr 40 min} \qquad \text{(E)}\ \text{2 hr}$

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Suppose that an interior point $O$ of a triangle $ABC$ is such that the angles $\angle BAO,\angle CBO, \angle ACO$ are all greater than or equal to $30^o$. Prove that the triangle $ABC$ is equilateral.

Knowing that the polynomials $$2x^5 - 13x^4 + 4x^3 + 61x^2 + 20x-25$$ $$x^5 -4x^4 - 13x^3 + 28x^2 + 85x+50$$ have two common double roots, determine all their roots.

For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots? $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 4$

Let $ABC$ be an acute-angled triangle with $AB<AC$, incentre $I$, and let $M$ be the midpoint of major arc $BAC$. Suppose the perpendicular line from $A$ to segment $BC$ meets lines $BI$, $CI$, and $MI$ at points $P$, $Q$, and $K$ respectively. Prove that the $A-$median line in $\triangle AIK$ passes through the circumcentre of $\triangle PIQ$. [i]Proposed by Pranjal Srivastava and Rohan Goyal[/i]

In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$.

There are $n$ cards on a table in a line, with a positive real written on eachcard. LTF and Sunny are playing a game where they take turns taking away the first or the last card in line. The player that has the bigger sum of all the numberson his cards wins. If LTF goes first, find all $n$ such that LTF can always prevent Sunny from winning, regardless of the numbers written on the cards.

Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.

Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.

A perpendicular bisector to the side $AC$ of triangle $ABC$ meets $BC,AB$ at points $A_1$ and $C_1$ respectively. Points $O,O_1$ are the circumcenters of triangles $ABC$ and $A_1BC_1$ respectively. Prove that $C_1O_1\perp AO$.

Describe all groups $ G $ which have the property that: $$ (\forall H\le G)(\forall x,y\in G)(xy\in H\implies (x,y\in H\vee xy=1)) $$

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done. [i]Proposed by Morteza Saghafian, Iran[/i]

In a square with side 1 are placed $n$ equilateral triangles (without having any parts outside the square) each with side greater than $\sqrt{\frac{2}{3}}$. Prove that all of the $n$ equilateral triangles have a common inner point.

On a soccer tournament with $n\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match. $(a)$ If $n=7$, find the smallest number of matches that must be played. $(b)$ Find the smallest number of matches in terms of $n$.

A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible? $ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

A subset of the set $\{1, 2, 3, ..., 30\}$ is called [i]delicious [/i ]if it doesn't contain elements a and b such that $a = 3b$. A [i]delicious[/i] subset It is called [i]super delicious[/i] if, in addition to being delicious, it is verified that no [i]delicious[/i] subset has more elements than it has. Determine the number of [i]super delicious[/i] subsets

A group of $ 12 $ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $ k^\text{th} $ pirate to take a share takes $ \frac{k}{12} $ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $ 12^{\text{th}} $ pirate receive? $ \textbf{(A)} \ 720 \qquad \textbf{(B)} \ 1296 \qquad \textbf{(C)} \ 1728 \qquad \textbf{(D)} \ 1925 \qquad \textbf{(E)} \ 3850 $

Construct a quadrilateral with three sides $1$, $4$ and $3$ so that a circle could be circumscribed around it.

Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational. [i]Proposed by A. Golovanov[/i]

Medians $AD$, $BE$, and $CF$ of triangle $ABC$ meet at $G$ as shown. Six small triangles, each with vertex at $G$, are formed. We draw the circles inscribed in triangles $AFG$, $BDG$, and $CDG$ as shown. Prove that if these three circles are all congruent, then $ABC$ is equilateral. [asy] size(200); defaultpen(fontsize(10)); pair C=origin, B=(12,0), A=(3,14), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C); draw(A--B--C--A--D^^B--E^^C--F); draw(incircle(C,G,D)^^incircle(G,D,B)^^incircle(A,F,G)); pair point=G; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(7));[/asy]

Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$, respectively. Find all possible values of the ratio $d_1:d_2$. [i](I. Voronovich)[/i]

Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.