An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$?

Find all real numbers $a$ such that the inequality $3x^2 + y^2 \ge -ax(x + y)$ holds for all real numbers $x$ and $y$.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

Each vertex of a finite graph can be coloured either black or white. Initially all vertices are black. We are allowed to pick a vertex $P$ and change the colour of $P$ and all of its neighbours. Is it possible to change the colour of every vertex from black to white by a sequence of operations of this type? Note: A finite graph consists of a finite set of vertices and a finite set of edges between vertices. If there is an edge between vertex $A$ and vertex $B,$ then $A$ and $B$ are neighbours of each other.

Compute the number of sequences of five positive integers $a_1,..., a_5$ where all $a_i \le 5$ and the greatest common divisor of all five integers is $1$.

In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$

A number $n$ is $\it{bad}$ if there exists some integer $c$ for which $x^x \equiv c \pmod n$ has no integer solutions for $x$. Find the number of bad integers between 2 and 42 inclusive.

How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two? $ \textbf{(A)}\ 96 \qquad \textbf{(B)}\ 104 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 256$

For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

Let $a_{ij}$, $1 \leq i,j \leq 3$, $b_1, b_2, b_3$, and $c_1, c_2, c_3$ be positive real numbers. Let $S$ be the set of triples of positive real numbers $(x, y, z)$ such that: \[ a_{11}x + a_{12}y + a_{13}z \leq b_1, \quad a_{21}x + a_{22}y + a_{23}z \leq b_2, \quad a_{31}x + a_{32}y + a_{33}z \leq b_3. \] Let $M$ be the largest possible value of $f(x, y, z) = c_1x + c_2y + c_3z$ for $(x, y, z) \in S$. Let $T$ be the set of triples $(x_0, y_0, z_0)$ from $S$ such that $f(x_0, y_0, z_0) = M$. Prove that if $T$ contains at least two distinct triples, then $T$ is an infinite set.

In $\triangle ABC$, $AD$ is the bisector of $\angle A$, and $E$, $F$ are the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. $H$ is the intersection of $BE$ and $CF$, and $G$, $I$ are the feet of the perpendiculars from $D$ to $BE$ and $CF$, respectively. Prove that both $AFEH$ and $AEIH$ are cyclic quadrilaterals.

[b]p1.[/b] One chilly morning, $10$ penguins ate a total of $50$ fish. No fish was shared by two or more penguins. Assuming that each penguin ate at least one fish, prove that at least two penguins ate the same number of fish. [b]p2.[/b] A triangle of area $1$ has sides of lengths $a > b > c$. Prove that $b > 2^{1/2}$. [b]p3.[/b] Imagine ducks as points in a plane. Three ducks are said to be in a row if a straight line passes through all three ducks. Three ducks, Huey, Dewey, and Louie, each waddle along a different straight line in the plane, each at his own constant speed. Although their paths may cross, the ducks never bump into each other. Prove: If at three separate times the ducks are in a row, then they are always in a row. [b]p4.[/b] Two computers and a number of humans participated in a large round-robin chess tournament (i.e., every participant played every other participant exactly once). In every game, the winner of the game received one point, the loser zero. If a game ended in a draw, each player received half a point. At the end of the tournament, the sum of the two computers' scores was $38$ points, and all of the human participants finished with the same total score. Describe (with proof) ALL POSSIBLE numbers of humans that could have participated in such a tournament. [b]p5.[/b] One thousand cows labeled $000$, $001$,$...$, $998$, $999$ are requested to enter $100$ empty barns labeled $00$, $01$,$...$,$98$, $99$. One hundred Dalmatians - one at the door of each barn - enforce the following rule: In order for a cow to enter a barn, the label of the barn must be obtainable from the label of the cow by deleting one of the digits. For example, the cow labeled $357$ would be admitted into any of the barns labeled $35$, $37$ or $57$, but would not admitted into any other barns. a) Demonstrate that there is a way for all $1000$ cows to enter the barns so that at least $50$ of the barns remain empty. b) Prove that no matter how they distribute themselves, after all $1000$ cows enter the barns, at most $50$ of the barns will remain empty. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1, a_2, \ldots a_{2n+1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_{1}=a_2 =\cdots=a_{2n+1}.$

