Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$.

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.

In how many ways can the numbers $0,1,2,\dots , 9$ be arranged in such a way that the odd numbers form an increasing sequence, also the even numbers form an increasing sequence? $ \textbf{(A)}\ 126 \qquad\textbf{(B)}\ 189 \qquad\textbf{(C)}\ 252 \qquad\textbf{(D)}\ 315 \qquad\textbf{(E)}\ \text{None} $

The lateral sides and diagonals of a trapezoid intersect a line $l$, determining three equal segments on it. Must $l$ be parallel to the bases of the trapezoid?

$a,b,c$ are positive real numbers such that $\max\{\frac{a(b+c)}{a^2+bc},\frac{b(c+a)}{b^2+ca},\frac{c(a+b)}{c^2+ab}\}\le \frac{5}{2}$. Prove inequality $$\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab}\le 3$$

Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.

Let $LOV ER$ be a convex pentagon such that $LOV E$ is a rectangle. Given that $OV = 20$ and $LO =V E = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$ .

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200.$ $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …, 200, 200, …, 200$$ What is the median of the numbers in this list? $\textbf{(A)}\ 100.5 \qquad\textbf{(B)}\ 134 \qquad\textbf{(C)}\ 142 \qquad\textbf{(D)}\ 150.5 \qquad\textbf{(E)}\ 167$

Using projective transformations prove the Pascal theorem (also find where the Pascal line intersects the circle).

$ABCD$ is a convex quadrilateral such that $AB<AD$. The diagonal $\overline{AC}$ bisects $\angle BAD$, and $m\angle ABD=130^\circ$. Let $E$ be a point on the interior of $\overline{AD}$, and $m\angle BAD=40^\circ$. Given that $BC=CD=DE$, determine $m\angle ACE$ in degrees.

Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.

If $N=1000^2-950^2$, what is the largest prime factor of $N$? $\textbf{(A) }5\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad\textbf{(E) }29$

[b]p1.[/b] Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$. Given that $\angle ABC$ is a right angle, determine the length of $AC$. [b]p2.[/b] Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$. Find the largest possible value of $m-n$. [b]p3.[/b] Six middle school students are sitting in a circle, facing inwards, and doing math problems. There is a stack of nine math problems. A random student picks up the stack and, beginning with himself and proceeding clockwise around the circle, gives one problem to each student in order until the pile is exhausted. Aditya falls asleep and is therefore not the student who picks up the pile, although he still receives problem(s) in turn. If every other student is equally likely to have picked up the stack of problems and Vishwesh is sitting directly to Aditya’s left, what is the probability that Vishwesh receives exactly two problems? [b]p4.[/b] Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$. Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done? [b]p5.[/b] You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between $10$ and $20$ cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be certain of appeasing Charon? [b]p6.[/b] Let $a, b$, and $c$ be positive integers such that $gcd(a, b)$, $gcd(b, c)$ and $gcd(c, a)$ are all greater than $1$, but $gcd(a, b, c) = 1$. Find the minimum possible value of $a + b + c$. [b]p7.[/b] Let $ABC$ be a triangle inscribed in a circle with $AB = 7$, $AC = 9$, and $BC = 8$. Suppose $D$ is the midpoint of minor arc $BC$ and that $X$ is the intersection of $\overline{AD}$ and $\overline{BC}$. Find the length of $\overline{BX}$. [b]p8.[/b] What are the last two digits of the simplified value of $1! + 3! + 5! + · · · + 2009! + 2011!$ ? [b]p9.[/b] How many terms are in the simplified expansion of $(L + M + T)^{10}$ ? [b]p10.[/b] Ben draws a circle of radius five at the origin, and draws a circle with radius $5$ centered at $(15, 0)$. What are all possible slopes for a line tangent to both of the circles? PS. You had better use hide for answers.

An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle. For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with $V$ as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of $n.$ [img]http://s3.amazonaws.com/classroom.artofproblemsolving.com/Images/Transcripts/497b4e1ef5043a84b433a5c52c4be3ae.png[/img]

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous function such that $$ \int_0^1 (x-1)f(x)dx =0. $$ Show that: [b]a)[/b] There exists $ a\in (0,1) $ such that $ \int_0^a xf(x)dx =0. $ [b]b)[/b] There exists $ b\in (0,1) $ so that $ \int_0^b xf(x)dx=bf(b). $

Let $g: \mathbb{C} \rightarrow \mathbb{C}$ be a function, $w\in\mathbb{C}$ and $w^3=1$. Show that there exists a function $f:\mathbb{C} \rightarrow \mathbb{C}$ such that $f(z)+f(wz+a)=g(z), \forall z\in\mathbb{C}$. When there is an unique function $f$ with this property? Find it.

Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.

Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers. Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.

Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$. Let $X$ be a point on the ray $AB$ such that $2 \angle CXA=\angle CMA$. Let $Y$ be a point on the ray $AC$ such that $2 \angle AYB=\angle AMB$. The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$, such that the points $P, B, C$, and $Q$ lie in this order on the line $BC$. Prove that $PB=QC$. [i]Proposed by Dominik Burek, Poland[/i]

We are given an acute triangle $ABC$. Let $D$ be the point on its circumcircle such that $AD$ is a diameter. Suppose that points $K$ and $L$ lie on segments $AB$ and $AC$, respectively, and that $DK$ and $DL$ are tangent to circle $AKL$. Show that line $KL$ passes through the orthocenter of triangle $ABC$.

Find all triples $(a, b, c)$ of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$ $b^2 + c^2 = n \cdot lcm(b, c) + n^2$ $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer $n$.

Find all 2-digit numbers$ n$ having the property: 'Number $n^2$ is 4-digit number of form $\overline{xxyy}$.

Let $\Omega$ be a circle disk with radius $1$. Determine the minimum $r$ that has the following property: You can select three points on $\Omega$ so that each circle disk located in $\Omega$ and has a radius greater than $r$ contains at least one of the three points.

Quadrilateral $ABCD$ is circumscribed around the circle with centre $I$. Let points $M$ and $N$ be the midpoints of sides $AB$ and $CD$ respectively and let $\frac{IM}{AB} = \frac{IN}{CD}$. Prove that $ABCD$ is either a trapezoid or a parallelogram.