In an acute angled triangle $ABC$ with $AB < AC$, let $I$ denote the incenter and $M$ the midpoint of side $BC$. The line through $A$ perpendicular to $AI$ intersects the tangent from $M$ to the incircle (different from line $BC$) at a point $P$> Show that $AI$ is tangent to the circumcircle of triangle $MIP$. [i]Proposed by Tejaswi Navilarekallu[/i]

In $ \triangle ABC$, we have $ \angle C \equal{} 3 \angle A$, $ a \equal{} 27$, and $ c \equal{} 48$. What is $ b$? [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(14,0), C=(10,6); draw(A--B--C--cycle); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$a$", B--C, dir(B--C)*dir(-90)); label("$b$", A--C, dir(C--A)*dir(-90)); label("$c$", A--B, dir(A--B)*dir(-90)); [/asy] $ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ \text{not uniquely determined}$

Let $n$ be a positive integer, and let $S_n$ be the set of all permutations of $1,2,...,n$. let $k$ be a non-negative integer, let $a_{n,k}$ be the number of even permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$ and $b_{n,k}$ be the number of odd permutations $\sigma$ in $S_n$ such that $\sum_{i=1}^{n}|\sigma(i)-i|=2k$. Evaluate $a_{n,k}-b_{n,k}$. [i]* * *[/i]

An integer $n$, unknown to you, has been randomly chosen in the interval $[1,2002]$ with uniform probability. Your objective is to select $n$ in an ODD number of guess. After each incorrect guess, you are informed whether $n$ is higher or lower, and you $\textbf{must}$ guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of winning is greater than $\tfrac{2}{3}$.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Compute the number of positive integers $n < 2012$ that share exactly two positive factors with 2012. [i]Proposed by Aaron Lin[/i]

There is an interval $[a, b]$ that is the solution to the inequality \[|3x-80|\le|2x-105|\] Find $a + b$.

Show that a line passing through the feet of two altitudes of an acute-angled triangle cuts off a similar triangle.

If a planet of radius $R$ spins with an angular velocity $\omega$ about an axis through the North Pole, what is the ratio of the normal force experienced by a person at the equator to that experienced by a person at the North Pole? Assume a constant gravitational field $g$ and that both people are stationary relative to the planet and are at sea level. $ \textbf{(A)}\ g/R\omega^2$ $\textbf{(B)}\ R\omega^2/g $ $\textbf{(C)}\ 1- R\omega^2/g$ $\textbf{(D)}\ 1+g/R\omega^2$ $\textbf{(E)}\ 1+R\omega^2/g $

[b]p1.[/b] $ x,y,z$ are required to be non-negative whole numbers, find all solutions to the pair of equations $$x+y+z=40$$ $$2x + 4y + 17z = 301.$$ [b]p2.[/b] Let $P$ be a point lying between the sides of an acute angle whose vertex is $O$. Let $A,B$ be the intersections of a line passing through $P$ with the sides of the angle. Prove that the triangle $AOB$ has minimum area when $P$ bisects the line segment $AB$. [b]p3.[/b] Find all values of $x$ for which $|3x-2|+|3x+1|=3$. [b]p4.[/b] Prove that $x^2+y^2+z^2$ cannot be factored in the form $$(ax + by + cz) (dx + ey + fz),$$ $a, b, c, d, e, f$ real. [b]p5.[/b] Let $f(x)$ be a continuous function for all real values of $x$ such that $f(a)\le f(b)$ whenever $a\le b$. Prove that, for every real number $r$, the equation $$x + f(x) = r$$ has exactly one solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the tribe of Zimmer, being able to hike long distances and knowing the roads through the forest are both extremely important, so a boy who reaches the age of manhood is not designated as a man by the tribe until he completes an interesting rite of passage. The man must go on a sequence of hikes. The first hike is a $5$ kilometer hike down the main road. The second hike is a $5\frac{1}{4}$ kilometer hike down a secondary road. Each hike goes down a different road and is a quarter kilometer longer than the previous hike. The rite of passage is completed at the end of the hike where the cumulative distance walked by the man on all his hikes exceeds $1000$ kilometers. So in the tribe of Zimmer, how many roads must a man walk down, before you call him a man?